Exploration of the Relationship Between Somatic and Otolith Growth

Exploration of the relationship between somatic and otolith growth,

and development of a proportionality-based back-calculation

approach based on traditional growth equations

Eloïse Claire Ashworth

M.Sc. (Distinction)

This thesis is presented for the degree of

Doctor of Philosophy of Murdoch University, Western Australia

2016

DECLARATION

I declare that this thesis is my own account of my research and contains, as its main content, work which has not previously been submitted for a degree at any tertiary education institution.

Eloïse Claire Ashworth

This thesis has been conducted under the supervision of

Professor Norm G. Hall (Primary supervisor)

Centre for Fish and Fisheries Research, School of Veterinary and Life Sciences

Murdoch University

Perth, Western Australia

Professor Ian C. Potter (Secondary supervisor)

Centre for Fish and Fisheries Research, School of Veterinary and Life Sciences

Murdoch University

Perth, Western Australia

To my Family and Raif L. Douthwaite,

for their love and support.

“Always bear in mind that your own resolution to succeed, is more important than any other one thing.”

Abraham Lincoln

“Blow, wind! Come, wrack! At least we’ll die with harness on our back.”

Macbeth, William Shakespeare

ABSTRACT

Back-calculation of lengths at ages prior to capture has been found to be a valuable tool for many fish studies. The approach relies on the relationship between fish length and measures of growth zones formed at validated, regular intervals in hard structures within the fish, such as otoliths. While it has been suggested that the inclusion of age in back-calculation procedures might improve the quality of the estimates that are produced, there are relatively few back-calculation approaches that have employed this variable, and it appears that none has made use of traditional growth curves when describing somatic and otolith growth.

This thesis employed data for six teleost species with different biological characteristics to determine whether results of analyses were broadly applicable to a wide range of species. The performance of a new proportionality-based back-calculation approach based on a model of somatic and otolith growth that employs traditional forms of growth curves and assumes a bivariate distribution of deviations from these two growth curves was explored. The forms of the curves used to describe fish length and otolith size at capture were selected from a suite of traditional and flexible growth curves on the basis of Akaike

Information Criteria for the fitted models. Coefficients of determination indicated good fits of the bivariate growth model for five of the six species. Deviations from the two growth curves were positively correlated and, for three of the six species, statistically significant.

The accuracy and precision of predictions made using the fitted bivariate growth model were assessed by comparing predicted lengths at capture given both age and otolith radius, and predicted otolith radius at capture given both age and fish length, with the corresponding observed values for fish that were not included when fitting the model. The resulting measures of root mean square error (RMSE) for six fish species with different biological characteristics were compared with those obtained using the methods employed in

I a set of existing proportionality-based back-calculation approaches that also incorporated age when describing the relationship between fish length and otolith radius. Based on the results from this cross-validation, the RMSEs of predictions of fish length and otolith size of the new bivariate model were found typically to be equal to or better than those produced using the regression equations of the alternative approaches.

The new bivariate growth model was extended to provide a proportionality-based back-calculation approach, with the option of constraining the growth curves to pass through a biological intercept, i.e., the length, otolith radius and age of recently-hatched larvae. Back- calculated estimates of lengths at ages prior to capture calculated for individuals from a population of Acanthopagrus butcheri using the bivariate growth model were compared with the estimates produced by other proportionality-based back-calculation approaches that employed age and with a constraint-based back-calculation approach that was known to have good performance. The resulting estimates of length at ages produced by the proportionality- based back-calculation approach developed using the bivariate growth model, when constrained to pass through the biological intercept for this species, were typically more consistent with mean observed lengths at corresponding ages than those of the alternative back-calculation approaches. In combination with the cross-validation results, these findings suggest that, for this population of A. butcheri, back-calculated lengths produced using the bivariate growth model are likely to be more reliable than those produced using the other back-calculation approaches.

A common assumption of mixed effects models of otolith growth suggests that, through inclusion of a random effect for different fish, the growth rate of otolith of an individual fish relative to that of other fish will persist throughout life. It was proposed that, throughout the life of an individual from a selected population of A. butcheri, the sizes of its otolith remain in constant proportion to the average sizes of the otoliths of fish of the same

II ages. This hypothesis was investigated by exploring the extent to which the natural logarithms of the ratios of otolith size for individual fish to average otolith size from

A. butcheri of the same age remained constant throughout life. Although, for individuals of this species, the hypothesis of constant proportionality with age was found to be invalid as the ratios of relative otolith size varied among different periods of life, these ratios became increasingly constant with increasing age.

Other factors likely to affect predictions derived from the new back-calculation approach, such as length-dependent selection and level of fishing mortality, were explored using simulation. Results from this simulation suggest that, due to the cumulative effect of fishing mortality on survival, the mean age of fish of a given length or otolith size is likely to decrease as length-dependent fishing mortality increases for fish with larger lengths or otolith sizes, with the effect apparently less on otolith size than on fish length. Similarly, mean lengths for fish with otoliths of a given size and, to a lesser extent, mean otolith sizes for a given fish length, decreased with increasing fishing mortality for fish with larger lengths or otolith sizes. Mean otolith sizes at age of younger fish, however, appeared little affected by reduced selectivity. Although otolith size at age of older fish predicted by bivariate models fitted to simulated otolith sizes at capture appeared little affected by increasing fishing mortality, predicted fish lengths at age of older fish and fish lengths at otolith size of fish with larger otoliths decreased with increasing fishing mortality, with the magnitude and direction of the effect varying among species with different levels of fishing mortality.

The model developed in this study provides a link between studies of somatic growth and investigations of the relationship between length and otolith size undertaken in traditional back-calculation approaches, thereby facilitating future investigation of factors affecting this relationship and, through this, improving our understanding of the influence of environmental factors on somatic and otolith growth.

III

ACKNOWLEDGEMENTS

“To wake in that desert dawn was like waking in the heart of an opal […] See the

desert on a fine morning and die - if you can.” Gertrude Bell in The Desert and the Sown:

Travels in Palestine and Syria.

Memories of scorching deserts, ephemeral wadis and hidden oases treasured from a golden childhood in the Sultanate of Oman remained long unrivalled. At the time, going on desert expeditions to the sounds of Beethoven and Mozart in search of the rare Oryx, exploring ancient Arabian forts, and fossil hunting were part of my life. Having now been exposed to Oceania, through study and work in Australia and New-Zealand, I count myself privileged to have had the opportunity to enjoy such wondrous parts of planet Earth.

My first encounter with fish ear bones occurred Down Under, in Perth (Western

Australia). During that time, I embarked upon an exciting M.Sc. research project on the quantitative diet analysis of mesopredatory fishes from Ningaloo. Publishing my first paper with Dr. Shaun Wilson (Department of Parks and Wildlife) was one of the most valuable experiences that I could wish for. My interest in otoliths grew further in windy Wellington

(New-Zealand), capital of Hobbits and Dragons, while given the chance to participate in the establishment of long term and large scale larval fish experiment with Dr. Jeff Shima

(Victoria University’s Coastal Ecology Laboratory). I am forever grateful to the trust and generosity shown to me, by both Shaun and Jeff, in those early months of my career.

Whilst in New Zealand, a window to dive deeper into the world of otoliths opened with the thrilling opportunity to pursue a PhD in Western Australia. I would like to express my endless gratitude for the contribution of all who came on this journey with me.

It has been my good fortune to be supervised and to work closely with two gurus in

Estuarine and Fisheries Sciences, Emeritus Professor Ian Potter and Emeritus Professor Norm

Hall. I thank you both for the chance to fulfil my PhD and for your constructive assessment of my work, which has helped me develop the art of scientific thinking and writing. Thank you Ian for your encouragement, the knowledge you imparted and for your sense of fairness when it was most needed. Thank you Norm for your guidance, morale support and for sharing your invaluable knowledge of statistics, modelling and programming (debugging included - when finding the light at the end of the tunnel was hard to see!). I am more than grateful for the weekends, which you and Jenny so generously surrendered towards my thesis. Your commendable mentorship has been an inspiration and is a model for me to aspire to in my future endeavours.

For their help and input, I wish to thank my colleagues at the Department of Fisheries and Murdoch University. I am indebted to Dr. Peter Coulson for providing continuous support and guidance in the laboratory, for sharing your extensive knowledge on fish biology, and for showing me the ropes in otolith ageing and sectioning. You have been a role model for high standards of work and perfectionism. Thank you for taking me squid fishing: I will forever remember the day when the head of a rather large Northwest Blowfish rose from the water to devour the first squid I had ever caught – what an experience! Many thanks Dr. Alex

Hesp for your supportive advice, for the time you spent giving me invaluable feedback on my chapters, for following my progress with genuine interest and, when the occasion arose, for taking on the role of surrogate supervisor. Dr. Alan Cottingham, thank you for your assistance and knowledge on the biology of the illustrious Black Bream. I am very grateful to

Dr. James Tweedley for your guidance (such as Ariadne's thread) through the administrative intricacies of the academic world at the start of my PhD, for providing me with experience opportunities in the field (long live the Prawn Stars!), and for being an example in leadership.

Thank you Dr. Chris Hallett for taking the time to answer a cold-letter, thus becoming an important instigator of this chapter in my life. Thank you Gordon Thompson for your advice on microscopy and particularly for your help in my quest to find ‘invisible’ otoliths in fish larvae. Special thanks to Michaël Manca for arranging a meeting with Bruce Ginboy at the

Australian Centre for Applied Aquaculture Research (Challenger, Institute of Technology), which resulted in a generous donation of Black Bream eggs – a stroke of luck! Thank you

Dale Banks for your administrative advice and for smoothing application tasks; I appreciate the time you devoted to assist and answer questions. Finally, I would also like to thank Paul

Day, Marine Marziac, Max Wellington and Lauren Taaffe, who kindly volunteered during the laboratory work of a broader Murdoch University study on the biology of Black Bream.

Despite the fishy smells and gore, I hope that you were still able to gain something useful from your experience with me.

To Raif Douthwaite, love of my life, I am most grateful for your moral support, for your affection, and for helping me maintain my courage and faith. Amongst the many gifts you have provided me with, expanding my mind to the world of entrepreneurship, growth mindset and systems thinking are the most precious. For unlocking an unaware but natural talent for public speaking and storytelling, I am forever indebted to you. Thank you for showing me that the world provides a billion opportunities and that the fastlane to boundless success is close at hand – ‘a thousand battles, a thousand victories’ (Sun Tzu). Your confidence and belief in me have helped me become a better and worldlier person in many different ways.

To my parents, Elisabeth and Malcolm Ashworth, I am endlessly thankful for your infinite love and unwavering faith in me. Thank you for your prayers, for being my most devoted supporters, for worrying when things were not so bright and for helping me believe that there is always a silver lining. With the passing of my grand-mother, Jacqueline

Destrieux, November 2013 was a time of mourning during my PhD. This thesis is dedicated to her, who understood, more than most, the value of honest hard work.

Last but not least, I would like to mention personal inspirations of fortitude, honour and perseverance. The spirits of my ancestors, who conquered the seas on their Drekkar, who reigned in Albion and who expanded the frontiers of Frankia, give me heart to pursue my goals and the strength to embrace my fears throughout life – Fac recte, nil time.

The three years of my PhD have been a most enjoyable and character building enterprise. For the achievements and for the friends made throughout my journey, I am most thankful. Obstacles, which invariably arose along the way, likewise participated in my personal growth. Failure is the greatest teacher. With confidence, I come stronger and wiser as a human being from these experiences. The adventure is just beginning.

A seadragon is naught but another seahorse without its leaf-like protrusions: I would not have accomplished as much were it not for the support that I received. Thank you!

VII

Note. All illustrations in this thesis are my own designs.

TABLE OF CONTENTS

Abstract ...... I

Acknowledgements ...... IV

List of Tables ...... XII

List of Figures ...... XIX

Chapter 1 – General Introduction ...... 1

1.1 Broad overview of the thesis ...... 1

1.2 Scientific importance and use of otoliths ...... 2

1.3 Otolith function, formation and growth ...... 5

1.4 Somatic and otolith growth relationship ...... 8

1.5 Back-calculation ...... 11

1.6 Contribution of the thesis ...... 16

Chapter 2 – Study species, sampling methods, growth models and statistical distributions ...... 20

2.1 Study species ...... 20

2.2 Acanthopagrus butcheri - Sampling regime ...... 23

2.3 Otolith ageing and measurements ...... 24

2.4 Modified von Bertalanffy growth equation and Schnute (1981) growth model ...... 28

2.5 Special cases of the Schnute (1981) growth equation ...... 30

2.6 Univariate and bivariate probability density functions ...... 32

Chapter 3 – Age and growth rate variation influence the functional relationship between somatic and otolith size ...... 39

3.1 Introduction ...... 39

3.2 Materials and Methods ...... 42

3.2.1 The six selected species ...... 42 3.2.2 Fish processing and otolith measurements ...... 43 3.2.3 Analyses ...... 44 3.3 Results ...... 49

3.3.1 Forms of growth curves providing best descriptions of length at age ...... 49

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3.3.2 Forms of growth curves providing best descriptions of otolith radius at age ...... 51 3.3.3 Distribution of lengths and otolith radii at age about growth curves ...... 52 3.3.4 Descriptions of somatic growth provided by the fitted bivariate growth models ...... 52 3.3.5 Descriptions of otolith growth provided by the fitted bivariate growth models ...... 53 3.3.6 Correlations between deviations from somatic and otolith growth curves...... 54 3.3.7 Relationships between expected total fish lengths and otolith sizes at age ...... 54 3.4 Discussion...... 57

3.4.1 Growth ...... 58 3.4.2 Somatic growth ...... 59 3.4.3 Otolith growth ...... 60 3.4.4 The modified von Bertalanffy growth model with oblique linear asymptote ...... 61 3.4.5 Distribution of lengths and otolith radii at age about growth curves ...... 62 3.4.6 Correlation between the deviations of fish lengths and otolith sizes at age from the two growth curves ...... 63 3.4.7 Relationships between expected total fish lengths and otolith sizes at age ...... 64 3.4.8 Summary and conclusions ...... 65 Chapter 4 – A new proportionality-based back-calculation approach, which employs traditional forms of growth equations, improves estimates of length at age ...... 99

4.1 Introduction ...... 100

4.2 Material and Methods ...... 104

4.2.1 The six study species ...... 104 4.2.2 Fish processing and otolith measurements for six species, and biological intercept for Acanthopagrus butcheri ...... 105 4.2.3 Bivariate growth model and associated back-calculation approach ...... 107 4.2.4 Proportionality-based and constraint-based back-calculation approaches ...... 110 4.2.5 Analyses ...... 113 4.3 Results ...... 116

4.3.1 Accuracy and precision of length and otolith radius estimates for six species ...... 116 4.3.2 Accuracy and precision of length and otolith radius estimates for Acanthopagrus butcheri, with and without a biological intercept ...... 117 4.3.3 Comparison of length predictions for Acanthopagrus butcheri from different back- calculation approaches ...... 119 4.4 Discussion...... 120

4.4.1 Cross-validation of estimates derived from relationship between length, otolith radius and age...... 120

4.4.2 Comparison of length predictions between back-calculation approaches ...... 124 4.4.3 Summary and conclusions ...... 126 Chapter 5 – The ratio of individual to average otolith size at age does not remain constant throughout life of Acanthopagrus butcheri ...... 139

5.1 Introduction ...... 140

5.2 Material and Methods ...... 143

5.2.1 Sampling, fish processing and otolith measurements ...... 143 5.2.2 Analyses ...... 145 5.3 Results ...... 151

5.3.1 Homogeneity of variance ...... 151 5.3.2 Correlation analysis ...... 151 5.3.3 Linear relationships between deviations at different ages ...... 152 5.3.4 Linear relationships between annual increment and initial otolith size ...... 153 5.4 Discussion...... 154

5.4.1 Otolith growth at first opaque zone ...... 155 5.4.2 Otolith growth at second opaque zone ...... 156 5.4.3 Otolith growth at third and subsequent opaque zones ...... 157 5.4.4 Implications ...... 158 Chapter 6 – The effects of length-dependent fishing mortality on relationships between fish length, otolith size and age at capture ...... 172

6.1 Introduction ...... 173

6.2 Material and Methods ...... 176

6.2.1 Selected species for the simulation ...... 176 6.2.2 Simulation ...... 176 6.2.3 Simulation scenarios ...... 180 6.2.4 Analyses ...... 180 6.3 Results ...... 183

6.3.1 Distributions of age, length and otolith size ...... 183 6.3.2 Effect of fishing mortality on mean age for a given length and otolith size ...... 184

6.3.3 Effect of increase in mean length at first capture (L50) on mean otolith size for a given age ...... 185 6.3.4 Effect of fishing mortality on mean fish length for a given otolith size ...... 186 6.3.5 Effect of fishing mortality on mean otolith size for a given fish length ...... 187

6.3.6 Effect of fishing mortality on predicted fish length and otolith size at age, and on predicted fish length at otolith size ...... 188 6.3.7 Changes in distribution of deviations of fish length and otolith size with age ...... 189 6.4 Discussion...... 191

6.4.1 Effect of fishing mortality on mean age for fish of a given fish length or otolith size ... 191 6.4.2 Effect of selectivity on mean otolith size for fish of a given age ...... 192 6.4.3 Effect of fishing mortality on mean fish length for fish of a given otolith size ...... 194 6.4.5 Effect of fishing mortality on predicted fish length and otolith size at age, and on predicted fish length at otolith size ...... 195 6.4.6 Changes in distribution of deviations of fish length and otolith size with age ...... 197 6.4.7 Implications ...... 198 Chapter 7 – General discussion and perspectives ...... 236

Appendices ...... 247

Bibliography ...... 250

Glossary of acronyms ...... 285

LIST OF TABLES

Table 3.1. Types of growth curves (and statistical distributions of errors) that, based on

Akaike Information Criterion corrected for small sample size, best described total lengths and otolith radii at age of capture for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. 푎 and 푏 are parameters of the Schnute (1981) growth model. The modified von Bertalanffy curve has an oblique asymptote. AW = Akaike Weight……………………………………………...p.68

Table 3.2. Types of growth curves (and statistical distributions of errors) that, based on frequency of occurrence in bootstrap trials, best described total lengths and otolith radii at age of capture for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. Curve types are the same as described for Table 3.1. Bootstrap percentage = percentage of 4 000 trials for which the curve and error types were selected as best describing the bootstrap sample of lengths and ages at capture…………………………………………………………………p.69

Table 3.3. Values of parameters (푎, 푏, 푦1and 푦2) and standard deviations (SD) for Schnute

(1981) (normal font) and modified von Bertalanffy (oblique linear asymptote, bold font) somatic and otolith growth curves of the fitted bivariate models for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Glaucosoma hebraicum. SD = estimated standard deviation of marginal distribution of deviations from the respective growth curve. Curves were the same as those of Table 3.1.

휏1= first reference age (youngest fish in the sample), 휏2= second reference age (oldest fish in the sample), 푎 and 푏 = parameters of growth model, 푦1 = size at age 휏1, 푦2= size at age 휏2.

Standard errors (SE) of the parameters and of the standard deviations are presented in parentheses…………………………………………………………………………….. p.70-71

Table 3.4. Negative log-likelihood (NLL) for the fitted bivariate growth models for

XII

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum, together with correlation 휌 of the bivariate distribution of deviations from the somatic and otolith growth curves. Standard errors (SE) of the correlation are presented in parentheses. ‘푃{ > 0}’ = P-value of one-tailed t-test that

 > 0. ‘Prop. of bootstraps > 0’ = proportion of 4 000 bootstrap trials for which the point estimate of the correlation coefficient exceeded zero…………………………………….. p.72

Table S3.1. Maximum ages and total lengths (TL), sexuality, and habitats of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum………………………………………………..…… p.77

Table S3.2. Location and sampling regimes for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum in estuarine and coastal waters along the western coast of Australia…………. p.79

Table S3.3. Birth dates assigned to each of Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum in estuarine and coastal waters along the coast of Western Australia…………………..… p.80

Table S3.4. The parameter space (a, b) for the Schnute (1981) growth curve was divided into nine regions such that, when fitting the model, parameters estimates could be constrained to each region. This avoided any discontinuity in derivatives that would have resulted if tests to determine whether to employ an equation with an alternative structure had been included in the Template Model Builder (TMB) code………………………………………………… p.83

Table S3.5. Values of corrected Akaike Information Criteria (AICc) and Akaike Weights

(AWs) for somatic growth curves of different functional forms (i.e., a modified version of the von Bertalanffy growth equation and, for all other curve types, the versatile growth model described by Schnute (1981)) fitted to total lengths (mm) at ages of capture for 50 individuals of Acanthopagrus butcheri collected from Wellstead Estuary in 2013, where the models

XIII assumed a normal distribution or lognormal distribution of deviations from expected lengths at age…………………………………………………………………………………..….. p.89

Table S3.6. Quantitatively similar somatic growth curves, i.e., curves with values of corrected Akaike Information Criteria (AICc) lying within 2 units of the AICc (in bold font) of the best-fitting (i.e., lowest AICc) curve, for each of Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Glaucosoma hebraicum………………………………………………………………..… p.90

Table S3.7. Values of corrected Akaike Information Criteria (AICc) and Akaike Weights

(AWs) for otolith growth curves of different functional forms (i.e., a modified version of the von Bertalanffy growth equation and, for all other curve types, the versatile growth model described by Schnute (1981)) fitted to otolith radii (mm) at ages of capture for 50 individuals of Acanthopagrus butcheri collected from Wellstead Estuary in 2013, where the models assumed a normal distribution or lognormal distribution of deviations from expected radii at age…………………………………………………………………………………….…. p.91

Table S3.8. Quantitatively similar otolith growth curves, i.e., curves with values of corrected

Akaike Information Criteria (AICc) lying within 2 units of the AICc (in bold font) of the best-fitting (i.e., lowest AICc) curve, for each of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum…………………………………………………………………………..……. p.92

2 Table S3.9. Values of adjusted coefficients of determination (푅adjusted) for the curves for total lengths and otolith radii at age of the fitted bivariate growth model for each of

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum, where the somatic and otolith growth curves were of the forms (and possessing the statistical distribution for deviations) that produced the lowest values of AICc…………………………………………...………… p.93

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Table S3.10. Ranges of negative log-likelihoods (NLL) for the bivariate growth models produced using the quantitatively similar somatic and otolith growth curves for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum, together with ranges of both the correlations (휌) between deviations from the growth curves for those bivariate models and the

P-values, 푃{ > 0}, of one-tailed t-tests that those correlations exceed zero. Quantitatively similar curves were those with AICc scores lying within 2 units of the lowest score, i.e., the

AICc of the fitted curve that best described the data………………………………….… p.94

Table 4.1. Regression equations of the back-calculation approaches described by Morita and

Matsuishi (2001), Finstad (2003) and Vigliola and Meekan (2009), or derived from those equations…………………………………………………………………………….…… p.128

Table 4.2. Regression equations, which pass through the biological intercept, modified (or developed) from the equations of Morita and Matsuishi (2001), Finstad (2003) and Vigliola and Meekan (2009)………………………………………………………………….…… p.129

Table 4.3a. Root Mean Square Error (RMSE) and mean error (ME) of total length estimates for the bivariate growth model and models derived from the Morita and Matshuishi (2001) and Finstad (2003) models calculated using ten-fold cross-validations for samples of 50 fish for each of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. The percentages by which the RMSE of the alternative models differ from that of the bivariate growth model are presented in parentheses………………………………………………………….….. p.130

Table 4.3b. Root Mean Square Error (RMSE) and mean error (ME) of otolith radius estimates for the bivariate growth model and for the Morita and Matshuishi (2001) and

Finstad (2003) models calculated using ten-fold cross-validations for samples of 50 fish for

XV each of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. The percentages by which the

RMSE of the alternative models differ from that of the bivariate growth model are presented in parentheses……………………………………………………………………….…… p.131

Table 4.4a. Root Mean Square Error (RMSE) and mean error (ME) of total length estimates for the bivariate growth model and for models derived from the Morita and Matshuishi

(2001) and Finstad (2003) models calculated using holdout and ten-fold cross-validations with and without the biological intercept for samples of Acanthopagrus butcheri. The percentages by which the RMSE of the alternative models differ from that of the bivariate growth model are presented in parentheses……………………………………………… p.132

Table 4.4b. Root Mean Square Error (RMSE) and mean error (ME) of total length estimates for the bivariate growth model and the Morita and Matshuishi (2001) and the Finstad (2003) models calculated using holdout and ten-fold cross-validations with and without the biological intercept for samples of Acanthopagrus butcheri. The percentages by which the

RMSE of the alternative models differ from that of the bivariate growth model are presented in parentheses………………………………………………………………………….… p.133

Table 4.5. Mean observed total length (mm) and mean age of fish within each age class in the 2013 sample of Acanthopagrus butcheri, together with mean back-calculated total lengths

(mm) for fish at ages at zones bounding the age classes, calculated using different back- calculation approaches……………………………………………………….……... p.134-135

Table 5.1a. Corrected Akaike Information Criteria (AICc) for the fitted models (Eqs 1-5) describing the relationships between the deviations, 푆푗,푎, at each pair of ages 푎1 and 푎2, and, where present in the equations, the slopes (β) of those models…………………….…… p.162

Table 5.1b. Corrected Akaike Information Criteria (AICc) for the fitted models (Eqs 1-5)

XVI describing the relationships between the deviations, 푆푗,푎, at each pair of ages 푎1 and 푎2, and, where present in the equations, the slopes (β) of those models…………………….…… p.163

Table 5.2. Corrected Akaike Information Criteria (AICc) for the fitted models (Eqs 6-8) describing the relationships between the increment in otolith radius ∆푅푗,푡=푎+1 between opaque zones a and a + 1 and the radius of the otolith at the first of these zones 푅푗,푡=푎, and, where present in the equations, the intercepts (α) and slopes (β) of those relationships... p.165

Table S5.1. Akaike Weights (AWs) for the fitted models describing the relationships between deviations, 푆푗,푎, at each pair of ages 푎1 and 푎2 for fish j (1 ≤ 푗 ≤ 53) (Eqs 1-5, as shown in column headings) for 7-year-old Acanthopagrus butcheri from the Wellstead

Estuary……………………………………………………………………………… p.168-169

Table S5.2. Akaike Weights (AWs) for the fitted models (Eqs 6-8, as shown in column headings) describing the relationships between the increment in otolith radius ∆푅푗,푡=푎+1 between opaque zones a and a + 1 for fish j (1 ≤ 푗 ≤ 53) and the radius of the otolith at the first of these zones 푅푗,푡=푎 for 7-year-old Acanthopagrus butcheri from the Wellstead

Estuary………………………………………………………………………………….... p.170

Table 6.1. Maximum ages and total lengths (TL), sexuality, and habitats of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum, the biological characteristics of which were employed in the simulation study………………………………………………………………...… p.201

Table 6.2. Parameters from the ‘true’ somatic and otolith growth curves used to generate simulated data for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……… p.202-203

Table 6.3. Parameters for the somatic and otolith growth curves of the bivariate growth model fitted to samples of length and otolith size at age obtained through simulations

XVII employing a level of fishing mortality of F = 0.1 year-1 for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum…………………………………………………………..…. p.204

Table 6.4. Parameters for the somatic and otolith growth curves of the bivariate growth model fitted to samples of length and otolith size at age obtained through simulations employing a level of fishing mortality of F = 0.4 year-1 for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………………………... p.205

Table 6.5. Parameters for the somatic and otolith growth curves of the bivariate growth model fitted to samples of length and otolith size at age obtained through simulations employing a level of fishing mortality of F = 0.8 year-1 for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………………………... p.206

XVIII

LIST OF FIGURES

Figure 1.1. Schema of the location of the inner ear, its components and the innervation of the otolith to the brain in teleost fish…………………………………………………………… p.6

Figure 2.1. Sectioned otolith of a 262 mm Acanthopagrus butcheri from the Wellstead

Estuary with seven opaque zones…………………………………………………………. p.25

Figure 2.2. Left, photograph of the head of a two day-old Acanthopagrus butcheri larva, showing larval otoliths (identified by arrows); top-right, photograph of a whole

Acanthopagrus butcheri larva; bottom-right, extracted larval otolith………………..…… p.28

Figure 3.1. Growth curves fitted to the total lengths (mm) at ages of capture (years) for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum……………………………………...…p.73

Figure 3.2. Growth curves fitted to the otolith radii (mm) at ages of capture (years) for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum……………………………………..… p.74

Figure 3.3. Total lengths (mm) and otolith radii (mm) at capture for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Glaucosoma hebraicum, and the relationships formed by the pairs of values of total length and otolith radius at each age (over the range of observed ages at capture) predicted for each species using the somatic and otolith growth curves of the fitted bivariate growth model for that species………………………………………………………………………………… p.75

Figure 3.4. Relative instantaneous rates of somatic and otolith growth versus age (years) at capture for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. Solid line (−)

XIX represents somatic growth and dashed line (---) represents otolith growth…………..…… p.76

Figure S3.1. Quantitatively similar somatic growth curves, i.e., curves with AICc scores lying within 2 units of the lowest score (the AICc of the best curve, in black), for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. Pütter = Pütter number 2, VB = traditional von

Bertalanffy, ‘a=0, b>0’ = Schnute (1981) growth curve with a = 0 and b > 0, ModVB = modified von Bertalanffy, ND = normal distribution, LND = lognormal distribution…... p.95

Figure S3.2. Quantitatively similar otolith growth curves, i.e., curves with AICc scores lying within 2 units of the lowest score (the AICc of the best curve, in black), for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. Pütter = Pütter number 2, VB = traditional von

Bertalanffy, GenVB = generalised von Bertalanffy, ‘a=0, b>0’ = Schnute (1981) growth curve with a = 0 and b > 0, ‘a<0, b>0’ = Schnute (1981) growth curve with a < 0 and b > 0, ModVB

= modified von Bertalanffy…………………………………………………………….… p.96

Figure S3.3. Relationships between total lengths and otolith radii at age formed by quantitatively similar somatic and otolith growth curves for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Glaucosoma hebraicum. Black = best fitting somatic and otolith growth curves. Pütter =

Pütter number 2, VB = traditional von Bertalanffy, GenVB = generalised von Bertalanffy,

‘a=0, b>0’ = Schnute (1981) growth curve with a = 0 and b > 0, ‘a<0, b>0’ = Schnute (1981) growth curve with a < 0 and b > 0, ModVB = modified von Bertalanffy, ND = normal distribution, LND = lognormal distribution. Quantitatively similar = AICc scores lying within

2 units of the lowest score…………………………………………………………...…… p.97

Figure 4.1. Comparison of mean observed total length (mm) versus mean age (years) of fish within each age class in the 2013 sample of Acanthopagrus butcheri, with the means of the total lengths (mm) at different ages at zones calculated using the different back-calculation approaches, i.e., the Bivariate Growth model (BG), the Age Effect model (AE) (i.e., the model described by Morita and Matsuishi (2001)), the Interaction Term model (IT) (i.e., model described by Finstad (2003)) and the Modified Fry model (MF) (i.e., model described by Vigliola et al. (2000)) with and without constraining the data through a biological intercept (BI) for Acanthopagrus butcheri. 95% confidence intervals are represented as error bars. Note that data for the single fish at 21 years of age was excluded………….… p.136-137

Figure 5.1. Box and whisker plot of the deviations 푆푗,푎 at each opaque zone. Mean values are given within each box, which represents the first two quartiles of data about the mean. Note that the mean is displayed rather than the more commonly used median. Standard deviations are displayed above the plotted data for each age……………………………………..… p.166

Figure 5.2. Scattergrams of the deviations for individual fish at each pair of opaque zones

(lower left triangle), histograms of deviations at each opaque zone (diagonal), and the correlations (and associated P-values) for the different pairs of opaque zones of individual fish (upper right triangle)……………………………………………………………...… p.167

Figure 6.1. Mean ages (years) of fish within length slices (mm) at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………………………………………...… p.207

Figure 6.2. Differences between the mean ages (years) of fish within corresponding length

(mm) classes for fishing mortalities, F = 0.4 and 0.8 year-1, and the mean age of the fish from

XXI the sample produced for the fishing mortality F = 0.1 year-1, using a one-tailed Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. Solid black line represents mean ages at length for fish in the samples generated using F = 0.1 year-1, blue circles represent the comparison between mean ages at length for fish in the samples generated using F = 0.1 and

0.4 year-1, and red circles represent the comparison between mean ages at length for fish in the samples generated using F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance (푃 < 0.05) of the comparisons……………………….…..… p.208-209

Figure 6.3. Mean ages (years) of fish within otolith radius slices (mm) at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………………………………………...… p.210

Figure 6.4. Differences between the mean ages (years) of fish within corresponding otolith radius (mm) classes for fishing mortalities, F = 0.4 and 0.8 year-1, and the mean age of the fish from the sample produced for the fishing mortality F = 0.1 year-1, using a one-tailed

Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. Solid black line represents mean ages at otolith radius for fish in the samples generated using F = 0.1 year-1, blue circles represent the comparison between mean ages at otolith radius for fish in the samples generated using F = 0.1 and 0.4 year-1, and red circles represent the comparison between mean ages at otolith radius for fish in the samples generated using F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance (푃 < 0.05) of the comparisons………………………………………………………………………… p.211-212

Figure 6.5. Mean otolith radius (mm) of fish within age (years) slices at increasing levels of

-1 mean length at first capture (퐿50) for a fishing mortality of F = 0.1 year for Acanthopagrus

XXII butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum………………………………………………...… p.213

Figure 6.6. Mean otolith radius (mm) of fish within age (years) slices at increasing levels of

-1 mean length at first capture (퐿50) for a fishing mortality of F = 0.8 year for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………...…… p.214

Figure 6.7. Differences between the mean otolith radius (mm) of fish within corresponding age (years) classes for larger values of 퐿50 and the mean otolith size of the fish from the sample produced for the smaller values of 퐿50, using a one-tailed Aspin-Welch t-test for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum at the highest fishing mortality,

F = 0.8 year-1. Solid black line represents mean otolith radii at age for fish in the samples generated using the smaller values of 퐿50 and red circles represent the comparison between mean otolith radii at age for fish in the samples generated using the larger and the smaller values of 퐿50. Large red circles indicate the statistical significance (푃 < 0.05) of the comparisons……………………………………………………………...……….… p.215-216

Figure 6.8. Mean length (mm) of fish within otolith radius slices (mm) at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum…………………………………………………………………………...…… p.217

Figure 6.9. Differences between the mean length (mm) of fish within corresponding otolith radius (mm) classes for fishing mortalities, F = 0.4 and 0.8 year-1, and the mean length of the fish from the sample produced for the fishing mortality F = 0.1 year-1, using a one-tailed

Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. Solid black line

XXIII represents mean lengths at otolith radius for fish in the samples generated using

F = 0.1 year-1, blue circles represent the comparison between mean lengths at otolith radius for fish in the samples generated using F = 0.1 and 0.4 year-1, and red circles represent the comparison between mean lengths at otolith radius for fish in the samples generated using

F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance

(푃 < 0.05) of the comparisons………………………………………………...…… p.218-219

Figure 6.10. Mean otolith radius (mm) of fish within length slices (mm) at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………………………………………...… p.220

Figure 6.11. Differences between the mean otolith radius (mm) of fish within corresponding length (mm) classes for fishing mortalities, F = 0.4 and 0.8 year-1, and the mean otolith radius of the fish from the sample produced for the fishing mortality F = 0.1 year-1, using a one- tailed Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. Solid black line represents mean otolith radii at length for fish in the samples generated using

F = 0.1 year-1, blue circles represent the comparison between mean otolith radii at length for fish in the samples generated using F = 0.1 and 0.4 year-1, and red circles represent the comparison between mean otolith radii at length for fish in the samples generated using

F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance

(푃 < 0.05) of the comparisons…………………………………………………...… p.221-222

Figure 6.12. Length (mm) at age (years) calculated using the ‘true’ growth curves and predicted using the bivariate growth model at different levels of fishing mortality (F = 0.1,

0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum…….……... p.223

XXIV

Figure 6.13. Otolith radius (mm) at age (years) calculated using the ‘true’ growth curves and predicted using the bivariate growth model at different levels of fishing mortality (F = 0.1,

0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum…………… p.224

Figure 6.14. Length (mm) at otolith radius (mm) calculated using the ‘true’ growth curves and predicted using the bivariate growth model at different levels of fishing mortality

(F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.…. p.225

Figure 6.15. Distribution of length (mm) deviations with age (years) from the ‘true’ somatic growth curve at the lowest level of fishing mortality, F = 0.1 year-1, for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………..….… p.226

Figure 6.16. Distribution of otolith radius (mm) deviations with age (years) from the ‘true’ otolith growth curve at the lowest level of fishing mortality, F = 0.1 year-1, for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum…………………………………...… p.227

Figure 6.17. Distribution of length (mm) deviations with age (years) from the ‘true’ somatic growth curve at the highest level of fishing mortality, F = 0.8 year-1, for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………...…… p.228

Figure 6.18. Distribution of otolith radius (mm) deviations with age (years) from the ‘true’ otolith growth curve at the highest level of fishing mortality, F = 0.8 year-1, for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum………………………………...... … p.229

Figure S6.1. Age (years) compositions of the fish in the simulated samples at different levels

XXV of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum……………………………………………………………………………...… p.230

Figure S6.2. Fish length (mm) compositions of the fish in simulated samples at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum…………………………………………………………...… p.231

Figure S6.3. Otolith radius (mm) compositions of the fish in the simulated samples at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum…………………………………………………………...… p.232

Figure S6.4. Age (years) compositions of the fish in the simulated samples at two levels of

Figure S6.5. Fish length (mm) compositions of the fish in the simulated samples at two

-1 levels of mean length at first capture (퐿50) for a fishing mortality of F = 0.8 year for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum…………………………………...… p.234

Figure S6.6. Otolith radius (mm) compositions of the fish in the simulated samples at two

-1 levels of mean length at first capture (퐿50) for a fishing mortality of F = 0.8 year for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum………………………………...…… p.235

XXVI

Figure SA. Sectioned otoliths of the six study species, showing the location of the radius (red line) from the primordium to the outer edge of the otolith, to the right of the sulcus and perpendicular to the opaque zones. Scale bar = 1 mm……………………………...…… p.248

Figure SB. Whole otoliths of adults and juveniles for the six study species. Scale bar = 1 mm………………………………………………………………………….…… p.249

XXVII

CHAPTER 1 – GENERAL INTRODUCTION

1.1 BROAD OVERVIEW OF THE THESIS

Back-calculation is an important tool in fisheries science and fish ecology for reconstructing growth histories of individual marine organisms based on the microstructure present within calcified structures such as statoliths, vertebrae and otoliths. For teleosts, the most commonly used hard structures in back-calculation studies are their otoliths, presumably because the results of numerous fish age validation studies have shown that these structures can be used to accurately determine the ages of individual fish, and thereby also facilitate reliable determination of their growth. Although back-calculation studies also rely on an accurate description of the relationship between fish length and otolith size, the factors that can influence this relationship remain poorly understood. Furthermore, current studies of back-calculation typically provide no link between traditional somatic growth models and their relationship to otolith growth. To address this omission, I developed a growth model that describes both somatic and otolith growth using traditional forms of growth curves and, assuming a bivariate distribution of deviations from these two growth curves, also relates fish lengths at age to otolith size. The performance of this model was compared with existing approaches using data for six fish species with widely varying biological characteristics. The bivariate growth model was then extended to produce a proportionality-based back- calculation approach, i.e., which assumed constant proportionality of the length of each fish from its expected length given the size of its otolith at each age throughout life. The estimates of back-calculated lengths produced using this approach were compared with those obtained using other contemporary back-calculation methods. The thesis also explored the hypothesis that, throughout the life of an individual of a selected fish species, the ratio of its otolith size relative to the average size of otoliths for individuals of the same age remains constant.

Finally, the effects of fishing mortality and length-dependent selectivity on mean and predicted otolith sizes at age and on the relationship between fish length and otolith size, the basis of all back-calculation approaches, were explored.

1.2 SCIENTIFIC IMPORTANCE AND USE OF OTOLITHS

Otoliths are hard calcium carbonate structures found in pairs within the heads of all teleosts, and are employed for balance, orientation and hearing (e.g., Campana and Neilson

1985; Campana 1999). The main interest that otoliths hold for biologists and fishery scientists lies in the fact that these structures store valuable information on age of fishes and the environment within which they lived at different ages throughout life, facilitating understanding of population dynamics, stock identity, fish systematics and evolution (Popper et al. 2005). The use of otoliths as indicators of fish age began with Reibisch’s (1899) observations of otolith annuli in Pleuronectes platessa. Since then, interest in otolith microstructure as a metabolically-inert timekeeper and environmental recorder has grown exponentially, with close to a million fish aged each year worldwide (Campana et al. 2000;

Campana and Thorrold 2001; Campana 2005).

The information related to growth and environment contained within the otolith microstructure and chemistry at different temporal scales is important for the management of fisheries and protected fish species around the world (Kalish 1989; Campana 1999). This information, which is employed in studies of age, growth, movement patterns and habitat interactions, makes these biological structures highly valuable to fishery scientists. Through records of daily events stored within their structure and chemical composition, otoliths can assist in elucidating the effects of changes in the environment on growth and survival in the early life stages of fish, thus resulting in an improved understanding of factors affecting recruitment success (Jones 1992). For adult fish, counts of the annuli that form in otoliths at

annual intervals are used to determine the ages individual fish for estimating growth, longevity, and mortality rates in fish populations (e.g., Campana 2001; Wilson and Nieland

2001; Laidig et al. 2003). This understanding of fish population biology is vital for producing reliable fish stock assessments required for determining effective management strategies to allow fish harvesting whilst ensuring that the risk of such harvesting to the long term sustainability of the stock remains at an appropriate level (Jones 1992; Campana 2001; Begg et al. 2005).

Somatic growth rate is calculated from growth models, which are fitted to lengths at capture using ages derived by examining the microstructure of otoliths (Campana and Jones

1992). The diversity of growth models available in the literature and their interpretation make the determination of the most appropriate model to describe length-at-age data for a species difficult (Ricker 1979). Thus, for example, a number of alternative curves have been used to describe growth rates of individuals at early life history stages, e.g., linear (e.g., Townsend and Graham 1981; Jones 1986; Victor 1987), exponential (e.g., Struhsaker and Uchiyama

1976), and Gompertz (e.g., Laroche et al. 1982; Lough et al. 1982; Warlen and Chester 1985;

McGurk 1987) models. The von Bertalanffy growth curve (von Bertalanffy 1938) has also been employed to describe the growth of young fish (e.g., Ralston 1976; Wild and Foreman

1980; Laroche et al. 1982), but to a much lesser extent than for adult fish (Ricker 1979;

Campana and Jones 1992; Jones 1992), largely due to its inability to represent sigmoidal growth data. Before selecting the final form of curve to be employed, the fit to length-at-age data provided by alternative growth curves should, therefore, be explored. The growth models should also be able to account for both accelerated growth at the juvenile stage and decelerated growth during the adult stage, when individuals approach a final size (Schnute

1981). To address this, Schnute (1981) proposed a comprehensive growth model, which

includes several historical models as special cases, and which can be related to both accelerated and decelerated growth.

A prominent application of otolith microstructure examination is growth back- calculation, which involves estimating the length of individual fish at successive ages throughout life (e.g., Francis 1990; Campana and Jones 1992). Back-calculation is an invaluable analytical method widely used in fisheries science and ecology around the world for reconstructing individual growth histories of teleosts (e.g., Francis 1990; Campana 2005;

Vigliola and Meekan 2009). The most common application of back-calculation is that of complementing the observations of lengths at the ages of capture with estimates of lengths at age for younger fish. Curves are then fitted to the combined set of observations and estimates to determine values of key growth parameters (e.g., the asymptotic length or the growth coefficient of the von Bertalanffy growth function) and to compare, describe or predict growth variations among individuals (e.g., Shafi and Jasim 1982; Jones 2000; Colloca et al.

2003; Lorance et al. 2003; Ballagh et al. 2011). For some species, growth histories within bony structures, such as otoliths, become particularly useful when samples sizes are small or information on life history at earlier ages is lacking (e.g., Campana 1989; Meekan et al.

1998a; Vigliola and Meekan 2009). Growth curves derived from back-calculated data have been used to compare growth rates between sexes, cohorts and populations of the same species (e.g., Frost and Kipling 1980; Thorrold and Williams 1989; Sirois et al. 1998;

Goldman and Musick 2006). Back-calculated lengths at age have also been used in models to account for individual variability in growth, thereby facilitating improvement of fish stock assessments (e.g., Pilling et al. 2002; Santiago and Arrizabalaga 2005). Other developments, which use the results of growth back-calculations, involve relating environmental conditions from a specific time period to estimates of length at age (e.g., Dutil et al. 1999). Back- calculation has also led to significant progress in assessment of the influence of year-to-year

variation of size-selective mortality on previous life history stages (e.g., Grønkjær and

Schytte 1999; Good et al. 2001; Sinclair et al. 2002a).

Another common application of information from otoliths is the use of otolith microchemistry for discriminating fish stocks and understanding population connectivity

(e.g., Gillanders 2002; Rooker et al. 2003; Fowler et al. 2005; Stransky et al. 2005; Thresher and Proctor 2007). Environmental histories and migration patterns and/or habitat shifts throughout the lives of individual fish have been reconstructed from otolith elemental concentrations, which change in a predictable manner with environmental variables (e.g.,

Elsdon and Gillanders 2004; Daverat et al. 2005; Dorval et al. 2005; Hobbs et al. 2005).

Furthermore, techniques developed initially for analysing tree-ring data (i.e., dendrochronology) have been employed to describe the long-term relationships between trends exhibited by otolith growth and aspects of environmental variability (e.g., water temperature) (e.g., Black 2009; Neuheimer et al. 2011; Gillanders et al. 2012; Black et al.

2013). Sclerochronology, or the study of calcified structures to reconstruct the past history of living organisms, which employs these relationships to predict the effects of climate change from otolith biochronologies, is becoming an important aspect of fisheries management (e.g.,

Panfili et al. 2002; Matta et al. 2010; Gillanders et al. 2012).

1.3 OTOLITH FUNCTION, FORMATION AND GROWTH

Otoliths play the role of a mechanoreceptor (e.g., involvement with the maintenance of equilibrium, the detection of gravity and sound) in the inner ear of fish. The nerve cells in the macula (a sensory epithelium which lies close to the otolith) of the labyrinth organ are sensitive to pressure, movement and sound vibrations (Lowenstein 1971), which are detected through the relative motion between the macula and the otolith (Popper and Lu 2000). The labyrinth (inner ear) consists of three epithelial chambers (i.e., the lagena, the utriculus and

the sacculus) connected by three semi-circular canals, filled with endolymph (an acellular medium similar in constitution to plasma and secreted by the inner ear epithelium) and containing one otolith each, i.e., the asteriscus, the lapillus and the sagitta (Lowenstein 1956;

Enger 1964; Payan et al. 2004) (see Fig. 1.1). The sagittae, the largest of the three pairs of otoliths, are used by fish for directional hearing and frequency responses (Lu and Xu 2002), while the lapilli are predominantly involved in balance and orientation (Riley and Moorman

2000).

Figure 1.1. Schema of the location of the inner ear, its components and the innervation of the

1 otolith to the brain in teleost fish.

1 The illustration, which was drawn by the author, was inspired from different sources, namely - Drawing of the left Merluccius spp. inner ear by A. Lombarte (see Morales-Nin 2000) - ‘Sensory systems’ refer to http://www.snre.umich.edu/~pwebb/NRE422-BIO440/lec16.html (viewed on 16/02/2016) - ‘Illustration of the anatomy of the inner ear of fish’ by C. Iverson

The acellular and metabolically inert nature of otoliths enables elements or compounds to accrete onto their surface and to be permanently retained, while continued otolith growth from before hatching to death records the entire life of the fish (Campana and

Neilson 1985). The general pathway of inorganic elements into the otolith is from water into the blood plasma via osmoregulation in the gills (in freshwater fish) or the intestine (in saltwater fish), then into the endolymph and, lastly, into the crystallising otolith (Olsson et al.

1998; Campana 1999).

Otolith composition is dominated by calcium carbonate (99% CaCO3 in the aragonite form) deposited onto a non-collagenous organic matrix (e.g., Asano and Mugiya 1993; Hoff and Fuiman 1993; Campana 1999; Payan et al. 2004). The calcification process (i.e., the deposition of mineral) is mainly dependent on the composition of the endolymphatic fluid surrounding the otolith and can thus be described on the basis of physical principles, one of the key regulating factors being the pH of the endolymph (Romanek and Gauldie 1996;

Payan et al. 1997, 1998, 2004; Campana 1999). The gradient of the pH, determined by the concentration of bicarbonate ions within the endolymph and resulting from high to low levels of metabolic activity, maintains the gradient in otolith calcification (Mugiya 1986; Gauldie and Nelson 1990a; Mugiya and Yoshida 1995; Payan et al. 2004). Since otolith growth is a biomineralisation process, the proteins, which represent a small percentage of the otolith and constitute the organic matrix, also play a pivotal role in the calcification process (e.g.,

Wheeler and Sikes 1984; Campana 1999). While water-insoluble otolith proteins constitute the structural framework of calcification, it is the water-soluble, calcium-binding proteins which regulate the rate of calcification and the type of calcium carbonate crystals which are formed (Wright 1991; Asano and Mugiya 1993; Campana 1999).

Otolith growth is a complex phenomenon encompassing both endogenous and exogenous factors, regulated by the physiology of the fish (Morales-Nin 2000). Within the

otolith, growth increments are laid down through differential deposition of calcium carbonate

(translucent concentric rings) and protein (opaque concentric rings) (Campana and Neilson

1985; Mugiya 1987; Gauldie and Nelson 1988). At the endogenous level, the proteins of the organic matrix regulate the rate of calcification and thus otolith growth (Wheeler and Sikes

1984; Wright 1991; Asano and Mugiya 1993). The periodicity of increment formation follows endogenous diel rhythms, entrained by photoperiod and synchronised to periodic environmental cycles, which characterise the neural activity of most animals (Campana and

Neilson 1985; Gauldie and Nelson 1988). Since Pannella’s (1971) discovery of daily growth increments, the circadian rhythm of calcium deposition has been shown to be controlled by endocrinological processes (e.g., Boeuf and Le Bail 1999; Shiao et al. 2008). Another important control and the main environmental factor affecting otolith growth rate is water temperature, which has been found to regulate the rate of calcification (e.g., Gutiérrez and

Morales-Nin 1986; Mosegaard and Titus 1987; Mosegaard et al. 1988; Lombarte and

Lleonart 1993; Folkvord et al. 2004). Subsequently, both the anatomy and function of the inner ear, together with environmental conditions (conveyed through physiological processes) regulate, otolith growth rates.

Note that in the present study, the term ‘otolith growth’ relates to the accretion of mineral onto the organic matrix of the otolith, rather than referring to growth in terms of cellular division.

1.4 SOMATIC AND OTOLITH GROWTH RELATIONSHIP

Understanding the relationship between somatic and otolith growth is fundamental for environmental studies and stock assessments. In effect, the validity of inferences drawn from the results of recent sclerochronological studies on otolith growth (e.g., Thresher et al. 2007;

Matta et al. 2010; Gillanders et al. 2012; Black et al. 2013) relies on the assumption that the

rate of somatic growth is directly proportional to the rate of otolith growth. Such an assumption also underpins most back-calculation methods (e.g., Campana 1990; Francis

1990; Sirois et al. 1998; Vigliola et al. 2000; Morita and Matsuishi 2001).

The rate of protein synthesis in fish is associated with metabolic rate, somatic growth rate and temperature (Mommsen 2001; Pörtner et al. 2001). As somatic growth reflects the rate of net protein synthesis, the rate of formation of a proteinaceous matrix on the surface of a growing otolith is consequently highly correlated with the rate of somatic growth (Campana

1999). In an experiment on the effect of a short period of starvation between two groups of rainbow trout (Oncorhynchus mykiss), Payan et al. (1998) selected starvation as a natural physiological situation characterised by a reduction in both somatic and otolith growth. A reduction in the rate of otolith growth in a group of starved trout was directly related to an overall decline in the alkalinity of the endolymph which, in turn, is known to be dependent on the acid-base balance in teleosts (Payan et al. 1997; Payan et al. 1998). Furthermore, as ~ 10-

30% of otolith carbon can be derived from metabolism, isotopic composition of otoliths can be used to trace changes in metabolic activity throughout the life of fish, with a tendency for protein synthesis and otolith growth to decline with age (Mulcahy et al. 1979; Schwarcz et al.

1998). Through this effect of metabolism on otolith growth, variation in somatic growth will be reflected, to some extent, in otolith microstructure, implying that increment width trajectory, i.e., along a line perpendicular to the widths between successive growth zones from the centre to the outer edge of the otolith, can provide information on changes in somatic growth (e.g., Thorrold and Williams 1989; Wang and Tzeng 1999; Oozeki and

Watanabe 2000; Fey 2006).

Despite the widely-accepted relationship between otolith and somatic growth, the rate of otolith accretion is also positively related to temperature, independent of somatic growth

(e.g., Mosegaard et al. 1988; Hoff and Fuiman 1993; Fey 2006). This finding resulted in a

general theory which relates otolith growth to metabolic rate (influenced by temperature, food intake, dominance status and other activity cycles), rather than somatic growth rate (e.g.,

Mosegaard et al. 1988; Wright et al. 1990; Titus and Mosegaard 1991; Wright 1991; Metcalfe et al. 1992; Yamamoto et al. 1998; Folkvord et al. 2004; Fey 2005). Consequently, changes in otolith increment width may not be well linked to variation in somatic growth, and particularly when otolith and somatic growths are affected by short-term fluctuations in temperature or food (e.g., Bradford and Geen 1992; Barber and Jenkins 2001). It is important to note, however, that otolith growth continuously scales to somatic growth in an age- dependent manner (Secor and Dean 1992). Variations in the strength of the relationship between otolith and somatic growth may also reflect the effects of measurement error associated with differing measures of otolith size and, in linear regression analyses, reduced contrast associated with the compression of the range of fish sizes as a result of, for example, size-selective mortality (Meekan et al. 1998a, b).

Descriptions of the existence of a partial ‘uncoupling’ between somatic and otolith growth rates have become commonplace in the literature (Mosegaard et al. 1988; Secor and

Dean 1989; Wright et al. 1990; Francis et al. 1993; Barber and Jenkins 2001; Fey 2005,

2006). Indeed, for a number of species, it has been demonstrated that, as fish grow older, the trends exhibited by inter-annual variations in otolith increment widths may not fully reflect those of somatic growth (e.g., Mosegaard et al. 1988; Secor and Dean 1989; Wright et al.

1990; Francis et al. 1993; Hoff and Fuiman 1993; Fey 2006). Two primary factors have been implicated in this ‘uncoupling’, a) slow-growing fish have larger otoliths at a given size than fast-growing fish of the same size, i.e., a ‘growth effect’ (e.g., Templeman and Squires 1956;

Reznick et al. 1989; Campana 1990; Francis et al. 1993; Strelcheck et al. 2003) and b) although somatic growth rate declines with age, calcification continues to occur onto the surface of the otolith due to the physiochemical and obligatory microincrementation

processes involved in daily physiological cycles, i.e., an ‘age effect’ (e.g., Mosegaard et al.

1988; Gauldie and Nelson 1990a; Secor and Dean 1992; Morales-Nin 2000). Indeed, while the rate of organic matrix formation might provide threshold limits for calcification rate in otoliths, the effect of temperature may promote calcification rate within those limits

(Mosegaard et al. 1988; Campana 1999). Both the age and growth effects are linked, as continued increase in otolith size can induce disproportionately larger otoliths in slow- growing individuals than those of fast-growing fish of the same size (Mosegaard et al. 1988;

Secor and Dean 1989; Wright et al. 1990; Fey 2006). The growth effect may also vary through time and therefore introduce curvature into otolith size trajectories for individual fish, as growth varies over time and particularly among ontogenetic stages (Campana 1990).

In a simulation, Campana (1990) demonstrated that individual variability in growth rates induced bias into the fish-otolith size relationship and could thus explain Lee’s Phenomenon, i.e., back-calculated lengths are smaller than observed lengths at age at capture.

The terms ‘uncoupling’, ‘decoupling’ and other such terminologies have been poorly defined in previous studies of the relationship between somatic and otolith growth rates and are likely to be inaccurate (Secor and Dean 1992; Xiao 1996). Their continued usage seems to serve the purpose of separating the specific problem of the relationship between somatic and otolith growth from the general context of allometry and from the basic assumptions of back-calculation (Hare and Cowen 1995). In this thesis, and as suggested by Xiao (1996), the relationships between somatic and otolith growth rates will be described through the life of fish.

1.5 BACK-CALCULATION

Growth back-calculation involves using longitudinal measurements made on otoliths, e.g., distances from the primordium to the outside edges of successive opaque zones and to

the otolith edge, to deduce body lengths for an individual fish at ages prior to capture, while assuming that there is a relationship between the length and otolith size at capture and the measured sizes (Carlander 1981; Campana 1990; Hare and Cowen 1995; Sirois et al. 1998;

Vigliola et al. 2000; Morita and Matsuishi 2001; Vigliola and Meekan 2009). Back- calculation requires 1) that the periodic deposition rate of increments is constant within the otolith (Geffen 1992); 2) that these increments can be read with accuracy and precision

(Campana 1992); and 3) that there is a relationship between somatic and otolith growth of fish, which is derived from strong correlations between body size of fish and otolith size

(Francis 1990; Vigliola and Meekan 2009). Field and experimental evidence as to partial

“uncoupling” between somatic and otolith growth rates (e.g., Mosegaard et al. 1988; Secor and Dean 1989; Wright et al. 1990; Fey 2006) may thus directly challenge the capability of back-calculation to produce reliable estimates of fish length. Indeed, because the effects of age and growth are not mutually exclusive (Morita and Matsuishi 2001), both factors can introduce biases into the fish length-otolith relationship and thus invalidate back-calculation.

To avoid potential bias associated with back-calculation, rather than predicting lengths at age of young fish using this approach, some authors employ otolith radius at age as a proxy for length at age when studying growth (Hare and Cowen 1995). Such an approach would avoid bias due to the effect of size-selective mortality on the fish-otolith size relationship (Ricker

1969; Gleason and Bengtson 1996; Grimes and Isely 1996). Recent experimental and theoretical findings, however, have strongly refuted this approach and results demonstrated that, for some species, back-calculated size at age is a better proxy of fish length at age than otolith radius at age (Wilson et al. 2009).

The first method used to back-calculate previous body lengths was based on fish scales and assumed that the scale grew in exact proportion to the length of individual fish

(Lea 1910; Francis 1990). This approach was later rejected by Lee (1920), as it required each

back-calculated line to pass through the origin (Francis 1990). The body length of fish at the time of bony structure formation (c, notation adopted from Francis 1990) was introduced into back-calculation methods by Fraser (1916) and Lee (1920), with proportionality assumed between subsequent growth increments from that point. A linear regression was employed to estimate c, using data for fish body length and otolith size at capture. Subsequently, use of regression to describe the linear or curvilinear relationship between fish and otolith size at capture has become the current practice when calculating the parameters of a back- calculation formula (BCF2). The choice of the dependent variable for this regression seems somewhat arbitrary, as both a regression of fish body length on bony structure size (Fraser

1916; Lee 1920) and a regression of bony structure size on fish body length (Francis 1990) have been proposed and used in back-calculation. The relationship described by the “simple regression” method (sensu Secor and Dean 1992) of fish length on otolith size typically ignores, however, the ‘errors-in-variables’ problem that results from variation in lengths at age and otolith radii at age (Francis 1990; Secor and Dean 1992; Vigliola and Meekan 2009).

In his review, Francis (1990) describes several non-linear BCF and their associated regression models. These models are commonly linearized without exploring the implicit assumption of normally-distributed residuals on the transformed dependent variable

(Kielbassa et al. 2011). An additive error for the dependent variable is often employed when applying non-linear regression directly to this variable. On the other hand, and to avoid heteroscedasticity, some studies have employed a logarithmic transformation of the dependent variable when using a linear regression, thus stabilising the variance of the log- transformed data and implying a multiplicative error for the size variable within arithmetic space (Francis 1990; Zhang and Beamish 2000; Rodríguez-Marín et al. 2002; Jessop et al.

2004).

2 Note that acronyms in Chapter 1 are listed within the Glossary of acronyms.

Due to its averaging effect, a regression method used to predict the lengths at age of fish prior to their age at capture, and which assumes that all fish follow the same back- calculation line, is indifferent to changes in individual growth histories (Secor et al. 1989).

Back-calculated estimates of length at age calculated using the fitted regression equation are biased as they fail to account for individual variation in growth (Vigliola and Meekan 2009).

By contrast, proportionality-based back-calculation methods take into account the proportional deviation of the length and otolith size at capture of each fish from the average fish-otolith size relationship to select a unique back-calculation line for each individual

(Francis 1990; Vigliola and Meekan 2009). As most back-calculation methods used are proportional, it has been suggested that the “simple regression” method is either used inadvertently or in ignorance of alternative traditional methods (Carlander 1981; Francis

1990; Vigliola and Meekan 2009).

The assumption of a constant proportional deviation from the mean size of the body or scale has led to the formulation of two fundamental hypotheses for back-calculation, i.e., the Scale Proportional Hypothesis (SPH) and the Body Proportional Hypothesis (BPH)

(Whitney and Carlander 1956; Francis 1990). Francis (1990, 1995) recommended back- calculation procedures based on either one of these hypotheses, which state that fish and otolith sizes differ from the average sizes for a fish of the same length, or of the same otolith size, by a constant proportion throughout the life of an individual fish. At the same time,

Campana (1990) introduced the Biological Intercept (BI), a biologically based value of c, in the Fraser-Lee linear back-calculation model (Fraser 1916; Lee 1920). The BI corresponds to the initiation of proportionality between fish and otolith growth, and is thus obtained by body and otolith size measurements on newly hatched larvae. It was developed to avoid using regressions and to reduce the influence of variable growth rates in the population (i.e., the growth effect) (Campana 1990). Nonetheless, the BI model remains sensitive to non-linear

effects and particularly those induced by growth rate variations through time, i.e., the time- varying growth effect. To address the latter, the Time-Varying Growth model (TVG, Sirois et al. 1998) was developed to incorporate a growth effect factor into the structure of the linear biological intercept model, by weighing the contribution of individual increments in the length back-calculation. A downside of the TVG model is that it assumes a linear fish-otolith size relationship and would consequently be inappropriate when, for reasons other than the growth effect, the underlying relationship is non-linear in form (Wilson et al. 2009). A different approach to non-linearity was employed by Vigliola et al. (2000), who revised the

Fry model (1943) to develop the Modified-Fry model (MF) by constraining an allometric fish length-otolith radius function to pass through a biological intercept. The MF model assumes non-linearity and is robust to growth effects.

Recent studies have considered the influence of ‘age’ and incorporated this independent variable into BCFs (e.g., Sirois et al. 1998; Morita and Matsuishi 2001; Finstad

2003; Kielbassa et al. 2011). Although it has been recognised that using ‘age’ would increase the precision of back-calculation methods (Francis 1990; Secor and Dean 1992; Vigliola and

Meekan 2009; Kielbassa et al. 2011), this covariate has largely been ignored in previous studies. The Age Effect model (AE, Morita and Matsuishi 2001) was the first BCF designed to remove bias due to the age effect and was based on the assumption that otolith radius is a linear combination of both length and age. The AE is obtained by applying a scale proportional hypothesis (SPH) to a three-dimensional back-calculation function that links length, otolith radius and age. Very large errors in predicted size, however, can be produced by the AE model, which was confirmed by Wilson et al.’s (2009) results. To overcome the vulnerability of the AE model to growth effects and its sensitivity to the accuracy and precision of the regression equation relating its three variables, Finstad (2003) included a length-age interaction term in the equation describing the relationship between length, otolith

radius and age. The interaction term contributed significantly to the fit of otolith size to the age and length data. Wilson et al. (2009) noted, however, that when age and time-varying growth effects were present in a dataset, the BCFs designed specifically to correct for these effects did not necessarily perform better than the MF model.

The assumptions underlying back-calculation of fish size from otoliths necessitate validation. In most cases, validation of the back-calculation approach for individuals of a species requires the ‘simple’ comparison of observed lengths at age with the corresponding back-calculated length-at-age estimates for the same fish. Such comparison, however, involves recording the lengths of individual fish at specific times within their lives, identifying the individual fish when they are subsequently recaptured or examined, and determining, for each fish, the size that the otolith had when its length was initially recorded.

This latter requirement may account for the paucity of studies reporting validation of back- calculation formulae (Francis 1990; Brothers 1995; Vigliola and Meekan 2009). Only nine peer reviewed studies have attempted to assess the ability of back-calculation models to reproduce realistic patterns of fish growth (Vigliola and Meekan 2009). Validating back- calculation should, however, also include testing the accuracy of the results from different hypotheses to compare different back-calculation methods (Francis 1990).

1.6 CONTRIBUTION OF THE THESIS

The overall objective of the thesis was to explore the relationship between lengths and otolith sizes of fish with respect to age and the use of that relationship for back-calculation. A proportionality-based back-calculation approach that employs traditional growth curves to describe the relationships between fish lengths, otolith sizes and ages at capture, assuming a bivariate distribution of deviations from the two curves was developed, explored and tested.

This thesis employs a combination of statistical and simulation-based approaches to analyse

the relationship between fish and otolith sizes with age, explores back-calculation methods, and examines the validity of an underlying proportionality-based assumption and factors likely to affect the length-otolith size relationship.

Six commercially and recreationally important species of fish with widely differing biological characteristics, and which occupy a variety of environments in Western Australian waters, were employed in the study. The species were Acanthopagrus butcheri (Sparidae),

Argyrosomus japonicus (Sciaenidae), Bodianus frenchii (Labridae), Epinephelus armatus

(Serranidae), Epinephelides coioides (Serranidae) and Glaucosoma hebraicum

(Glaucosomatidae).

This thesis is structured as follows:

 Chapter two provides a description of the study species, sampling methods, otolith

processing, and the growth models and statistical distributions used for various analyses

in this thesis. Note that brief details of some aspects of these methods are repeated in

different chapters to assist the reader and, by making these chapters more self-contained,

to facilitate subsequent publication.

 Chapter three: The relationship between fish length and otolith size, which is the basis of

all back-calculation approaches, results from increases in both fish length and otolith size

with age. Somatic growth is well studied, but relatively few studies have focussed on the

form of the relationship of otolith size with age or included age in back-calculation

formulae. Because otolith growth results from an accretionary process that is partially

regulated by metabolic processes, it is likely that, for fish of a given age, a positive

relationship exists between fish length and otolith size. The deviations of length and

otolith size from their expected sizes for fish of the same age would therefore be expected

to exhibit positive correlation. To test this hypothesis, a model describing both somatic

and otolith growth, assuming a bivariate distribution of deviations from expected values,

was fitted to lengths and otolith sizes at age of juveniles and adults of six fish species

with different biological characteristics. Results of the fitted bivariate growth model were

used to examine the relationship between fish length and otolith size.

 Chapter four: The bivariate growth model was modified to develop a proportionality-

based back-calculation approach, which was then extended to allow the option of

constraining the growth curves to pass through a biological intercept. Using data for six

different species, the accuracy and precision of the estimates of lengths and otolith sizes

at age produced using this model were compared with those calculated using the length-

otolith size-age relationships of several contemporary back-calculation approaches. Fish

lengths determined by the new back-calculation approach, both with and without

constraining the curves to the biological intercept, were compared with those produced

using contemporary approaches.

 Chapter five: It is commonly assumed, e.g., in mixed-effects models, that otolith sizes at

different ages throughout the life of an individual fish are correlated. That is, throughout

life, the rates of otolith growth of fish with faster growing otoliths will exceed those of

fish with slower-growing otoliths. This study explored the hypothesis that, throughout the

life of an individual from a selected population of a fish species, the sizes of its otolith

remained in constant proportion to the average sizes of the otoliths of fish of the same

ages, i.e., a hypothesis of ‘proportionality of otolith radius with age’ (PORA).

 Chapter six: Predicted lengths at age of older fish tend to decrease with increasing fishing

mortality due to length-dependent selectivity and the cumulative effect of the resulting

length-dependent mortality (Taylor et al. 2005). Because increases in fishing mortality

reduce the abundance of older fish, and there is variation around the otolith growth curve,

the expected age at which fish attain a given otolith size will also be reduced. This

suggests that otolith sizes and the fish length-otolith size-age relationship are likely to

experience similar effects as lengths at age in response to the combined effects of

selectivity and increase in fishing mortality. Simulated data for samples of fish collected

from populations with growth characteristics similar to those of the six species studied

earlier in the thesis were generated assuming different selectivity patterns and levels of

exploitation. The hypothesis that otolith sizes and fish length-otolith size age relationships

would experience the same effect as fish length when fish were subjected to these

different selectivity patterns and fishing mortalities was tested using these simulated data.

 Lastly, in the seventh and final chapter of this thesis, the implications of the findings are

discussed in the context of back-calculation and the study of the relationship between

somatic and otolith growth of fish with age. The contribution made by this study is

identified.

CHAPTER 2 – STUDY SPECIES, SAMPLING METHODS, GROWTH

MODELS AND STATISTICAL DISTRIBUTIONS

2.1 STUDY SPECIES

Data for six species from different families, environments and life cycle characteristics were employed in this study to determine whether results of analyses were applicable to a wide range of teleost species. For several specific components of the study, however, analysis focuses on Black Bream Acanthopagrus butcheri (Munro 1949) from the

Wellstead Estuary, for which data were collected during this study, and for which more detailed otolith measurements were recorded. The six species used in the major elements of the study were collected from estuarine and both temperate and tropical marine waters, and possessed both gonochoristic and hermaphroditic reproductive modes. The main biological characteristics of these species are described below.

The Black Bream Acanthopagrus butcheri (Munro 1949) - Fam. Sparidae

The endemic estuarine species, Acanthopagrus butcheri, occupies many of the temperate estuaries in southern Australia, where it completes its entire life cycle and represents an important commercial and recreational species (e.g., Potter and Hyndes 1999;

Farrington et al. 2000; Sarre and Potter 2000; Jenkins et al. 2006). The maximum total length

(TL) and age recorded for this sparid are ~ 530 mm and ~ 31 years, respectively

(Hutchins and Thompson 2001; Jenkins et al. 2006; Potter et al. 2008). The growth of

A. butcheri varies markedly among estuarine systems, reflecting differences in the environment, the abundance and quality of food source, and/or the density of the population of this species within the estuary (e.g., Sarre and Potter 2000; Chuwen 2009; Gardner et al.

2010). Acanthopagrus butcheri is a gonochoristic, broadcast spawner. Adults move upstream

into the riverine reaches of the estuaries to spawn between the middle of spring and early summer (Sarre and Potter 1999). As rudimentary hermaphrodites, both females and males possess ovotestes without undergoing either a protogynous or protandrous sex change (Sarre

1999). Ages and lengths at maturity of this sparid vary markedly between estuarine systems, making A. butcheri highly plastic in term of its reproductive biology (Sarre and Potter 1999).

The birth dates, corresponding to the approximate midpoints of the spawning period for this species, are 1 October for the Wellstead Estuary and 1 November in the Swan River, Moore

River and Walpole-Nornalup estuaries (Sarre and Potter 2000).

The Mulloway Argyrosomus japonicus (Temminck and Schlegel 1843) - Fam. Sciaenidae

In Australia, Argyrosomus japonicus is distributed along the eastern, southern and western coasts, i.e., from the Burnett River in southern Queensland to North West Cape in

Western Australia (Kailola et al. 1993; Griffiths and Heemstra 1995; Farmer 2008). This scianid is important to commercial and recreational fishers (Farmer et al. 2005;

Silberschneider and Gray 2008), and occupies a wide range of marine environments with some individuals seasonally entering estuaries (Kailola et al. 1993; Griffiths and Heemstra

1995; Griffiths 1996). The relative use of habitats by A. japonicus differs with life history stage. Juveniles use nearshore coastal waters as nursery habitats (e.g., estuaries, protected embayments, etc.), whereas adults of A. japonicus migrate between both nearshore and offshore waters, including reefs down to depths of ~ 200 m (Griffiths and Heemstra 1995;

Griffiths 1996; Farmer et al. 2005; Silberschneider and Gray 2008). This gonochoristic species attains a maximum TL of ~ 2 000 mm and maximum age of ~ 31 years (Farmer et al.

2005; Gomon et al. 2008). Birth dates for this species, inferred from the midpoint of the spawning period, are 1 July on the upper west coast, 1 December on the lower west coast and

1 November of the south coast of Western Australia (Farmer 2008).

The Foxfish Bodianus frenchii (Klunzinger 1880) - Fam. Labridae

Bodianus frenchii, whish is endemic to Australia and most abundant on the lower west and south coasts of Western Australia, occurs in and around limestone and granitic temperate reefs (Cossington et al. 2010). This labrid is an important recreational species, attaining a maximum TL of ~ 480 mm and maximum age of ~ 78 years (Gomon et al. 2008;

Cossington et al. 2010). Bodianus frenchii is a monandric protogynous hermaphrodite, with all males derived from functional females (Cossington 2006; Cossington et al. 2010). The mean birth date assigned to individuals of this species is 1 January (Cossington et al. 2010).

The Breaksea Cod Epinephelides armatus (Castelnau 1875) - Fam. Serranidae

Epinephelides armatus, which is endemic to the continental shelf of south-western

Australia, is an important recreational species, which lives over and around temperate limestone and coral reefs (Hutchins and Swainston 1986; Moore et al. 2007). A gonochoristic multiple spawner, E. armatus reaches a maximum TL of ~ 510 mm and age of ~ 19 years

(Moore et al. 2007; Gomon et al. 2008). The approximate midpoint of the spawning period,

1 February, is considered to represent the birth date of E. armatus (Moore et al. 2007).

The Goldspotted Rockcod Epinephelus coioides (Hamilton-Buchanan 1822) - Fam.

Serranidae

The subtropical-tropical Epinephelus coioides occupies mangrove and nearshore rocky areas as nursery habitats before migrating offshore to coastal reefs as adults, with a distribution extending from the central coast of Western Australia around the tropical north of

Australia and then southward to New South Wales (Pember et al. 2005). In north-western

Australia, this serranid represents one of the most important recreational and commercial fish species along the Pilbara and Kimberly coasts (Pember et al. 2005). This species attains a

maximum TL of ~ 1 110 mm and maximum age of ~ 22 years and is a protogynous hermaphrodite, i.e., individuals change sex to males after first maturing as a female (Quinitio et al. 1997; Pember et al. 2005; Gomon et al. 2008). Epinephelus coioides spawns predominantly from October to January, with 1 January being assigned as its birth date

(Pember et al. 2005).

The West Australian Dhufish Glaucosoma hebraicum Richardson, 1845 - Fam.

Glaucosomatidae

Glaucosoma hebraicum, which is endemic to south-western Australia, lives in temperate coastal marine waters, around reefs, where it reaches a maximum TL of

~ 1 120 mm and maximum age of ~ 41 years (Hesp et al. 2002; Lenanton et al. 2009). This glaucosomid represents one of the most important exploited species in the West Coast

Bioregion of Western Australia, which extends from Kalbarri (27°72’S) to Augusta

(115°14’E) (Hesp et al. 2002; Wise et al. 2007; Lenanton et al. 2009). A multiple spawning gonochorist, G. hebraicum migrates from nursery habitats of low-relief hard limestone substrates to the prominent limestone reefs that it occupies during its adult life (Hesp et al.

2002). Glaucosoma hebraicum spawns from November to March, with 1 February assigned as its birth date (Hesp et al. 2002; Lenanton et al. 2009).

2.2 ACANTHOPAGRUS BUTCHERI - SAMPLING REGIME

Random samples of A. butcheri were caught in the Wellstead Estuary (34°50’S latitude and 118°60’E longitude) in May 2013. The seine used to capture fish was 21.5 m long and comprised two 10 m long wings (4 m of 3 mm mesh and 6 m of 9 mm mesh) and a

1.5 m wide bunt of 3 mm mesh. This net, which was deployed during daylight, was laid parallel to the shore and dragged onto the estuary bank. It fished to a depth of ~ 1.5 m and swept an area of ~ 116 m2. Sunken composite multifilament gill nets were also employed to

catch fish in deeper, offshore waters. Each gill net comprised seven 20 m long panels, each with a height of 2 m and containing a different stretched mesh size, i.e., 51, 63, 76, 89, 102,

115 or 127 mm mesh. On each night of sampling, three gill nets were set parallel to the shore in ~ 2 m deep water prior to dusk and retrieved ~ 2 to 3 h later. The location of each seine and gill net was marked by GPS location. The sampling sites within the Wellstead Estuary, at which A. butcheri were collected, were those described in earlier studies of this species (Sarre

1999; Sarre and Potter 2000). As each gill net was hauled onto the boat, each gilled or enmeshed fish was tagged for future identification of the mesh in which it had been caught, except when the numbers in a panel were so numerous that they would have exceeded the numbers allowed by the permits issued by the Western Australian Department of Fisheries and by the Murdoch University Animal Ethics Committee. In the latter case, when the number of fish in a panel was excessive, a random subsample of twenty fish was retained from that panel. Retained fish were euthanized in an ice slurry (75:25 ice/water ratio) immediately after capture. They were then transported to the laboratory at Murdoch

University, where they were frozen until they were processed. Fish were measured and total lengths (TL) recorded to the nearest 1 mm.

This aspect of the study reported in this thesis was conducted in accordance with conditions in permit R2561/13 issued by the Murdoch University Animal Ethics Committee.

3 2.3 OTOLITH AGEING AND MEASUREMENTS

The two sagittal otoliths of each A. butcheri were extracted, cleaned, dried and stored in paper envelopes. The left sagittal otolith from each fish was embedded in clear epoxy resin and, using an Isomet® low-speed saw (Buehler Ltd., Lake Bluff, Illinois), cut transversely

3 Note that, in order to publish the data Chapters of this thesis, there will be duplication of the general material and methods section to ensure that the methods presented in each data Chapter stands alone.

into approximately 0.3 mm sections through its primordium and perpendicular to the sulcus acusticus. The sections were cleaned, polished with wet and dry carborundum paper (grade

1 200) under tap water and mounted on microscope slides under coverslips using DePX mounting adhesive. Otolith sections were examined under reflected light against a black background using a high-resolution digital microscope camera (Leica DFC 425) mounted on a dissecting microscope (Leica MZ7.5). High-contrast digital images of the sectioned otoliths were analysed using the computer imaging package Leica Application Suite version 3.6.0

(Leica Microsystems).

The number of opaque zones in each sectioned otolith of A. butcheri was counted independently and on different occasions by E. Ashworth and P. Coulson (Fig. 2.1). On the few occasions (< 4%) when the counts of the two readers disagreed, discussions between these readers always resulted in a mutually-agreed value for the number of opaque zones.

Figure 2.1. Sectioned otolith of an individual of Acanthopagrus butcheri measuring 262 mm

TL from the Wellstead Estuary with seven opaque zones.

The age of each A. butcheri was determined using 1) the number of opaque zones in its sectioned otolith, 2) the date of capture, 3) the assigned birth date (i.e., the approximate mid-point of the spawning season), and 4) the typical date when newly-formed opaque zones become delineated from the otolith periphery in this species (Sarre and Potter 2000). In the otoliths of some individuals, delineation of the opaque zone at the periphery may occur earlier or later than the assigned date. To account for this, for fish with a narrow translucent edge visible on the outer edge of its otolith, caught within two months prior to the assigned date at which the new opaque zone typically becomes delineated, the number of (delineated) opaque zones was adjusted downwards by a year. For those fish with an opaque periphery (in which case that non-delineated outer opaque zone is excluded from the count of delineated opaque zones) or wide translucent outer margin, caught within two months following that assigned date, the number of opaque zones was adjusted upwards by a year. The age (t) of each A. butcheri was then determined using the following equation:

푚푐 + (푚푑 − 푚푏) + 1 푑푐 if mc < md 푡 = 푧 + + 12 365.25

푚푐 + (푚푑 − 푚푏) − 11 푑푐 if mc ≥ md 푡 = 푧 + + 12 365.25 where z is the number of opaque zones, mc is the month at capture, dc is the day of the month at capture, md is the month of opaque zone delineation and mb is the birth month.

Sectioned otoliths from 50 individuals of each of the other five species were randomly selected from the otolith collections, which were housed at Murdoch University. Preparation of these sections had followed the same procedure as that used for A. butcheri. These species comprised the sciaenid A. japonicus studied by Farmer et al. (2005), the labrid B. frenchii by

Cossington et al. (2010), the serranids E. armatus and E. coioides by Moore et al. (2007) and

Pember et al. (2005) respectively, and the glaucosomid G. hebraicum by Hesp et al. (2002).

Total lengths recorded and ages assigned in the studies for which these fish had originally been collected were accepted for use in this study, as the same procedures as used for

A. butcheri had been employed when recording lengths, ageing, and comparing the ages assigned by two independent readers.

It should be noted that mean monthly marginal increment analyses have validated the annual deposition of opaque zones within the otoliths of all six species, confirming that counts of these zones could be employed for determination of the ages of fish (Sarre and

Potter 2000; Hesp et al. 2002; Pember et al. 2005; Moore et al. 2007; Farmer 2008;

Cossington et al. 2010).

For individuals of each of the six species, the ‘radius’ of its otolith at the age at which it was caught was measured from its primordium to the outer edge of the otolith along a line perpendicular to the opaque zones (see Appendix, Fig. SA for the location of the radius on sectioned otoliths for each species). Each otolith radius was measured on three separate occasions to the nearest 0.1 µm along the same axis under reflected light, employing the computer imaging package Leica Application Suite v3.6.0 analysis software (Leica

Microsystems). Note that the morphologies of the otoliths vary markedly between these species (see Appendix, Fig. SB) but, for each species, the same axis provided the clearest delineation of opaque zones and was thus employed for both ageing and measurement of otolith radii.

To provide data for back-calculation and other analyses, the distance for each

A. butcheri from the primordium of its otolith to the outside edge of the first opaque zone and the increments between the outside edges of successive opaque zones were measured under reflected light to the nearest 0.1 µm. All distances were measured along an axis perpendicular to the opaque zones adjacent to the posterior edge of the sulcus of the otolith.

To obtain a biological intercept (Campana 1990) for use in the back-calculation component of this study, A. butcheri eggs from the Australian Centre for Applied

Aquaculture Research (ACAAR, Challenger Institute of Technology, Western Australia) were hatched overnight in the laboratory, Murdoch University. The TLs of thirty larvae (two- days old) were measured to the nearest 0.01 mm under transmitted light. The left otolith of each larva was collected and measured under a high-resolution digital microscope camera

Leica DFC 425 mounted on a high-performance dissecting microscope Leica MZ7.5 (7.9:1 zoom). The radii of whole otoliths were measured to the nearest 0.1 µm under transmitted light (Fig. 2.2).

Figure 2.2. Left, photograph of the head of a two day-old Acanthopagrus butcheri larva, showing larval otoliths (identified by arrows); top-right, photograph of a whole

Acanthopagrus butcheri larva; bottom-right, extracted larval otolith.

2.4 MODIFIED VON BERTALANFFY GROWTH EQUATION AND SCHNUTE

(1981) GROWTH MODEL

Total fish length, L, and otolith radius, R, were each assumed to grow in accordance

with a growth curve of the form, 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆), which describes size S (either L or R)

as a function of age t, i.e., 푆(푡) = 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆), where 푆휏1 and 푆휏2 are the expected sizes at two specified reference ages 휏1 and 휏2, and 푎푆 and 푏푆 are parameters that determine the shape of the curve. The minimum and maximum ages at capture of each species were used as the reference ages 휏1 and 휏2, respectively, in this study. Thus, the expected length 퐿̂푗

̂ of fish j at its age at capture 푡푐,푗 was calculated as 퐿푗 = 푔퐿(푡푐,푗| 퐿휏1, 퐿휏2, 푎퐿, 푏퐿) and the radius

̂ of its otolith as 푅푗 = 푔푅(푡푐,푗| 푅휏1, 푅휏2, 푎푅, 푏푅). The expected size at age t, i.e., 푆(푡), was calculated using either a modified version of the von Bertalanffy growth equation that allowed for an oblique linear asymptote (as suggested by an anonymous reviewer) or the versatile growth model described by Schnute (1981).

The modified von Bertalanffy equation is:

푆(푡) = 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆) = 푐{1 − exp[−푎푠(푡 − 푏푠)]} + 푑(푡 − 푏푠), where

푦 ⁄(휏 − 푏 ) − 푦 ⁄(휏 − 푏 ) 푐 = 2 2 푠 1 1 푠 , {1 − exp[−푎푠(휏2 − 푏푠)]}⁄(휏2 − 푏푠) − {1 − exp[−푎푠(휏1 − 푏푠)]}⁄(휏1 − 푏푠) and

푦 − 푐{1 − exp[−푎 (휏 − 푏 )]} 푑 = 1 푠 1 푠 . 휏1 − 푏푠

The Schnute (1981) model, which comprises the following four equations, is:

−1 ( ) 푏 if 푎 ≠ 0, 푏 ≠ 0 1 − 푒−푎 푡−휏1 푆 푆 [푆푏 + (푆푏 − 푆푏 ) ] 휏1 휏2 휏1 −푎(휏 −휏 ) 1 − 푒 2 1

−푎(푡−휏1) 푆휏2 1 − 푒 if 푎푆 ≠ 0, 푏푆 = 0 푆휏 exp [ln ( ) ( )] 1 −푎(휏2−휏1) 푆휏1 1 − 푒 푆(푡) = 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆) = 푏−1 if 푎 = 0, 푏 ≠ 0 푏 푏 푏 푡 − 휏1 푆 푆 [푆휏1 + (푆휏2 − 푆휏1) ( )] 휏2 − 휏1

푆휏2 푡 − 휏1 if 푎푆 = 0, 푏푆 = 0 푆휏1 exp [ln ( ) ( )] 푆휏1 휏2 − 휏1

Sixteen alternative growth models were considered when fitting growth curves to the

lengths or otolith radii at age for each species. These comprised the modified von Bertalanffy

curve and 15 alternative forms of the Schnute (1981) growth curve, each of which was

formed by constraining 푎푆 and 푏푆 to specific values or ranges and thereby ensuring that no

discontinuity was present in the derivatives of the negative log-likelihood in the ranges over

which, for that model, those two parameters extended. These alternative model forms

included many of the growth curves commonly used in fisheries science, e.g., von

Bertalanffy, Richards, Gompertz, etc. (Schnute 1981).

2.5 SPECIAL CASES OF THE SCHNUTE (1981) GROWTH EQUATION

Curves of a range of different forms, encapsulated within the Schnute model and

fitted to the lengths and otolith radii at ages of capture, were compared to determine the

growth curve which provided the best description for each of these variables. The growth

curves tested included the following:

The generalised von Bertalanffy growth curve, obtained by setting values of a > 0 and

b > 0 in the Schnute model, may be written as

푝 푌(푡) = 푦∞(1 − 푒푥푝[−푔(푡 − 푡0)]) (1) where, in this equation and the equations that follow, 푌(푡) represents size at age 푡, and 푦∞, 푔, p and 푡0 are parameters with 푦∞ > 0, 푔 > 0 and 푝 > 0 (Schnute 1981). When p = 1, i.e., when a > 0, b = 1, the above equation becomes the traditional von Bertalanffy curve used to describe the growth of fish. When p = 3, the curve is the Pütter number 2 growth curve, which is often used to describe growth of fish in terms of mass.

The Richards growth curve, obtained using values of a > 0 and b < 0, is

1 −푝 (2) 푌(푡) = 푦 (1 − 푒푥푝[−푔(푡 − 푡 )]) ∞ 푝 0

The Gompertz growth curve, obtained using values of a > 0 and b = 0, is

푌(푡) = 푦∞푒푥푝(−푒푥푝[−푔(푡 − 푡0)]) (3)

The logistic growth curve, obtained using values of a > 0 and b = −1, is

−1 푌(푡) = 푦∞(1 + 푒푥푝[−푔(푡 − 푡0)]) (4)

The linear growth curve, obtained using values of a = 0 and b = 1, is

푌(푡) = 푔(푡 − 푡0) (5)

1 The quadratic growth curve, obtained using values of a = 0 and b = , with 훽 > 0, is 2

푌(푡) = (훼 + 훽푡)2 (6)

The tth power growth curve, obtained using values of a = 0 and b = 0, with 훼 > 0 and 훽 > 0, is

푌(푡) = 훼훽푡 (7)

The exponential growth curve, obtained using values of a < 0 and b = 1, with 훽 > 0 and

훾 > 0, is

푌(푡) = 훼 + 훽exp(훾푡) (8)

2.6 UNIVARIATE AND BIVARIATE PROBABILITY DENSITY FUNCTIONS

The following candidate bivariate distributions were employed when fitting growth models to recorded lengths and otolith radii of fish at their different ages of capture:

(1) a bivariate normal distribution, where

2 2 (퐿푗, 푅푗)~BVN(퐿̂푗, 푅̂푗, 휎휀 , 휎휂 , 휌),

(2) a bivariate normal (fish length) - lognormal (otolith radius) distribution, where

2 2 2 (퐿푗, 푅푗)~NLN(퐿̂푗, ln(푅̂푗) − 휎휂 ⁄2 , 휎휀 , 휎휂 , 휌),

(3) a bivariate lognormal (fish length) - normal (otolith radius) distribution, where

2 2 2 (퐿푗, 푅푗)~LNN(ln(퐿̂푗) − 휎휀 ⁄2 , 푅̂푗, 휎휀 , 휎휂 , 휌), or

(4) a bivariate lognormal distribution, where

2 2 2 2 (퐿푗, 푅푗)~BVLN(ln(퐿̂푗) − 휎휀 ⁄2 , ln(푅̂푗) − 휎휂 ⁄2 , 휎휀 , 휎휂 , 휌),

As shown above, the means of the log-transformed values in the above distributions were assumed to be offset from zero by − 휎2⁄2 to ensure that the expected values of

퐿푗 = 퐿̂푗exp(휀푗) or 푅푗 = 푅̂푗exp(휂푗) are 퐿̂푗 or 푅̂푗, respectively.

Univariate normal distribution

If 푦 is normally-distributed with mean  and standard deviation , i.e., 푦~N(휇, 휎2), the probability density function may be written as

1 ( 푦 − 휇)2 pdf(푦|휇, 휎) = exp [− ]. √2휋휎2 2휎2

The negative log-likelihood of observed values 푦푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 1 2 휆 = (푛⁄2)[ln(2휋) + ln(휎2)] + ∑(푦 − 휇) . 2휎2 푗 푗=1

Univariate lognormal distribution

If 푦 has a lognormal distribution with mean  and standard deviation , i.e.,

푦~LN(휇, 휎2), the probability density function may be written (Burnham and Anderson 2002) as

1 ( ln(푦) − 휇)2 pdf(푦|휇, 휎) = exp [− ]. 푦√2휋휎2 2휎2

The negative log-likelihood of observed values 푦푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 푛 1 2 휆 = (푛⁄2)[ln(2휋) + ln(휎2)] + ∑(ln(푦 ) − 휇) + ∑ ln(푦 ). 2휎2 푗 푗 푗=1 푗=1

Bivariate normal distribution

If 푦1 and 푦2 have a bivariate normal distribution with means of 1 and 2, respectively, and standard deviations 휎1 and 휎2, respectively, and with correlation coefficient

2 2 , i.e., (푦1, 푦2)~BVN(휇1, 휇2, 휎1 , 휎2 , 휌), then the probability density function of the joint distribution of these two variables is

1 푧 pdf(푦1, 푦2|휇1, 휇2, 휎1, 휎2, 휌) = exp [− ] 2 2 2 2(1 − 휌2) 2휋√(1 − 휌 )휎1 휎2 where

푧 = (퐴2 − 2휌퐴퐵 + 퐵2),

푦 − 휇 퐴 = 1 1 휎1 and

푦 − 휇 퐵 = 2 2 휎2

The negative log-likelihood of observed pairs of values 푦1푗 and 푦2푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 1 λ = 푛 ln (2휋휎 휎 √1 − 휌2) + ∑[푧 ]. 1 2 2(1 − 휌2) 푗 푗=1

The conditional distribution of y1 given y2 is the normal distribution with:

휎1 2 2 Mean: 휇1 + 휌 (푦2 − 휇2). Variance: 휎1 (1 − 휌 ). 휎2

Bivariate lognormal distribution

If 푦1 and 푦2 have a bivariate lognormal distribution (e.g., Cheng 1986) with means of

휇1 and 휇2, respectively, and standard deviations 휎1 and 휎2, respectively, and with correlation

2 2 coefficient , i.e., (푦1, 푦2)~BVLN(휇1, 휇2, 휎1 , 휎2 , 휌), then the probability density function of the joint distribution of these two variables is

1 푧 pdf(푦1, 푦2|휇1, 휇2, 휎1, 휎2, 휌) = exp [− ] 2 2 2 2(1 − 휌2) 2휋푦1푦2√(1 − 휌 )휎1 휎2 where

푧 = (퐴2 − 2휌퐴퐵 + 퐵2),

ln(푦 ) − 휇 퐴 = 1 1 휎1

and

ln(푦 ) − 휇 퐵 = 2 2. 휎2

The negative log-likelihood of observed pairs of values 푦1푗 and 푦2푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 푛 푛 1 λ = 푛 ln (2휋휎 휎 √1 − 휌2) + ∑[푧 ] + ∑ ln(푦 ) + ∑ ln(푦 ). 1 2 2(1 − 휌2) 푗 1푗 2푗 푗=1 푗=1 푗=1

The conditional distribution of 푦1, given 푦2, is the lognormal distribution

휎1 2 2 푦1~LN (휇1 + 휌 (ln(푦2) − 휇2), 휎1 (1 − 휌 )), 휎2 where the probability density function of this distribution is

pdf(푦1|푦2, 휇1, 휇2, 휎1, 휎2, 휌)

2 휎1 [ln(푦1) − [휇1 + 휌 (ln(푦2) − 휇2)]] 1 1 1 휎2 = exp − . 2 푦 2 휎2(1 − 휌2) √2휋(1 − 휌 )휎1 1 1 [ ]

Bivariate normal-lognormal distribution

If 푦1 and 푦2 have a bivariate normal-lognormal distribution, where 푦1 is normally distributed with mean of 휇1 and standard deviations 휎1 and 푦2 has a lognormal distribution with mean of 휇2 and standard deviations 휎2, and with correlation coefficient between 푦1 and

2 2 푙푛(푦2) of , i.e., (푦1, 푦2)~푁퐿푁(휇1, 휇2, 휎1 , 휎2 , ), then, as advised by Chen and Holtby

(2002), the probability density function of the joint distribution of these two variables is

1 푧 pdf(푦1, 푦2|휇1, 휇2, 휎1, 휎2, 휌) = exp [− ] 2 2 2 2(1 − 휌2) 2휋푦2√(1 − 휌 )휎1 휎2

where

푧 = (퐴2 − 2휌퐴퐵 + 퐵2),

푦 − 휇 퐴 = 1 1 휎1 and

ln(푦 ) − 휇 퐵 = 2 2. 휎2

The negative log-likelihood of observed pairs of values 푦1푗 and 푦2푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 푛 1 λ = 푛 ln (2휋휎 휎 √1 − 휌2) + ∑[푧 ] + ∑ ln(푦 ). 1 2 2(1 − 휌2) 푗 2푗 푗=1 푗=1

The negative log-likelihood of the bivariate lognormal-normal distribution may be obtained from the above equations for the bivariate normal-lognormal distribution by reversing the order of the two variables.

The conditional distribution of 푦1, given 푦2, is the normal distribution with:

휎1 2 2 Mean: 휇1 + 휌 (ln(푦2) − 휇2). Variance: 휎1 (1 − 휌 ). 휎2

The conditional distribution of 푦2, given 푦1, is the lognormal distribution

휎2 2 2 푦2~LN(휇2 + 휌 (푦1 − 휇1), 휎2 (1 − 휌 )), 휎1 where the probability density function of this lognormal distribution is:

pdf(푦2|푦1, 휇1, 휇2, 휎1, 휎2, 휌) 2 휎2 [ln(푦2) − [휇2 + 휌 (푦1 − 휇1)]] 1 1 1 휎1 = exp − . 2 푦 2 휎2(1 − 휌2) √2휋(1 − 휌 )휎2 2 2 [ ]

To aid future publication, material described in this section, which is directly related and pertinent to subsequent Chapters in this study, will be reported and discussed in greater detail within those Chapters. Details of methods, which relate specifically to the approach used within a specific element of the study, are reported in the Chapter dealing with that aspect of the thesis.

CHAPTER 3 – AGE AND GROWTH RATE VARIATION INFLUENCE THE

FUNCTIONAL RELATIONSHIP BETWEEN SOMATIC AND OTOLITH SIZE

The material in this chapter has been accepted for publication as: Ashworth, E. C., N. G. Hall, S. A.

Hesp, P. G. Coulson, and I. C. Potter. Age and growth rate variation influence the functional

relationship between somatic and otolith size. Can. J. Fish. Aquat. Sci. In press.

Abstract

Curves describing the length-otolith size relationships for juveniles and adults of six fish species with widely differing biological characteristics were fitted simultaneously to fish length and otolith size at age, assuming that deviations from those curves are correlated rather than independent. The trajectories of the somatic and otolith growth curves throughout life, which reflect changing ratios of somatic to otolith growth rates, varied markedly among species and resulted in differing trends in the relationships formed between fish and otolith size. Correlations between deviations from predicted values were always positive.

Dependence of length on otolith growth rate (i.e., ‘growth effect’) and ‘correlated errors in variables’ introduce bias into parameter estimates obtained from regressions describing the allometric relationships between fish lengths and otolith sizes. The approach taken in this study to describe somatic and otolith growth accounted for both of these effects and that of age to produce more reliable determinations of the length-otolith size relationships used for back-calculation and assumed when drawing inferences from sclerochronological studies.

3.1 INTRODUCTION

Data derived from the analysis of biological and environmental records preserved within the microstructure and chemistry of otoliths play a crucial role in the assessment and management of fish stocks and in studies of their biology (e.g., Campana 1999, 2005; Begg et

al. 2005). Thus, for example, since their inception, both the Alaska Fisheries Science Centre

(AFSC) and the Centre for Environment, Fisheries and Aquaculture Science in the United

Kingdom (CEFAS) have accumulated data for over a million otoliths (CEFAS 2014; Burke

Museum 2015), and large numbers of these hard structures continue to be collected and processed each year by fisheries agencies worldwide to produce data for stock assessments

(Campana and Thorrold 2001; Campana 2005). Studies of the effects of changes to fish habitat and environment on fish are of growing importance, particularly in view of the need to understand the implications of climate change. Because otoliths store information on interannual growth, these hard structures are being employed to explore the effects of variation in environmental factors on otolith growth (e.g., Pilling et al. 2007; Morrongiello et al. 2011; Coulson et al. 2014) and thereby draw inferences about how certain environmental factors influence somatic growth. Despite the value of the data derived from otoliths in such studies, however, it is somewhat surprising that the relationship between somatic and otolith growth remains poorly understood (Xiao 1996; Neuman et al. 2001; Fey and Hare 2012).

Otoliths are calcified structures used by fish for balance, hearing and orientation (e.g.,

Campana and Neilson 1985; Campana 1999; Popper et al. 2005). The growth of these structures, which vary widely in shape and size, is controlled by the combination of two processes: 1) the formation of an organic matrix, regulated by metabolic influences and, thus, to some extent linked to somatic growth; and 2) a physicochemical process, i.e., calcification

(e.g., Gauldie and Nelson 1990a; Mugiya and Tanaka 1992; Campana 1999). Consequently, if fish growth becomes negligible (e.g., hampered by starvation), otoliths continue to grow due to the physiochemical and obligatory microincrementation processes involved in the daily physiological cycle (e.g., Mosegaard et al. 1988; Gauldie and Nelson 1990b; Morales-

Nin 2000). These physiochemical processes also produce an ‘age effect’ in the relationship between fish length and otolith size, where reduction in somatic growth as fish age is not

accompanied by a proportional change in otolith growth (e.g., Wright et al. 1990; Morita and

Matsuishi 2001; Vigliola and Meekan 2009).

Variation in rates of somatic growth for individuals of a species, i.e., a ‘growth rate effect’, also influences otolith size and thus the relationship between fish length and otolith size (Templeman and Squires 1956; Campana 1990; Somarakis et al. 1997). For example,

Secor and Dean (1989) found that larval and juvenile striped bass (Morone saxatilis) of a given length from a pond in which slow somatic growth had been recorded had larger otoliths than faster-growing fish of the same size from a second pond in which faster somatic growth had been recorded. In another study, Hare and Cowen (1995) found that larval and juvenile bluefish (Pomatomus saltatrix) that were larger than the expected mean length at age possessed otoliths larger than the size expected for fish of that age, and vice versa. These and other studies (e.g., Strelcheck et al. 2003; Munday et al. 2004; Takasuka et al. 2008) indicate that, because otolith growth is partially linked to metabolic processes, a positive, species- dependent correlation is likely to exist between the deviations of length and otolith radius from their expected sizes for fish of the same age, i.e., faster-growing fish of a given age will have faster-growing otoliths. It appears likely that this correlation will be less for species where the lengths at age of individuals approach their maximum sizes at relatively early ages, than for species that continue to grow throughout life.

For a given population, the form of the length-otolith size relationship is a function of the somatic and otolith growth curves (Hare and Cowen 1995). The forms of those growth curves thus determine the manner in which the ratios of fish length to otolith radius, and rate of change of fish length to rate of change of otolith radius, vary with age. Because of the association of otolith growth with physiochemical as well as metabolic processes, growth of the otolith continues despite the decline in somatic growth as fish age, i.e., otolith growth becomes increasingly decoupled from somatic growth. As a consequence, the relative rates of

predicted increase in fish length and otolith radius would be expected to decline with age, and, ultimately, given sufficient longevity, approach asymptotes of zero.

The overall objective of this study was to test the above hypotheses using data for juveniles and adults of six fish species, which belong to different families, possess widely varying biological characteristics, and occupy a range of environments. Thus, for each species, a model was fitted to describe somatic and otolith growth and to determine the bivariate distribution of the deviations of the lengths and otolith radii at age from those curves. For each species, the correlation of the resulting bivariate distribution of deviations was then tested to determine whether, as hypothesised, it was positive and significantly greater than zero. Trends in the predicted relative rates of somatic and otolith growth with age were examined to determine whether, as expected, these declined with age. The implications of the findings for back-calculation and sclerochronological studies are discussed.

3.2 MATERIALS AND METHODS

3.2.1 THE SIX SELECTED SPECIES

The six species selected for this study were Black Bream, Acanthopagrus butcheri

(Munro 1949); Mulloway, Argyrosomus japonicus (Temminck and Schlegel 1843); Foxfish

Bodianus frenchii (Klunzinger 1880); Breaksea Cod Epinephelides armatus (Castelnau

1875); Goldspotted Rockcod Epinephelus coioides (Hamilton-Buchanan 1822) and West

Australian Dhufish Glaucosoma hebraicum Richardson, 1845. Maximum ages and lengths for these temperate and subtropical species ranged from ~ 19 to 78 years and from ~ 480 to

2 000 mm, respectively (for further details of these species, see Table S3.1).

The data for A. butcheri were derived from samples collected during the present study

(Appendix S3.1), whereas those for each of the other species were derived using samples

collected in previous studies in Western Australia by staff and research students at Murdoch

University. The methods employed and locations from which individuals of each species were collected are listed in Table S3.2. Details of the sampling regimes for A. japonicus,

B. frenchii, E. armatus, E. coioides, and G. hebraicum are provided in the associated references describing those respective studies (Table S3.2).

This study was conducted in accordance with conditions in permit R2561/13 issued by the Murdoch University Animal Ethics Committee.

3.2.2 FISH PROCESSING AND OTOLITH MEASUREMENTS

The total length (TL) of each A. butcheri was measured to the nearest 1 mm and its two sagittal otoliths removed and stored. The left sagittal otolith of each of 50 randomly selected individuals was embedded in clear epoxy resin and, using an Isomet® low-speed saw

(Buehler Ltd., Lake Bluff, Illinois), cut transversely into ~ 0.3 mm sections through its primordium and perpendicular to the sulcus acusticus. The sections were cleaned, polished with wet and dry carborundum paper (grade 1 200) using tap water, dried, and mounted on microscope slides under coverslips using DePX mounting adhesive. Essentially the same procedure had been employed to prepare the otolith sections of the other five species, although for several species (i.e., A. japonicus, E. coioides, and G. hebraicum), the sections were not polished. For each of these species, 50 otolith sections were randomly selected for the current study, noting that a common sample size was employed to facilitate comparability among species.

Otolith sections for individuals of all species were examined under reflected light against a black background using a high-resolution digital microscope camera (Leica DFC 425) with

5 Mpixel resolution mounted on a dissecting microscope (Leica MZ7.5) with a magnification range of 6.3x to 50x. High-contrast digital images of the sectioned otoliths were analysed

using the computer imaging package Leica Application Suite version 3.6.0 (Leica

Microsystems Ltd 2001).

The opaque zones in each sectioned otolith for A. butcheri were counted independently and on different occasions by E. Ashworth and P. Coulson, without knowledge of the size of the fish. On the few occasions for these 50 A. butcheri when the counts of these two readers disagreed (< 2%), the two readers discussed the basis for the discrepancy and determined a mutually agreed value. As for the other five fish species, the age of each A. butcheri was determined using the number of opaque zones in its otoliths, its date of capture and the assumed birth date (i.e., the approximate mid-point of the spawning season) (Table S3.3), and knowledge of when the new opaque zone becomes delineated from the otolith periphery for that species. The ages assigned in earlier studies to the individuals of the other species were accepted for use in the current study. For all six species, opaque zones have been shown to be formed annually in their otoliths.

For each species, the ‘radius’ of each otolith, i.e., the distance between the primordium and the outer edge of the otolith, was measured on three occasions to the nearest 0.1 µm along a line perpendicular to the opaque zones along the posterior edge of the sulcus of the otolith using digital images of the sectioned otoliths, taken under reflective light. The mean of these three measurements for each otolith was used as the radius of that otolith in subsequent analyses.

3.2.3 ANALYSES

The bivariate model used to describe somatic and otolith growth

Total fish length L and otolith radius R were each assumed to grow in accordance

with a growth curve of the form 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆), which describes size S (either L or R)

as a function of age t, where 푆휏1 and 푆휏2 are the expected sizes at two specified reference ages

휏1 and 휏2, and 푎푆 and 푏푆 are parameters that determine the shape of the curve. The minimum

and maximum ages at capture of each species were used as the reference ages 휏1 and 휏2,

respectively, in this study. Thus, the expected length 퐿̂푗 of fish j at its age at capture 푡푐,푗 was

̂ calculated as 퐿푗 = 푔퐿(푡푐,푗| 퐿휏1, 퐿휏2, 푎퐿, 푏퐿) and the radius of its otolith as

̂ 푅푗 = 푔푅(푡푐,푗| 푅휏1, 푅휏2, 푎푅, 푏푅). The expected size at age t, was calculated using either a

modified version of the von Bertalanffy growth equation that allowed for an oblique linear

asymptote or the versatile growth model described by Schnute (1981). The modified von

Bertalanffy equation is:

푆(푡) = 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆) = 푐{1 − exp[−푎푠(푡 − 푏푠)]} + 푑(푡 − 푏푠),

where

푦 ⁄(휏 − 푏 ) − 푦 ⁄(휏 − 푏 ) 푐 = 2 2 푠 1 1 푠 , {1 − exp[−푎푠(휏2 − 푏푠)]}⁄(휏2 − 푏푠) − {1 − exp[−푎푠(휏1 − 푏푠)]}⁄(휏1 − 푏푠)

and

푦 − 푐{1 − exp[−푎 (휏 − 푏 )]} 푑 = 1 푠 1 푠 . 휏1 − 푏푠

The following four equations comprise the Schnute (1981) model:

−1 ( ) 푏 if 푎 ≠ 0, 푏 ≠ 0 1 − 푒−푎 푡−휏1 푆 푆 [푆푏 + (푆푏 − 푆푏 ) ] 휏1 휏2 휏1 −푎(휏 −휏 ) 1 − 푒 2 1

푆휏2 푡 − 휏1 if 푎푆 = 0, 푏푆 = 0 푆휏1 exp [ln ( ) ( )] 푆휏1 휏2 − 휏1

Sixteen alternative growth models were considered. These comprised the modified von Bertalanffy curve and 15 alternative forms of the Schnute (1981) growth curve, each of which was formed by constraining 푎푆 and 푏푆 to specific values or ranges and thereby ensuring that no discontinuity was present in the derivatives of the negative log-likelihood in the ranges over which, for that model, those two parameters extended. These alternative model forms included many of the growth curves commonly used in fisheries science, e.g., von Bertalanffy, Richards, Gompertz, etc. (Schnute 1981; Appendix S3.2 and Table S3.4).

Deviations of total length L and otolith radius R for fish of age t years about the length

퐿̂ and radius 푅̂ predicted by the growth curves for these variables were assumed to be drawn from one of four candidate bivariate distributions. Thus, for each analysis, for fish j at its age of capture 푡푐,푗, the deviations of its recorded values of length 퐿푗 and otolith radius 푅푗 from the expected values for fish of that age were drawn from either the bivariate normal, normal- lognormal, lognormal-normal, or bivariate lognormal distributions. The form of this distribution determines the forms of the associated marginal distributions for the deviations of lengths and otolith radii from their expected values. That is, the recorded length 퐿푗 and otolith radius 푅푗 of fish j at its age at capture 푡푐,푗 around their respective expected values at that age, i.e., 퐿̂푗 and 푅̂푗, were drawn from either normal distributions, i.e., 퐿푗 = 퐿̂푗 + 휀푗 or

푅푗 = 푅̂푗 + 휂푗, or lognormal distributions, i.e., 퐿푗 = 퐿̂푗exp(휀푗) or 푅푗 = 푅̂푗exp(휂푗), where, in each case, the correlation between values of 휀 and 휂 is 휌.

Specifically, the study considered the following candidate bivariate distributions when fitting growth models to recorded lengths and otolith radii of fish at their different ages of capture:

(1) a bivariate normal distribution, where

2 2 (퐿푗, 푅푗)~BVN(퐿̂푗, 푅̂푗, 휎휀 , 휎휂 , 휌),

(2) a bivariate normal (fish length) – lognormal (otolith radius) distribution, where

2 2 2 (퐿푗, 푅푗)~NLN(퐿̂푗, ln(푅̂푗) − 휎휂 ⁄2 , 휎휀 , 휎휂 , 휌),

(3) a bivariate lognormal (fish length) – normal (otolith radius) distribution, where

2 2 2 (퐿푗, 푅푗)~LNN(ln(퐿̂푗) − 휎휀 ⁄2 , 푅̂푗, 휎휀 , 휎휂 , 휌), or

(4) a bivariate lognormal distribution, where

2 2 2 2 (퐿푗, 푅푗)~BVLN(ln(퐿̂푗) − 휎휀 ⁄2 , ln(푅̂푗) − 휎휂 ⁄2 , 휎휀 , 휎휂 , 휌), and where the probability density functions for these distributions are presented in Appendix

S3.3. As shown above, the means of log-transformed values in the above distributions were assumed to be offset from zero by −휎2⁄2 to ensure that the expected values of

퐿푗 = 퐿̂푗exp(휀푗) or 푅푗 = 푅̂푗exp(휂푗) are 퐿̂푗 or 푅̂푗, respectively.

A phased approach, which entailed first exploring the forms of growth models and statistical distributions that best described somatic and otolith growth, facilitated fitting of the full bivariate model by providing initial estimates of parameters and identifying the structural forms of the two growth models and their associated marginal distributions to be employed in that final model. The Akaike information criterion corrected for small sample sizes (AICc;

Akaike 1974; Hurvich and Tsai 1989; Anderson and Burnham 2002) and Akaike weight

(AW; Burnham and Anderson 2002; Katsanevakis and Maravelias 2008) were calculated to determine the models and statistical distributions that best described the lengths at age and otolith sizes at age. The fitted model that produced the minimum value of AICc, and thus the greatest value of AW, was selected as providing the best description of fish lengths or otolith radii at ages at capture. Models with AICc scores that differ by less than ~ 2 from that of the model with the lowest AICc are strongly supported by the data and provide representations of the data of almost similar quality to that provided by the model with the lowest AICc

(Anderson and Burnham 2002; Okamura and Semba 2009).

Models were developed in Template Model Builder (package ‘TMB’, Kristensen et al. 2015), in combination with the function ‘nlminb’, within R (R Development Core Team

2011). Thus, for each of the four alternative bivariate distributions and for each pair (i.e., somatic and otolith) of the 16 alternative growth models, the parameters that minimised the negative log-likelihood, 휆퐿,푅, of the deviations of the pair of lengths and otolith radii at their ages of capture from their expected values given the bivariate distribution of those deviations were calculated. Note that the equations for the negative log-likelihoods of the bivariate distributions scale the variables appropriately, thereby accounting for their very different magnitudes (Appendix S3.3).

The parameters estimated for each bivariate model were those for the somatic growth

equation, i.e., 퐿휏1, 퐿휏2, 푎퐿, 푏퐿, and 푠휀, those for the otolith growth equation, i.e., 푅휏1, 푅휏2, 푎푅,

푏푅, and 푠휂, and the correlation 휌 of the bivariate distribution. Note also that 푠휀 and 푠휂 are estimates of the standard deviations 휎휀 and 휎휂, respectively, and that an appropriate subset of parameters was estimated when one or more of the shape parameters of the Schnute (1981) model(s) was set to zero or to specific fixed values for particular forms of this curve. Values

2 of the adjusted coefficient of determination, 푅adjusted, were calculated for the fits provided to lengths and otolith sizes at age by the somatic and otolith growth curves, respectively, of the fitted bivariate growth model.

To provide further information on the form of the model (and associated statistical distribution of deviations from that model) that best described the lengths and otolith radii at age for each species, a bootstrapping analysis was undertaken by resampling 4 000 random data sets, with replacement, from the observed data for each species. For each random data set, the bivariate model was fitted and the resulting parameter estimates were stored. The proportions of bootstrap results that resulted in the selection of each model form were calculated.

Correlation between deviations from somatic and otolith growth

The proportion of the 4 000 bootstrap estimates of correlation that were less than or equal to zero for each species was calculated. A one-tailed t-test, calculated using the estimated standard error (SE) produced as output by TMB when fitting the model, was used to determine whether the correlation for each species was significantly greater than zero (i.e.,

P < 0.05).

Relationship between expected length and expected otolith radius

The expected lengths and otolith radii were calculated for ages extending over the observed range of ages for each species. The relationship formed by plotting the resulting expected lengths at age against associated otolith radii at age for each species was compared with the recorded fish lengths and otolith radii. To aid this exploration, instantaneous rates of somatic and otolith growth at each age within the age range for each species were calculated from the fitted curves using a forward difference approximation of the derivative (e.g.,

Hoffman 2001). To adjust for the different magnitudes of the two variables, instantaneous relative rates of somatic and otolith growth at each age were then calculated by dividing the instantaneous rates by predicted length and otolith radius at age respectively.

3.3 RESULTS

3.3.1 FORMS OF GROWTH CURVES PROVIDING BEST DESCRIPTIONS OF LENGTH AT

AGE

The curves that best described the lengths at ages of capture of the sampled fish in the bivariate growth models fitted to the length and otolith data for the different species varied considerably (Table 3.1; Fig. 3.1). The somatic growth curves identified as best representing

lengths at age included the Pütter number 2 growth curve for A. butcheri (Table S3.5), the traditional von Bertalanffy curve for both A. japonicus and E. coioides, the modified von

Bertalanffy curve (i.e., with oblique linear asymptote) for B. frenchii, a logistic model for

E. armatus, and a Schnute (1981) model with parameters 푎 = 0 and 푏 > 0 for G. hebraicum.

For five of the six species, the same somatic growth curves were identified using AICc and the bootstrapping procedure (Table 3.2). In the case of A. butcheri, however, the traditional von Bertalanffy model was most frequently (36% of samples) identified as the most appropriate somatic growth curve. It should be noted that, for A. japonicus, the modified von

Bertalanffy curve provided the best representation of lengths at age of the bootstrapped samples almost as frequently as the traditional von Bertalanffy curve, differing by only 0.3%.

For four of the six species (i.e., except E. armatus and G. hebraicum), the AICc scores calculated for several alternative somatic growth curves when fitting the bivariate growth model to the observed length and otolith data were only slightly greater (differing by

< 2 units) than the AICc of the selected model (Table S3.6). For A. butcheri, the AICc scores of the traditional von Bertalanffy, Gompertz curve and logistic curves, all with normally distributed deviations, lay close to and within 2 units of that of the Pütter number 2 curve

(Table S3.6). For A. japonicus and E. coioides, the AICc scores of the Pütter number 2 curve approached those (and within two units) of the traditional von Bertalanffy curves. The modified von Bertalanffy curve, with the lognormal distribution, provided a description of the lengths at age of B. frenchii that was of similar quality (and with an AICc that differed by

< 2 units) to that produced by the same curve with normally distributed deviations. The quantitatively similar lengths at age predicted for the different species by those somatic growth curves with AICc scores exceeding those of the best models by < 2 units differed only slightly from the values calculated using those latter models, with differences most evident at the ends of the age ranges or, within those ranges, where data were sparse (Fig. S3.1).

3.3.2 FORMS OF GROWTH CURVES PROVIDING BEST DESCRIPTIONS OF OTOLITH

RADIUS AT AGE

The curves that were identified as best describing otolith growth when fitting the bivariate growth model showed greater consistency among the different species than those that described somatic growth (Table 3.1). Schnute (1981) growth curves with parameters a = 0 and b > 0, which have no inflection point or finite asymptote, best represented the otolith sizes at age for A. butcheri, A. japonicus, and B. frenchii (Table 3.1; Fig. 3.2; Tables

S3.7 and S3.8). The traditional von Bertalanffy growth curve best represented otolith sizes within the age range of the sample for both E. armatus and E. coioides. For the former species, the rate of growth slowed markedly by ~ 7 years of age, whereas in the case of

E. coioides, the radii of its otoliths continued to increase in size without slowing markedly as they approached the upper end of their age range. Finally, otolith sizes at age for

G. hebraicum were best described by the modified von Bertalanffy model, where, following rapid growth until ~ 4 years of age, the radii continued to increase, approximately linearly, with age over the remainder of the age range of the sample. The same growth curves were identified using AICc and the bootstrapping procedure (Table 3.2).

For B. frenchii, the modified von Bertalanffy and a Schnute (1981) curve with a < 0 and b > 0 differed by less than 2 units from the AICc best model (Table S3.8). The AICc scores of the Pütter number 2, the Gompertz, a Schnute (1981) curve with a = 0 and b > 0, the logistic, and the modified von Bertalanffy curves describing otolith size at age of E. armatus were similar to the traditional von Bertalanffy curve identified as best representing those otolith sizes at age. For E. coioides, the AICc scores of the Schnute (1981) curve with a = 0 and b > 0, the modified and the generalised von Bertalanffy curves were within two units of the traditional von Bertalanffy curve. A Schnute (1981) model with parameters a < 0 and

b > 0 provided another alternative model with a quantitatively similar description of otolith growth for G. hebraicum. As with the somatic growth curves, differences for each species between the otolith radii at age predicted by the models with AICc scores exceeding that of the best model by < 2 units and the values predicted by that latter model differed only slightly and mainly at the ends of the age range or, within that range, where data were sparse (Fig.

S3.2).

3.3.3 DISTRIBUTION OF LENGTHS AND OTOLITH RADII AT AGE ABOUT GROWTH

CURVES

The somatic growth curves for A. butcheri, A. japonicus, B. frenchii, and E. coioides that employed a normal rather than lognormal marginal distribution provided the best description of the deviations of observed lengths at age from the values predicted by the growth curve (Table 3.1). For both E. armatus and G. hebraicum, however, lognormal distributions were the best.

Lognormal distributions provided the most appropriate descriptions of the deviations from the otolith growth curves for five of the six species, with the exception being

E. coioides, for which the marginal distribution of the deviations was best represented by the normal distribution (Table 3.1).

3.3.4 DESCRIPTIONS OF SOMATIC GROWTH PROVIDED BY THE FITTED BIVARIATE

GROWTH MODELS

The quality of the fit to lengths at age provided by the bivariate growth model was

2 high for five of the six species (i.e., 푅adjusted > 0.88), with E. armatus being the exception, for

2 which 푅adjusted was 0.79 (Table 3.3; Fig. 3.1; Table S3.9). The lengths at age of individuals of

A. butcheri and A. japonicus predicted by the somatic growth curves of the fitted bivariate

models approached horizontal asymptotes as age increased, whereas those of individuals of

B. frenchii, E. armatus, E. coioides, and G. hebraicum continued to increase throughout the range of observed ages (Fig. 3.1). In contrast to E. armatus, E. coioides and, G. hebraicum, for which the rates at which predicted lengths at age were positive but continued to slow with age, those for B. frenchii approached an oblique linear asymptote. The total lengths at age for younger A. japonicus and B. frenchii, (i.e., fish with ages less than ~ 8 and ~ 10 years, respectively) increased rapidly, but individuals of these species then exhibited much slower growth over the remainder of their lives (Figs 3.1b and c), noting that, in samples used in this study, these two species had the greatest maximum ages, i.e., ~ 30 years for A. japonicus and

~ 61 years for B. frenchii. In the case of E. armatus, the curve describing the lengths at age exhibited a slightly sigmoid shape with an apparent point of inflection at an age of 3.9 years

(Fig. 3.1d).

3.3.5 DESCRIPTIONS OF OTOLITH GROWTH PROVIDED BY THE FITTED BIVARIATE

GROWTH MODELS

2 For the five species for which 푅adjusted of lengths at age were high, similarly high quality fits to otolith radii at age were produced by the fitted bivariate growth models (i.e.,

2 2 푅adjusted > 0.87). As with its somatic growth curve, a poorer fit (푅adjusted = 0.53) of otolith sizes at ages was obtained for E. armatus (Table 3.3; Fig. 3.2d; Table S3.9).

Values of otolith radii predicted using the fitted growth curves for A. butcheri,

A. japonicus, B. frenchii, E. coioides and G. hebraicum continued to increase markedly with ages of capture throughout the age ranges of the different species (Fig. 3.2). For E. armatus, however, the rate of growth of the otolith radii slowed more appreciably as the expected values of the radii appeared to approach an asymptote at a relatively early age within the range of ages, and thus, by age ~ 4 years, the radii had attained 76% of their predicted

asymptotic size. The initial rapid rate of growth of the otolith radii of G. hebraicum, i.e.,

1.20 mm.year-1 at ~ 2 years, was reduced from age ~ 5 years to a much lower but more constant rate of growth, i.e., 0.1 mm.year-1.

3.3.6 CORRELATIONS BETWEEN DEVIATIONS FROM SOMATIC AND OTOLITH

GROWTH CURVES

The maximum likelihood estimates of the correlations of the bivariate distributions of deviations from the fitted somatic and otolith growth curves were positive for all species, although the values of correlations estimated for some bootstrap trials, and particularly those of B. frenchii, fell below zero (Table 3.4). Although results of the one-tailed t-test demonstrated that the point estimates of the correlation were significantly greater than zero for A. butcheri (P < 0.05), E. coioides (P < 0.001), and G. hebraicum (P < 0.001), this was not the case for A. japonicus, B. frenchii and E. armatus (all P > 0.05). The correlations of the bivariate distributions of deviations from lengths and otolith radii at age predicted for the different species by those somatic and otolith growth curves with AICc scores exceeding those of the best models by < 2 units were all positive, and the one-tailed test demonstrated that the point estimates of the correlations for these quantitatively similar bivariate models were significantly greater than zero for E. coioides (P < 0.001) and G. hebraicum (P < 0.001)

(Table S3.10).

3.3.7 RELATIONSHIPS BETWEEN EXPECTED TOTAL FISH LENGTHS AND OTOLITH

SIZES AT AGE

The curves relating the expected values of total length to otolith radius for five of the six species, i.e., A. butcheri, A. japonicus, B. frenchii, E. coioides, and G. hebraicum, appeared to match the trends exhibited by the observed total lengths and otolith radii at

capture (Fig. 3.3). The curve formed by the predicted values for E. armatus provided a poor representation of the rather unusual ‘sigmoid’ pattern exhibited by the observed lengths and otolith radii at age of the individuals for this species (Fig. 3.3d).

The expected lengths at age of A. japonicus increased approximately linearly with the associated expected sizes of their otoliths for smaller otoliths but then approached a well- defined asymptotic length as otolith sizes continued to increase (Fig. 3.3b). A similar slowing of the rate of increase in expected length relative to expected otolith radius as the latter increased towards their maxima was apparent for both A. butcheri and G. hebraicum, but the lengths at age of the latter species had not approached an asymptote (Figs 3.3a and f). There appeared to be no similar decline in the rates at which predicted lengths at age increased relative to expected otolith sizes as the latter increased towards their maxima for B. frenchii,

E. armatus, and E. coioides (Figs 3.3c, d, and e). The relationship between predicted length and otolith size at age for G. hebraicum displayed an obvious point of inflection at an otolith radius of ~ 2 mm (Fig. 3.3f). Similar but less obvious inflection appeared present in the length-otolith radius relationships for A. butcheri and A. japonicus (Figs 3.3a and b).

Throughout the ranges of their otolith sizes, the relationships between expected length and associated expected otolith size for both E. armatus and E. coioides displayed increasing trends (Figs 3.3d and e).

The alternative curves for each species, which related the predicted lengths and otolith radii at the same age for different combinations of the somatic and otolith growth curves that lay within 2 AICc units of the AICc of the best fitting curves, were quantitatively similar to that produced by the fitted bivariate growth model, with slight differences at the ends of the ranges of lengths and otolith radii or, within those ranges, where data were sparse (Fig. S3.3).

For all six species, predicted relative instantaneous rates of somatic and otolith growth declined with age towards zero (Fig. 3.4). In the case of A. butcheri, the relative rate of otolith growth was initially greater than that of somatic growth (Fig. 3.4a). It subsequently declined slightly more rapidly, being overtaken at an age of ~ 6 years by the declining relative rate of somatic growth. The relative rate of otolith growth slowed to become almost constant, but non-zero, at older ages within the range of observed ages, while that of somatic growth appeared to approach an asymptote of zero. The initial relative rate of otolith growth for A. japonicus was also greater than that of somatic growth, but both exhibited a very rapid and similar marked decline towards much reduced levels (Fig. 3.4b). For the remainder of the observed age range, the relative rates of growth for this species remained approximately constant with that for otolith growth remaining at a slightly greater level than that for somatic growth, which became close to zero for ages greater than ~ 10 years. For B. frenchii, the initial relative rate of somatic growth greatly exceeded that of otolith growth, but declined rapidly to fall below the latter at an age of ~ 5 years, approaching an apparent asymptote of zero (Fig. 3.4c). From the age when the relative otolith growth rate overtook the relative rate of somatic growth, it remained above but gradually declined towards that latter growth rate through subsequent ages of the observed age range.

For E. armatus, the initial relative rate of otolith growth was greater than that of somatic growth and, within the observed age range, declined far more rapidly than the latter towards an apparent asymptote of zero (Fig. 3.4d). Although differing in magnitude and rate, this decline had a similar form to that of the other five species. In contrast, however, the instantaneous relative rate of somatic growth of this species declined approximately linearly until an age of ~ 8 years, before the rate of decline began to slow as ages approached their maximum. The relative rate of somatic growth of E. armatus exceeded that for otolith growth over virtually all of its observed age range.

As with B. frenchii, the initial relative rate of somatic growth for E. coioides greatly exceeded that of its otolith growth (Fig. 3.4e). Both growth rates declined towards an apparent asymptote of zero with age, with the somatic growth rate remaining greater than the otolith growth rate throughout the entire observed age range. For G. hebraicum, the relative rate of otolith growth declined rapidly from its initial level, which was only slightly less than that of somatic growth, to become markedly less than that rate of growth by the age of

~ 3 years (Fig. 3.4f). Both curves appeared to decline towards a zero asymptote.

3.4 DISCUSSION

The inclusion of age improves the precision of the estimated relationship between fish length and otolith size for back-calculation (e.g., Francis 1990; Kielbassa et al. 2011). Thus, for example, Sirois et al. (1998) extended the ‘biological intercept’ method of calculation proposed by Campana (1990) to include age, while Morita and Matsuishi (2001) modified the linear relationship between otolith size and fish length to incorporate this variable. The latter model was further extended by Finstad (2003), who added an interaction term. Xiao (1996) coupled somatic growth curves (von Bertalanffy, logistic, and Gompertz) with an allometric relationship between otolith size and fish length, assuming constant allometric parameters, to form equations relating otolith size to fish length. To the best of my knowledge, however, the current study is the first to fit growth curves to lengths and otolith radii at ages of capture for juveniles and adults of different species, to explore the forms of those curves that best describe the otolith data, and to use predictions from those fitted curves over the range of observed ages to relate expected fish length at any given age to expected otolith radius at that age. It is noted, however, that Hare and Cowen (1995) fitted first- to fourth-order polynomials to explore relationships between length, otolith radius, and age for larvae and juveniles of Pomatomus saltatrix.

The inclusion of age as an explanatory variable in the equation relating fish length to otolith radius accounts directly for the ‘age effect’, in which otolith size continues to increase when somatic growth is little or null (e.g., Mugiya 1990; Secor and Dean 1992; Morita and

Matsuishi 2001). By recognising the bivariate distribution of the deviations of fish lengths and otolith radii from the expected lengths and radii of the individual fish at their ages at capture, the bivariate model has also taken the ‘growth effect’ into account. This effect, which results from the relationship between the size of the otolith of an individual fish and its rate of somatic growth, is exemplified by the fact that slow-growing fish have larger otoliths than faster-growing fish of the same size (e.g., Reznick et al. 1989; Campana 1990).

Correlation between deviations from the somatic and otolith growth curves converts an

‘errors in variables’ issue, commonly encountered in allometric relationships such as that between length and otolith radius (Laws and Archie 1981; Xiao 1996; Katsanevakis et al.

2007), into a ‘correlated errors in variables’ problem, adding to the statistical issues encountered when fitting equations to describe the relationship (Fuller 1980; Thoresen and

Laake 2007). The bivariate growth model developed for this study takes such individual variation and potential for correlation of deviations into account, thereby directly addressing this issue. It is important to note that assumptions made regarding the error structure of the model affect the parameters of the fitted growth curves. The approach, which has been used in this study, is not unique and alternative approaches should also be considered and compared in future studies.

3.4.1 GROWTH

The analyses demonstrated that the bivariate growth model provided good fits

2 (푅adjusted > 0.87 ) to the lengths and otolith sizes at age of the individuals of five of the six teleost species considered in this study. As these species had different biological

characteristics, it was expected that the functional forms of the growth curves and the error distributions would differ among the species.

3.4.2 SOMATIC GROWTH

The von Bertalanffy growth curve had been employed in the earlier studies of

A. japonicus, B. frenchii, E. armatus, E. coioides, and G. hebraicum, from which the samples for the current study were drawn, and in earlier studies for A. butcheri (for references to previous studies, see Table S3.2). In the current study, however, only the lengths at age for

A. japonicus and E. coioides were described best by such a curve.

In theory, because energy is directed towards reproduction and cell maintenance rather than somatic growth as fish approach maturity and continue to age, length is expected to approach an asymptote as age increases (Charnov et al. 2001; West et al. 2001; Lester et al.

2004). While the forms of Schnute (1981) growth curves that best described lengths at age for four of the six species in this study possessed a finite (horizontal) asymptote, the fitted somatic growth curves for B. frenchii (i.e., a modified von Bertalanffy curve) and

G. hebraicum (i.e., a Schnute (1981) curve with a = 0 and b > 0) did not. It is possible that for these two species, the samples possessed few individuals of sufficient age to facilitate selection of a form of model that included a finite asymptote. For example, the age range for

G. hebraicum was 0.7 to 24.2 years in this study, yet its maximum recorded age is 41 years

(Hesp et al. 2002). Although the ages of B. frenchii employed in this study, i.e., 1.1 to 61.9 years, provided better coverage of the expected age range (noting that the maximum age of this species is 78 years), few fish within the sample had ages in excess of 42 years

(Cossington et al. 2010).

For two of the six species, the growth curves identified as best representing lengths at age possessed points of inflection (at 0.69 years for A. butcheri and at 3.9 years for E. armatus).

The ages associated with these points of inflection are only slightly greater than the minimum ages of the fish in the samples. Although it is possible that the fitted curves reflect the acceleration of fish growth at young ages, and the subsequent decline in rate of growth at older ages (Campana and Jones 1992), the paucity of young fish in the samples for these two species may have influenced the forms of the somatic growth curves fitted to the data (e.g.,

Beamesderfer and Rieman 1988; Arreguín-Sánchez 1996; Helidoniotis and Haddon 2013).

The recommendation of Katsanevakis and Maravelias (2008) that, when fitting growth models, it is appropriate to explore the form of curve that best describes the size at age, is supported by the different forms of growth curves that were identified as providing the best representations of the data for the six species considered in the current study.

3.4.3 OTOLITH GROWTH

The analyses showed that the patterns for otolith growth were more consistent among the studied species than those for somatic growth for those species, with otoliths of individuals of all but E. armatus exhibiting continued marked growth over the range of observed ages. This finding is in accordance with the concept that although somatic growth decreases with age, deposition of material on the surface of the otolith continues to occur, with the rate of deposition varying in response to the different exogenous factors (e.g., temperature) involved (e.g., Bradford and Geen 1992; Morales-Nin 2000; Fey 2006). Indeed, for A. butcheri, A. japonicus, B. frenchii and G. hebraicum, the annual increment in otolith radius had become almost constant with age over the range of observed ages. Although the form of the fitted growth curve for E. coioides, i.e., a traditional von Bertalanffy curve, suggested that otolith radius was approaching an asymptote as age increased, the predicted annual increment still remained well above 0 mm.year-1 for fish of the maximum observed

age. In contrast to the other five species, the otolith radii for older individuals of E. armatus appeared to have closely approached their asymptotic size.

There would be value in investigating whether growth in otolith mass or distances along other axes of measurement produce otolith growth curves that, although possibly of different forms, are consistent with those obtained using the axis of measurement employed in this study. In such a future study, there would also be value in collecting samples containing greater numbers of older individuals to assess whether over a broader range of ages, the otoliths of older individuals of the different species continue to grow, although possibly not in the dimension considered in this study (e.g., Secor and Dean 1989; Francis and Campana 2004).

3.4.4 THE MODIFIED VON BERTALANFFY GROWTH MODEL WITH OBLIQUE LINEAR

ASYMPTOTE

Of the alternative somatic and otolith growth curves that were considered, the modified von Bertalanffy growth model with oblique linear asymptote provided the best representations of only the somatic growth of B. frenchii and the otolith growth of

G. hebraicum, for both of which an initial rapid rate of growth was followed by a much reduced and more constant rate of growth. As noted above, the finding that somatic growth of

B. frenchii was best represented by a modified von Bertalanffy growth curve is possibly explained by the small number of older fish within the observed age range of the sample to which the model was fitted, noting that the longevity of this species is 78 years (Table S3.1).

Although it had been anticipated that the modified von Bertalanffy growth curve would provide the best representation of otolith radii at age for the different species, the flexible form of the Schnute (1981) model proved more capable of describing the data within the samples employed in this study.

The findings that a particular form of growth curve best described the data for a particular species may relate to the range of alternative model forms that were compared, the error structures that were assumed, the sample size to which the models were fitted, and the representativeness of samples to the population from which those samples were drawn. While the forms of the somatic and otolith growth curves fitted to E. armatus (Figs 3.1d and 3.2d) appear sound, the resultant relationship between length and otolith size at capture (Fig. 3.3d) does not appear biologically plausible. This finding may result from the flexibility and independence of the forms of the two growth curves in the fitted bivariate model, where no constraint was placed on the curves to ensure that the relationship between length and otolith size at capture remained feasible. It may also result from the forms of the variation about the two growth curves that were assumed.

3.4.5 DISTRIBUTION OF LENGTHS AND OTOLITH RADII AT AGE ABOUT GROWTH

CURVES

It is typically assumed in studies of somatic growth of fish that deviations about the fitted length-at-age curve are additive and normally distributed with constant variance

(Bowker 1995). In the case of G. hebraicum, however, Hesp et al. (2002) did fit a somatic growth curve that assumed increasing variance with age. This study has demonstrated, however, that for some species, the assumption that deviations are additive and normally distributed with constant variance is invalid. Alternative statistical distributions, such as the lognormal, may provide a better description of sampled lengths at age, as found (by comparison of AICc scores) for E. armatus and G. hebraicum. While the statistical distributions that better described the variation around the otolith radius-at-age curves also varied among species, the lognormal distribution provided a better fit, i.e., lower AICc, than the normal distribution for five of the six species.

In this study, the relationship between fish length and otolith radius at age was that formed by the fish lengths and otolith radii predicted for fish of the same age using the somatic and otolith growth curves of the fitted bivariate model, rather than that formed empirically by observed lengths and ages at capture. This contrasts with the approach typically employed in more recent back-calculation studies where a relationship is fitted directly to lengths, otolith sizes, and ages at capture for a sample of fish (Morita and

Matsuishi 2001; Finstad 2003; Vigliola and Meekan 2009). Such relationships appear to have been fitted without accounting for the ‘growth effect’, which Campana (1990) addressed by introducing the concept of a ‘biological intercept’. The improvement in fit provided by the inclusion in the bivariate growth model of such an intercept, as a constraint on the somatic and otolith growth curves, is the subject of a current investigation.

3.4.6 CORRELATION BETWEEN THE DEVIATIONS OF FISH LENGTHS AND OTOLITH

SIZES AT AGE FROM THE TWO GROWTH CURVES

Although the maximum likelihood estimates of the correlations of the bivariate distributions were positive for all species and significantly greater than zero for A. butcheri,

E. coioides, and G. hebraicum, this was not the case for A. japonicus, B. frenchii, and

E. armatus. It may be pertinent to this finding that, compared with the first three species, the lengths at age of individuals of A. japonicus and B. frenchii exhibited a far more rapid increase then more marked reduction in rate of increase relatively early within the ranges of their observed ages. For the last of these species, it may also be pertinent that deviations about the fitted somatic and otolith growth curves were considerably greater than for the other species.

The finding of a significant positive correlation between the deviations of the lengths and otolith radii of individuals of A. butcheri, E. coioides, and G. hebraicum from their predicted

values parallels the results obtained for larval and juvenile stages of Pomatomus saltatrix by

Hare and Cowen (1995), who, after allowing for age, demonstrated a positive correlation between deviations of fish size at age and otolith size at age. It should be noted, however, that the data considered in the current study comprised a wider age range that included both juveniles and adults of each species.

3.4.7 RELATIONSHIPS BETWEEN EXPECTED TOTAL FISH LENGTHS AND OTOLITH

SIZES AT AGE

The trend exhibited by the relationship between fish length and otolith radius reflects the rates at which the relative instantaneous rates of somatic and otolith growth decline with age. When the relative instantaneous rate of somatic growth lies below the corresponding rate of otolith growth, the rate of increase in fish length slows with increasing otolith radius, and the converse is true when the rate of somatic growth lies above that for otolith growth. If an inflection is apparent in the relationship between fish length and otolith radius (as with

G. hebraicum), it results from the relative instantaneous rate of otolith growth declining from above the corresponding relative rate of somatic growth to below that rate, before then again increasing to above that rate.

Although the predicted relative instantaneous rates of somatic and otolith growth of all species declined with age towards zero, the growth curves that best described the lengths and otolith radii at age for a number of species were of forms that did not have finite asymptotes. It is suggested that for those species, the paucity of older fish within the samples may have reduced the information available to determine reliably the form of the underlying relationship between size and age for older fish.

3.4.8 SUMMARY AND CONCLUSIONS

Numerous studies have recognized that the relationship between observed lengths of fish and the size of their otoliths is influenced by a ‘growth’ effect, which is associated with individual variation in somatic and otolith growth rates (e.g., Secor and Dean 1989; Reznick et al. 1989; Mugiya and Tanaka 1992). Thus, when fitting empirical relationships directly to lengths and otolith radii of fish at capture, as is typical in many back-calculation studies, an

‘errors in variables’ problem is encountered, which, if not taken into account, introduces bias into the parameters of the fitted relationship (Xiao 1996). The ‘errors in variables’ problem is addressed by the bivariate growth model by fitting growth curves to length and otolith size at age, and the correlation likely to exist between these errors is recognised by use of a bivariate statistical distribution of deviations from the respective growth curves. Through this approach, the bivariate growth model developed in the current study may produce a more reliable description of the relationship between fish length and otolith size than is produced by many of the approaches employed in earlier back-calculation studies.

Sclerochronological studies explore interannual variation in the ‘average’ widths of the annual growth increments in otoliths, after accounting for age effects and individual variation, and correlate the resulting otolith biochronologies with other time series of environmental or biological data (e.g., Guyette and Rabeni 1995; Black 2009; Ong et al.

2015). Such studies provide no direct information regarding interannual variation in somatic growth or its relationship with environmental variables, however, and thus, when drawing inferences regarding somatic growth, assume that otolith growth rate is an index of somatic growth rate and may be used as a proxy for that variable (e.g., Xiao 1996; Morrongiello et al.

2011; Black et al. 2013). There would be value, in future studies, in quantifying the extent to which the relationship between fish lengths and otolith sizes varies in response to different

environmental variables, e.g., temperature. This might possibly be accomplished by extending the approach developed in this study by considering, within a mixed effects context, the interannual variation in growth parameters when fitting the somatic and otolith growth curves (e.g., Szalai et al. 2003; He and Bence 2007) and employing data from different systems or time periods and thereby taking the influence of environmental variables into account.

To summarise, this study has confirmed that the form of the relationship between expected fish length and otolith radius at age for juveniles and adults of a species is determined by the growth curves relating those two variables to age and that the effect of age should therefore be included when describing this relationship. A versatile growth curve, such as that of Schnute (1981) or chosen from a suite of alternative growth curves and which allows for continued growth with age, should be employed when describing otolith size at age. The current study has demonstrated that because of individual variation in both somatic and otolith growth rates, there is likely to be a positive correlation between the deviations of the lengths at age and otolith radii at age from their expected values. Because such correlation might exist, somatic and otolith growth curves should be fitted simultaneously, assuming a bivariate distribution of the deviations from the growth curves and exploring alternative forms of marginal distributions, e.g., normal or lognormal.

Acknowledgements

This research was sponsored by Murdoch University and Recfishwest. Gratitude is expressed to Murdoch University for logistical support whilst in the field, Fisheries Research and Development Corp. for financial support for several of the earlier studies, and J.

Williams and D. Yeoh for partaking in the collection of fish (i.e., A. butcheri) for this study. I gratefully acknowledge the advice of two anonymous referees and the associate editor of

Canadian Journal of Fisheries and Aquatic Sciences, whose constructive suggestions greatly improved the content of this paper.

Table 3.1. Types of growth curves (and statistical distributions of errors) that, based on

Akaike Information Criterion corrected for small sample size, best described total lengths and

otolith radii at age of capture for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus

frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. 푎 and 푏

are parameters of the Schnute (1981) growth model. The modified von Bertalanffy curve has

an oblique asymptote. AW = Akaike Weight.

Species Somatic growth Otolith growth

Curve type Error type AW (%) Curve type Error type AW (%)

Acanthopagrus butcheri Pütter number 2 normal 24 푎 = 0 and 푏 > 0 lognormal 48 Argyrosomus japonicus Traditional von normal 38 푎 = 0 and 푏 > 0 lognormal 62 Bertalanffy Bodianus frenchii Modified von normal 47 푎 = 0 and 푏 > 0 lognormal 38 Bertalanffy Epinephelides armatus Logistic lognormal 37 Traditional von lognormal 20 Bertalanffy Epinephelus coioides Traditional von normal 46 Traditional von normal 25 Bertalanffy Bertalanffy Glaucosoma hebraicum 푎 = 0 and 푏 > 0 lognormal 48 Modified von lognormal 48 Bertalanffy

Table 3.2. Types of growth curves (and statistical distributions of errors) that, based on

frequency of occurrence in bootstrap trials, best described total lengths and otolith radii at age

of capture for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. Curve types are

the same as described for Table 3.1. Bootstrap percentage = percentage of 4 000 trials for

which the curve and error types were selected as best describing the bootstrap sample of

lengths and ages at capture.

Species Somatic growth Otolith growth

Bootstrap Bootstrap Curve type Error type percentage Curve type Error type percentage (%) (%)

Acanthopagrus butcheri Traditional von normal 36 푎 = 0 and 푏 > 0 lognormal 81 Bertalanffy Argyrosomus japonicus Traditional von normal 32 푎 = 0 and 푏 > 0 lognormal 85 Bertalanffy Bodianus frenchii Modified von normal 85 푎 = 0 and 푏 > 0 lognormal 60 Bertalanffy Epinephelides armatus Logistic lognormal 74 Traditional von lognormal 33 Bertalanffy Epinephelus coioides Traditional von normal 65 Traditional von normal 41 Bertalanffy Bertalanffy

Glaucosoma hebraicum 푎 = 0 and 푏 > 0 lognormal 67 Modified von lognormal 75 Bertalanffy

Table 3.3. Values of parameters (푎, 푏, 푦1and 푦2) and standard deviations (SD) for Schnute (1981) (normal font) and modified von Bertalanffy

(oblique linear asymptote, bold font) somatic and otolith growth curves of the fitted bivariate models for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. SD = estimated standard deviation of marginal distribution of deviations from the respective growth curve. Curves were the same as those of Table 3.1. 휏1= first reference age

(youngest fish in the sample), 휏2= second reference age (oldest fish in the sample), 푎 and 푏 = parameters of growth model, 푦1 = size at age 휏1,

푦2= size at age 휏2. Standard errors (SE) of the parameters and of the standard deviations are presented in parentheses.

Species Growth type 풂 풃 풚ퟏ 풚ퟐ SD

Acanthopagrus butcheri Somatic 0.19 0.33 98 327 17

(휏1 = 0.62, 휏2 = 19.62) (SE = 0.02) (SE = 6.69) (SE = 7.25) (SE = 1.67) Otolith 0 2.03 0.41 1.65 0.06 (SE = 0.16) (SE = 0.01) (SE = 0.04) (SE = 0.01)

Argyrosomus japonicus Somatic 0.24 1 154 1214 79

(휏1 = 0.22, 휏2 = 31.03) (SE = 0.02) (SE = 39.37) (SE = 27.21) (SE = 7.92) Otolith 0 1.64 0.57 10.21 0.09 (SE = 0.05) (SE = 0.05) (SE = 0.30) (SE = 0.01)

Bodianus frenchii Somatic 0.28 -0.38 90 420 21

(휏1 = 1.13, 휏2 = 61.91) (SE = 0.05) (SE = 0.53) (SE = 15.42) (SE = 12.55) (SE = 2.07) Otolith 0 1.78 0.36 2.09 0.09 (SE = 0.16) (SE = 0.02) (SE = 0.08) (SE = 0.01)

Epinephelides armatus Somatic 0.38 -1 149 482 0.11

(휏1 = 1.71, 휏2 = 13.04) (SE = 0.05) (SE = 9.75) (SE = 21.85) (SE = 0.01) Otolith 0.33 1 0.52 1.06 0.10 (SE = 0.08) (SE = 0.04) (SE = 0.04) (SE = 0.01)

Epinephelus coioides Somatic 0.12 1 70 1059 61

(휏1 = 0.15, 휏2 = 14.70) (SE = 0.02) (SE = 19.55) (SE = 29.36) (SE = 6.13) Otolith 0.15 1 0.35 1.87 0.11 (SE = 0.02) (SE = 0.03) (SE = 0.05) (SE = 0.01)

Glaucosoma hebraicum Somatic 0 1.90 105 992 0.12

(휏1 = 0.71, 휏2 = 24.20) (SE = 0.19) (SE = 7.79) (SE = 43.73) (SE = 0.01) Otolith 1.31 0.22 0.74 3.42 0.09 (SE = 0.38) (SE = 0.16) (SE = 0.04) (SE = 0.12) (SE = 0.01)

Table 3.4. Negative log-likelihood (NLL) for the fitted bivariate growth models for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum, together with correlation 휌 of the bivariate

distribution of deviations from the somatic and otolith growth curves. Standard errors (SE) of

the correlation are presented in parentheses. ‘푃{ > 0}’ = P-value of one-tailed t-test that

 > 0. ‘Prop. of bootstraps > 0’ = proportion of 4 000 bootstrap trials for which the point

estimate of the correlation coefficient exceeded zero.

Species NLL  푷{흆 > ퟎ} Prop. of bootstraps > 0

Acanthopagrus butcheri 141 0.31 0.04 0.97

(SE = 0.13)

Argyrosomus japonicus 299 0.13 0.43 0.89

(SE = 0.14)

Bodianus frenchii 169 0.06 0.73 0.64

(SE = 0.14)

Epinephelides armatus 199 0.22 0.15 0.95

(SE = 0.13)

Epinephelus coioides 228 0.50 0.00 1

(SE =0.11)

Glaucosoma hebraicum 253 0.63 0.00 1

(SE = 0.09)

Figure 3.1. Growth curves fitted to the total lengths (mm) at ages of capture (years) for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum.

Figure 3.2. Growth curves fitted to the otolith radii (mm) at ages of capture (years) for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum.

Figure 3.3. Total lengths (mm) and otolith radii (mm) at capture for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Figure 3.4. Relative instantaneous rates of somatic and otolith growth versus age (years) at capture for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum. Solid line (−) represents somatic growth and dashed line (---) represents otolith growth.

Supplemental materials for Chapter 3

Table S3.1. Maximum ages and total lengths (TL), sexuality, and habitats of Acanthopagrus

butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus

coioides and Glaucosoma hebraicum.

Species Max. age Max. TL Sexuality Habitat References (years) (mm)

Acanthopagrus butcheri 31 530 Gonochorist Temperate estuaries Sarre and Potter (2000) Jenkins et al. (2006) Potter et al. (2008)

Argyrosomus japonicus 31 2000 Gonochorist Coastal marine waters, Farmer et al. (2005) seasonally entering Gomon et al. (2008) estuaries

Bodianus frenchii 78 480 Protogynous Over and around coastal Gomon et al. (2008) hermaphrodite temperate reefs Cossington et al. (2010) Platell et al. (2010)

Epinephelides armatus 19 510 Gonochorist Over and around coastal Moore et al. (2007) temperate reefs Gomon et al. (2008) Platell et al. (2010)

Epinephelus coioides 22 1110 Protogynous Subtropical/tropical Heemstra and Randall (1993) hermaphrodite mangrove nursery Heemstra (1995) habitats and coastal reefs Pember et al. (2005)

Glaucosoma hebraicum 41 1120 Gonochorist Temperate coastal Hesp et al. (2002) marine waters, around Lenanton et al. (2009) reefs Platell et al. (2010)

Appendix S3.1. Sampling regime used for the collection of Acanthopagrus butcheri.

Acanthopagrus butcheri was sampled in the Wellstead Estuary at 34°50’S latitude and

118°60’E longitude on the south coast of Western Australia in May 2013 using seine and gill nets. The seine net was 21.5 m long and comprised two 10 m long wings (6 m of 9 mm mesh and 4 m of 3 mm mesh) and a 1.5 m wide bunt of 3 mm mesh. This net, which was deployed during daylight, fished to a depth of ~ 1.5 m and swept an area of ~ 116 m2. The sunken composite multifilament gill net comprised seven 20 m long panels, each with a height of 2 m and containing a different stretched mesh size, i.e., 51, 63, 76, 89, 102, 115 or 127 mm mesh.

Gill nets were set parallel to the shore at dusk and retrieved ~ 2 to 3 h later. Fish were euthanized in an ice slurry immediately after capture and transported to the laboratory where they were frozen until processed.

Table S3.2. Location and sampling regimes for Acanthopagrus butcheri, Argyrosomus

japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma

hebraicum in estuarine and coastal waters along the western coast of Australia.

Species Location Method References

Acanthopagrus butcheri Wellstead Estuary (34°50’S, 118°60’E) Seine and gill netting Present study

Argyrosomus japonicus Coastal waters between Carnarvon Gill netting Farmer et al. (2005)

(24°53’S, 113°39’E) and Augusta Rod and line angling

(34°19’S, 115°10’E)

Bodianus frenchii Coastal marine waters along the lower Gill netting Cossington et al. (2010)

west coast (between 30°18’S, 115°02’E Rod and line angling

and 32°30’S, 115°42’E) Spear fishing

Epinephelides armatus Coastal marine waters off the lower west Fish traps Moore et al. (2007)

coast of Australia (between 30°18’S, Rod and line angling

115°02’E and 32°30’S, 115°42’E)

(Murray Reef, Rottnest Island)

Epinephelus coioides Kimberley and Pilbara coast (between Fish traps Pember et al. (2005)

16°00’S, 126°00’E and 21°00’S, Rod and line angling

119°00’E) Trawl

Glaucosoma hebraicum Lower west coast of Australia between Rod and line angling Hesp et al. (2002)

Mandurah (32°32’S) and the Houtman Spear fishing

Abroholos (28°35’S) Trawl

Table S3.3. Birth dates assigned to each of Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and Glaucosoma hebraicum in estuarine and coastal waters along the coast of Western Australia.

Species Date of birth References

Acanthopagrus butcheri 1 October Sarre and Potter (2000)

Argyrosomus japonicus 1 July on upper west coast Farmer (2008)

1 December on lower west coast

Bodianus frenchii 1 January Cossington et al. (2010)

Epinephelides armatus 1 February Moore et al. (2007)

Epinephelus coioides 1 January Pember et al. (2005)

Glaucosoma hebraicum 1 February Hesp et al. (2002)

Appendix S3.2. Special cases of the Schnute (1981) growth equation that are equivalent to common growth curves.

Curves of a range of different forms, encapsulated within the Schnute model and fitted to the lengths and otolith radii at ages of capture, were compared to determine the growth curve which provided the best description for each of these variables. The growth curves tested included the following:

The generalised von Bertalanffy growth curve, obtained by setting values of a > 0 and b > 0 in the Schnute model, may be written as

The Richards growth curve, obtained using values of a > 0 and b < 0, is

1 −푝 (2) 푌(푡) = 푦 (1 − 푒푥푝[−푔(푡 − 푡 )]) ∞ 푝 0

The Gompertz growth curve, obtained using values of a > 0 and b = 0, is

푌(푡) = 푦∞푒푥푝(−푒푥푝[−푔(푡 − 푡0)]) (3)

The logistic growth curve, obtained using values of a > 0 and b = −1, is

−1 푌(푡) = 푦∞(1 + 푒푥푝[−푔(푡 − 푡0)]) (4)

The linear growth curve, obtained using values of a = 0 and b = 1, is

푌(푡) = 푔(푡 − 푡0) (5)

1 The quadratic growth curve, obtained using values of a = 0 and b = , with 훽 > 0, is 2

푌(푡) = (훼 + 훽푡)2 (6)

The tth power growth curve, obtained using values of a = 0 and b = 0, with 훼 > 0 and 훽 > 0, is

푌(푡) = 훼훽푡 (7)

The exponential growth curve, obtained using values of a < 0 and b = 1, with 훽 > 0 and

훾 > 0, is

푌(푡) = 훼 + 훽exp(훾푡) (8)

Region Parameter space

1 a > 0 and b > 0

2 a > 0 and b < 0

3 a < 0 and b > 0

4 a < 0 and b < 0

5 a > 0 and b = 0

6 a < 0 and b = 0

7 a = 0 and b > 0

8 a = 0 and b < 0

9 a = 0 and b = 0

Note. Several regions described in this table, and defined by the constraints imposed on the parameters a and b, were the same as those of the specific common growth curves described by Schnute (1981), i.e., the Region 1 curve (above) is equivalent to the generalized von

Bertalanffy curve, the Region 2 curve is equivalent to the Richards growth curve, the Region

5 curve is equivalent to the Gompertz growth curve and the Region 9 curve is equivalent to the tth power curve (Appendix S3.2).

Appendix S3.3. Univariate and bivariate probability density functions

Univariate normal distribution

If 푦 is normally-distributed with mean  and standard deviation , i.e., 푦~N(휇, 휎2), the probability density function may be written as

1 ( 푦 − 휇)2 pdf(푦|휇, 휎) = exp [− ]. √2휋휎2 2휎2

The negative log-likelihood of observed values 푦푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 1 2 휆 = (푛⁄2)[ln(2휋) + ln(휎2)] + ∑(푦 − 휇) . 2휎2 푗 푗=1

Univariate lognormal distribution

If 푦 has a lognormal distribution with mean  and standard deviation , i.e.,

푦~LN(휇, 휎2), the probability density function (Burnham and Anderson 2002) may be written as

1 ( ln(푦) − 휇)2 pdf(푦|휇, 휎) = exp [− ]. 푦√2휋휎2 2휎2

The negative log-likelihood of observed values 푦푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 푛 1 2 휆 = (푛⁄2)[ln(2휋) + ln(휎2)] + ∑(ln(푦 ) − 휇) + ∑ ln(푦 ). 2휎2 푗 푗 푗=1 푗=1

Bivariate normal distribution

If 푦1 and 푦2 have a bivariate normal distribution with means of 1 and 2, respectively, and standard deviations 휎1 and 휎2, respectively, and with correlation coefficient

2 2 , i.e., (푦1, 푦2)~BVN(휇1, 휇2, 휎1 , 휎2 , 휌), then the probability density function of the joint distribution of these two variables is

1 푧 pdf(푦1, 푦2|휇1, 휇2, 휎1, 휎2, 휌) = exp [− ] 2 2 2 2(1 − 휌2) 2휋√(1 − 휌 )휎1 휎2 where

푧 = (퐴2 − 2휌퐴퐵 + 퐵2),

푦 − 휇 퐴 = 1 1 휎1 and

푦 − 휇 퐵 = 2 2 휎2

The negative log-likelihood of observed pairs of values 푦1푗 and 푦2푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 1 λ = 푛 ln (2휋휎 휎 √1 − 휌2) + ∑[푧 ]. 1 2 2(1 − 휌2) 푗 푗=1

The conditional distribution of y1 given y2 is the normal distribution with:

휎1 2 2 Mean: 휇1 + 휌 (푦2 − 휇2). Variance: 휎1 (1 − 휌 ). 휎2

Bivariate lognormal distribution

If 푦1 and 푦2 have a bivariate lognormal distribution (e.g., Cheng 1986) with means of

휇1 and 휇2, respectively, and standard deviations 휎1 and 휎2, respectively, and with correlation

2 2 coefficient , i.e., (푦1, 푦2)~BVLN(휇1, 휇2, 휎1 , 휎2 , 휌), then the probability density function of the joint distribution of these two variables is

1 푧 pdf(푦1, 푦2|휇1, 휇2, 휎1, 휎2, 휌) = exp [− ] 2 2 2 2(1 − 휌2) 2휋푦1푦2√(1 − 휌 )휎1 휎2 where

푧 = (퐴2 − 2휌퐴퐵 + 퐵2),

ln(푦 ) − 휇 퐴 = 1 1 휎1 and

ln(푦 ) − 휇 퐵 = 2 2. 휎2

The negative log-likelihood of observed pairs of values 푦1푗 and 푦2푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 푛 푛 1 λ = 푛 ln (2휋휎 휎 √1 − 휌2) + ∑[푧 ] + ∑ ln(푦 ) + ∑ ln(푦 ). 1 2 2(1 − 휌2) 푗 1푗 2푗 푗=1 푗=1 푗=1

The conditional distribution of 푦1, given 푦2, is the lognormal distribution

휎1 2 2 푦1~LN (휇1 + 휌 (ln(푦2) − 휇2), 휎1 (1 − 휌 )), 휎2 where the probability density function of this distribution is

pdf(푦1|푦2, 휇1, 휇2, 휎1, 휎2, 휌)

2 휎1 [ln(푦1) − [휇1 + 휌 (ln(푦2) − 휇2)]] 1 1 1 휎2 = exp − . 2 푦 2 휎2(1 − 휌2) √2휋(1 − 휌 )휎1 1 1 [ ]

Bivariate normal-lognormal distribution

2 2 푙푛(푦2) of , i.e., (푦1, 푦2)~푁퐿푁(휇1, 휇2, 휎1 , 휎2 , ), then, as advised by Chen and Holtby

(2002), the probability density function of the joint distribution of these two variables is

1 푧 pdf(푦1, 푦2|휇1, 휇2, 휎1, 휎2, 휌) = exp [− ] 2 2 2 2(1 − 휌2) 2휋푦2√(1 − 휌 )휎1 휎2 where

푧 = (퐴2 − 2휌퐴퐵 + 퐵2),

푦 − 휇 퐴 = 1 1 휎1 and

ln(푦 ) − 휇 퐵 = 2 2. 휎2

The negative log-likelihood of observed pairs of values 푦1푗 and 푦2푗, where 푗 = 1, 2, … , 푛, may be calculated as

푛 푛 1 λ = 푛 ln (2휋휎 휎 √1 − 휌2) + ∑[푧 ] + ∑ ln(푦 ). 1 2 2(1 − 휌2) 푗 2푗 푗=1 푗=1

Note that the negative log-likelihood of the bivariate lognormal-normal distribution may be obtained from the above equations for the bivariate normal-lognormal distribution by reversing the order of the two variables.

The conditional distribution of 푦1, given 푦2, is the normal distribution with:

휎1 2 2 Mean: 휇1 + 휌 (ln(푦2) − 휇2). Variance: 휎1 (1 − 휌 ). 휎2

The conditional distribution of 푦2, given 푦1, is the lognormal distribution

휎2 2 2 푦2~LN(휇2 + 휌 (푦1 − 휇1), 휎2 (1 − 휌 )), 휎1 where the probability density function of this lognormal distribution is:

pdf(푦2|푦1, 휇1, 휇2, 휎1, 휎2, 휌) 2 휎2 [ln(푦2) − [휇2 + 휌 (푦1 − 휇1)]] 1 1 1 휎1 = exp − . 2 푦 2 휎2(1 − 휌2) √2휋(1 − 휌 )휎2 2 2 [ ]

Table S3.5. Values of corrected Akaike Information Criteria (AICc) and Akaike Weights

Normal distribution Lognormal distribution

Curve type AICc AW AICc AW

Modified von Bertalanffy 434.70 0.01 442.64 < 0.01

Generalised von Bertalanffy 434.57 0.07 442.61 < 0.01

Traditional von Bertalanffy 432.22 0.23 440.17 < 0.01

Pütter number 2 432.19 0.24 440.57 < 0.01

Gompertz 432.46 0.21 441.25 < 0.01

Richards 434.94 0.06 443.73 < 0.01

Logistic 433.92 0.10 444.44 < 0.01

Linear 489.04 0.00 493.38 0.00

Quadratic 500.87 0.00 515.23 0.00

tth power 509.73 0.00 532.14 0.00

Exponential 491.46 0.00 495.81 0.00

a < 0 and b > 0 441.08 < 0.01 448.27 0.00

a < 0 and b < 0 514.61 0.00 591.94 0.00

a < 0 and b = 0 512.13 0.00 534.55 0.00

a = 0 and b > 0 438.60 0.01 445.79 < 0.01

a = 0 and b < 0 512.10 0.00 534.51 0.00

Note. Bold font is used to identify the growth curve with the lowest AICc.

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Glaucosoma hebraicum.

Species Curve type Distribution type AICc

Acanthopagrus butcheri Pütter number 2 normal 432.19

Traditional von Bertalanffy normal 432.22

Gompertz normal 432.46

Logistic normal 433.92

Argyrosomus japonicus Traditional von Bertalanffy normal 587.93

Pütter number 2 normal 589.12

Bodianus frenchii Modified von Bertalanffy normal 456.11

Modified von Bertalanffy lognormal 456.25

Epinephelides armatus Logistic lognormal 508.08

Epinephelus coioides Traditional von Bertalanffy normal 562.38

Pütter number 2 normal 564.37

Glaucosoma hebraicum a = 0 and b > 0 lognormal 564.78

Table S3.7. Values of corrected Akaike Information Criteria (AICc) and Akaike Weights

Normal distribution Lognormal distribution

Curve type AICc AW AICc AW

Modified von Bertalanffy -115.79 < 0.01 -123.51 0.09

Generalised von Bertalanffy -115.94 < 0.01 -124.48 0.14

Traditional von Bertalanffy -118.26 0.01 -123.87 0.10

Pütter number 2 -117.34 < 0.01 -120.02 0.02

Gompertz -116.62 < 0.01 -117.84 < 0.01

Richards -114.15 < 0.01 -115.37 < 0.01

Logistic -113.80 < 0.01 -111.33 < 0.01

Linear -87.04 0.00 -88.83 0.00

Quadratic -67.23 0.00 -57.83 0.00

tth power -52.15 0.00 -32.88 0.00

Exponential -84.60 0.00 -86.39 0.00

a < 0 and b > 0 -115.45 < 0.01 -124.56 0.15

a < 0 and b < 0 -47.25 0.00 47.22 0.00

a < 0 and b = 0 -49.73 0.00 -30.46 0.00

a = 0 and b > 0 -117.93 0.01 -126.95 0.48

a = 0 and b < 0 -49.78 0.00 -30.51 0.00

Note. Bold font is used to identify the growth curve with the lowest AICc.

Table S3.8. Quantitatively similar otolith growth curves, i.e., curves with values of corrected

Species Curve type Distribution type AICc

Acanthopagrus butcheri a = 0 and b > 0 lognormal -126.95

Argyrosomus japonicus a = 0 and b > 0 lognormal 29.49

Bodianus frenchii a = 0 and b > 0 lognormal -97.07

Modified von Bertalanffy lognormal -96.35

a < 0 and b > 0 lognormal -95.82

Epinephelides armatus Traditional von Bertalanffy lognormal -91.97

Pütter number 2 lognormal -91.77

Gompertz lognormal -91.64

a = 0 and b > 0 lognormal -91.24

Logistic lognormal -91.21

Modified von Bertalanffy lognormal -90.63

Epinephelus coioides Traditional von Bertalanffy normal -74.41

a = 0 and b > 0 normal -74.31

Modified von Bertalanffy normal -74.20

Generalised von Bertalanffy normal -73.77

Glaucosoma hebraicum Modified von Bertalanffy lognormal -13.15

a < 0 and b > 0 lognormal -11.36

2 Table S3.9. Values of adjusted coefficients of determination (푅adjusted) for the curves for total lengths and otolith radii at age of the fitted bivariate growth model for each of

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

2 푅adjusted Species Total length Otolith radius

Acanthopagrus butcheri 0.92 0.94

Argyrosomus japonicus 0.92 0.96

Bodianus frenchii 0.91 0.94

Epinephelides armatus 0.79 0.53

Epinephelus coioides 0.95 0.94

Glaucosoma hebraicum 0.88 0.87

Table S3.10. Ranges of negative log-likelihoods (NLL) for the bivariate growth models produced using the quantitatively similar somatic and otolith growth curves for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum, together with ranges of both the correlations (휌) between deviations from the growth curves for those bivariate models and the

P-values, 푃{ > 0}, of one-tailed t-tests that those correlations exceed zero. Quantitatively similar curves were those with AICc scores lying within 2 units of the lowest score, i.e., the

AICc of the fitted curve that best described the data.

Species NLL  푷{흆 > ퟎ}

Acanthopagrus butcheri 141-142 0.31-0.31 0.04-0.05

Argyrosomus japonicus 299-300 0.12-0.13 0.42-0.45

Bodianus frenchii 168-169 0.04-0.14 0.38-0.83

Epinephelides armatus 198-199 0.21-0.24 0.12-0.17

Epinephelus coioides 226-229 0.50-0.54 0.00-0.00

Glaucosoma hebraicum 253-255 0.61-0.63 0.00-0.00

Figure S3.1. Quantitatively similar somatic growth curves, i.e., curves with AICc scores lying within 2 units of the lowest score (the AICc of the best curve, in black), for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides and Glaucosoma hebraicum. Pütter = Pütter number 2, VB = traditional von Bertalanffy, ‘a=0, b>0’ = Schnute (1981) growth curve with a = 0 and b > 0,

ModVB = modified von Bertalanffy, ND = normal distribution, LND = lognormal distribution.

Bertalanffy, GenVB = generalised von Bertalanffy, ‘a=0, b>0’ = Schnute (1981) growth curve with a = 0 and b > 0, ‘a<0, b>0’ = Schnute (1981) growth curve with a < 0 and b > 0,

ModVB = modified von Bertalanffy.

Figure S3.3. Relationships between total lengths and otolith radii at age formed by quantitatively similar somatic and otolith growth curves for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides and

Glaucosoma hebraicum. Black = best fitting somatic and otolith growth curves.

Pütter = Pütter number 2, VB = traditional von Bertalanffy, GenVB = generalised von

Bertalanffy, ‘a=0, b>0’ = Schnute (1981) growth curve with a = 0 and b > 0,

‘a<0, b>0’ = Schnute (1981) growth curve with a < 0 and b > 0, ModVB = modified von

Bertalanffy, ND = normal distribution, LND = lognormal distribution.

Quantitatively similar = AICc scores lying within 2 units of the lowest score.

CHAPTER 4 – A NEW PROPORTIONALITY-BASED BACK-

CALCULATION APPROACH, WHICH EMPLOYS TRADITIONAL FORMS

OF GROWTH EQUATIONS, IMPROVES ESTIMATES OF LENGTH AT AGE

The material in this chapter has been accepted for publication as: Ashworth, E. C., S. A. Hesp, and N.

G. Hall. A new proportionality-based back-calculation approach, which employs traditional forms of

growth equations, improves estimates of length at age. Can. J. Fish. Aquat. Sci. In press.

Abstract

The performance of a new proportionality-based back-calculation approach, describing the relationship between length, otolith size and age using traditional growth curves and assuming a bivariate distribution of deviations from those curves, was evaluated. Cross- validation was used for six teleost species to compare predictions of expected lengths or otolith sizes at age, given otolith size or length, respectively, with those of other proportionality-based approaches that incorporate age. For four species, and particularly

Acanthopagrus butcheri when using a biological intercept, better estimates were produced using the new model than were produced using the regression equations in the other back- calculation approaches. Back-calculated lengths for A. butcheri estimated using this model were more consistent with observed lengths, particularly when employing a biological intercept, than those obtained using other proportionality-based approaches and also a constraint-based approach known to produce reliable estimates. By selecting somatic and otolith growth curves from a suit of alternatives to better describe the relationships between length, otolith size, and age, the new approach is likely to produce more reliable estimates of back-calculated length for other species.

4.1 INTRODUCTION

Back-calculation is an invaluable tool used by fisheries scientists around the world for reconstructing individual growth histories of fish from the microstructures present within their hard body parts, such as otoliths (e.g., Campana 1990, 2005; Vigliola and Meekan

2009). The development of a back-calculation model is a two-step process, which involves 1) fitting an appropriate regression equation to describe the relationship between fish length, otolith size and, in some recent approaches, age (Morita and Matsuishi 2001; Finstad 2003), and 2) developing a back-calculation formula which, using the results of the regression analysis, may be used to estimate the lengths of an individual fish at a given age (Francis

1990; Vigliola and Meekan 2009). If a proportionality-based back-calculation approach is to produce reliable back-calculated estimates of length, the regression equations, fitted in the first of these two steps, must produce accurate estimates of the expected length or otolith radius for a fish, given observed values for its independent variables. Although several studies have attempted to validate the final lengths estimated using various back-calculation formulae (e.g., see Table 2 in Vigliola and Meekan 2009), apparently none has used cross- validation to directly explore the accuracy and precision of estimates of fish length and otolith radis at capture predicted by the regression equations fitted to those variables and age, prior to employing those equations in the back-calculation formulae.

Two main back-calculation methods, which have led to the development of different back-calculation formulae, emerge from the literature. The first approach assumes that, throughout the life of a fish, particular measurements of somatic and otolith size retain constant proportionality with respect to the values expected for fish within the population

(e.g., Whitney and Carlander 1956; Francis 1990). The second approach constrains the equation relating length and otolith size for individual fish to pass through one or more

100

points, e.g., a common intercept (e.g., Campana 1990; Vigliola and Meekan 2009). More recently-developed proportionality-based back-calculation approaches, such as those of

Morita and Matsuishi (2001) and Finstad (2003), also recognise the influence of age in the relationship between fish length and otolith size. Constraint-based back-calculation approaches, such as the ‘biological intercept’ method proposed by Campana (1990), were developed to improve the accuracy of predicted lengths for younger fish and to reduce the influence on the reliability of length estimates of variation in somatic growth rate, where slow-growing fish have larger otoliths than faster-growing fish of the same size, i.e., the growth effect. The biological intercept method assumes a common fish length and otolith size for fish at “the initiation of proportionality between fish and otolith growth” and, when employed in back-calculation, is typically taken to be the length and otolith size of newly- hatched fish.

In their recent review, based on theoretical considerations and on the results of the comparative study by Wilson et al. (2009), Vigliola and Meekan (2009) recommended use of the constraint-based approach of Fry (1943), as modified by Vigliola et al. (2000) and which uses the biological intercept, as this better accommodated possible allometry of fish length and otolith size than the Biological Intercept model of Campana (1990).

Theoretically, Vigliola and Meekan (2009) based their decision on a requirement that the back-calculation formula must (1) assume proportionality of otolith-fish growth, and (2) generate realistic estimates of sizes at age (through use of a biological intercept). The former requirement, which was expressed mathematically as

푑(퐿 − 푎) 푑푅 = 푐 (퐿 − 푎)푑푡 푅푑푡 where L and R represent the length and otolith radius of a fish, a is the body length when 푅 = 0 and c is the constant of proportionality, is more restrictive than either of the

101

assumptions of the Scale and Body Proportional Hypotheses (SPH or BPH). These last hypotheses describe the relationships between the particular measurements of length or otolith size for individual fish and the average values of expected length or otolith size for fish within the population given the observed values of the other variables, i.e., otolith size and fish length, respectively. They impose no specific constraint on the mathematical form of the equation(s) relating fish length and otolith size throughout the lives of the fish in the population, leaving this to be determined by the mathematical forms of the regression equations used to represent the relationships between those variables. In contrast, the criterion of Vigliola and Meekan (2009), as expressed in the above equation, specifies not only the proportionality between measures for each fish and those expected for the population but also the form of the relationship between the expected values of variables over the lives of the fish in the population. As the processes of fish and otolith growth differ, with the latter also involving the physico-chemical process of accretion of material on the surfaces of the otoliths, the relationship between the relative growth rates of the fish and their otoliths is likely to vary throughout life. It is thus suggested that it is the extent to which the regression equations accurately describe the relationships between fish and otolith size that should be the criterion for acceptance rather than the strict requirement that those regression equations are consistent with above differential equation of Vigliola and Meekan (2009).

Although models based on the SPH or BPH do not constrain the trajectories of length and otolith radius for individual fish such that these pass through a specific, pre-determined biological intercept, the functions describing the relationships between expected length and otolith radius and covariates can be constrained to pass through such an intercept. This would reduce the influence of the growth effect on estimates of back-calculated length, thereby addressing the second requirement of Vigliola and Meekan (2009), i.e., that realistic estimates of sizes at age are produced by the approach. The introduction of such constraints

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into the regression equations of the proportionality-based back-calculation approaches requires further exploration, however, as it will affect the estimates of the expected values of fish length and otolith size produced by those equations.

All back-calculation formulae assume a relationship between fish and otolith growth

(Vigliola and Meekan 2009), typically describing that relationship by a function that directly relates those variables. None of the regression equations employed in existing back- calculation approaches appears to recognise explicitly, however, that the allometric relationship between fish lengths and otolith sizes for individual fish is the result of the somatic and otolith growth that those fish have experienced. Note also that, although otolith growth is essentially a physico-chemical process, it is partially governed by fish metabolism and thus the two growth curves are not independent. Through explicit incorporation of these growth equations in the descriptions of fish length-otolith size allometry, it will be possible to draw on the very considerable body of knowledge of somatic growth, and factors that affect this, to improve back-calculation approaches and better inform growth studies. A bivariate growth model developed by Chapter 34, which links the predictions of somatic and otolith growth curves for fish of the same age, offers an opportunity to explore how such a model might be extended for use in a proportionality-based back-calculation approach and to examine how its estimates compare with those of various existing approaches.

The overall objective of this study was to develop a proportionality-based back- calculation approach based on the bivariate growth model (Chapter 3), and to assess whether, for one selected species, the back-calculated estimates of length produced by this model are equally reliable, or more reliable, than those produced by other contemporary back- calculation approaches. Firstly, the bivariate growth model and the regression models

4 Ashworth, E. C., N. G. Hall, S. A. Hesp, P. G. Coulson, and I. C. Potter. Age and growth rate variation influence the functional relationship between somatic and otolith size. Can. J. Fish. Aquat. Sci. In press.

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described by Morita and Matsuishi (2001) and Finstad (2003), which also included age when predicting expected otolith size, were fitted to fish lengths and otolith sizes at capture for individuals from each of six fish species with differing biological characteristics. Cross- validation was then employed to compare the accuracy and precision of estimates of expected length and otolith radius at capture calculated using the different fitted models when applied to data for fish that had been excluded when fitting those models. Secondly, data for one species (Acanthopagrus butcheri) were used to test, using cross-validation, whether constraining the various models to pass through a biological intercept improved their predictive performance in estimating lengths and otolith radii at capture, and ascertained whether the bivariate growth model performed better than other approaches. Finally, the three proportionality-based back-calculation approaches, i.e., the fitted bivariate growth model and the approaches of Morita and Matsuishi (2001) and Finstad (2003), both with and without constraining the curves to a biological intercept, and the constraint-based approach of

Vigliola et al. (2000), were used to estimate lengths at ages associated with opaque zones delineated prior to the age at capture (subsequently termed ‘age at zone’) for A. butcheri, which were then compared.

4.2 MATERIAL AND METHODS

4.2.1 THE SIX STUDY SPECIES

Data for six fish species from different families and environments, and with varying life cycle characteristics were used for this study. The species, which were studied, were the sparid Black Bream, Acanthopagrus butcheri (Munro 1949); the sciaenid Mulloway,

Argyrosomus japonicus (Temminck and Schlegel 1843); the labrid Foxfish, Bodianus frenchii (Klunzinger 1880); the serranids Breaksea Cod, Epinephelides armatus (Castelnau

1875), and Goldspotted Rockcod, Epinephelus coioides (Hamilton-Buchanan 1822); and the

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glaucosomatid West Australian Dhufish, Glaucosoma hebraicum Richardson, 1845.

Maximum ages and total lengths of these species ranged from ~ 19 to 78 years and from

~ 480 to 2 000 mm, respectively, with habitats extending from temperate estuaries to tropical marine waters (for further details, see Tables S3.1 and S3.2 in Chapter 3).

During the present study, 128 A. butcheri were collected in May 2013 from the Wellstead

Estuary at 34°50’S latitude and 118°60’E longitude on the south coast of Western Australia.

This study was conducted in accordance with conditions in permit R2561/13 issued by the

Murdoch University Animal Ethics Committee. For the five other species, permits were issued during earlier studies by the Murdoch University Animal Ethics Committee. Details of date and location of capture for each of these five species can be found in Table S2.

4.2.2 FISH PROCESSING AND OTOLITH MEASUREMENTS FOR SIX SPECIES, AND

BIOLOGICAL INTERCEPT FOR ACANTHOPAGRUS BUTCHERI

The total length (TL) of each A. butcheri was measured to the nearest 1 mm and its two sagittal otoliths removed and stored. High-contrast digital images of the sectioned otoliths

(i.e., the left sagittal otolith) of each individual of A. butcheri were taken under reflected light and analysed using the computer imaging package Leica Application Suite version 3.6.0

(Leica Microsystems Ltd. 2001) (for details regarding the sectioning of otoliths refer to

Chapter 3). The ages for A. butcheri were determined for each fish from the number of opaque zones in a section from its otolith, its date of capture, and the birth date assigned to

A. butcheri in the Wellstead Estuary (corresponding to the approximate mid-point of the spawning season) (Sarre and Potter 2000). Opaque zones in otoliths of A. butcheri were counted independently and on different occasions by E. Ashworth and P. Coulson. On the few occasions for the 128 A. butcheri when the counts of these two readers differed (< 4%),

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the two readers discussed the basis for the discrepancy and determined a mutually agreed value which was used in following analyses.

Sectioned otoliths from 50 individuals of each of the other five species were randomly selected from the otolith collected in previous studies in Western Australia by staff and research students at Murdoch University (see Table S3.2 in Chapter 3). Preparation of these sections had followed the same procedure as that used for A. butcheri. The lengths recorded for, and ages assigned to the individuals of each species in those earlier studies (Table S3.1 in

Chapter 3) were accepted for use in the current study. Note that a common sample size of 50 randomly-selected fish of each species, other than A. butcheri, was used to facilitate comparability among results.

For all species, the ‘radius’ of each otolith, i.e., the distance between the primordium and the outer edge of the sectioned otolith, was measured under reflected light on three occasions to the nearest 0.1 µm along a line perpendicular to the opaque zones near the posterior edge of the sulcus acusticus of the otolith. The mean of these three measurements for each otolith was used as the radius of that otolith in subsequent analyses. In the case of A. butcheri, the distance along the same axis from the primordium of each otolith to the outside edge of the first opaque zone and the increments between the outside edges of successive opaque zones were also measured. For this species, the relative distinctness of the opaque zones in its otoliths made it possible to accurately measure the widths of increments between the outer margins of successive opaque zones.

For A. butcheri, eggs from the Australian Centre for Applied Aquaculture Research

(ACAAR, Challenger Institute of Technology, Western Australia) were hatched overnight in the laboratory at Murdoch University to provide data to be used when employing a biological intercept (BI) in back-calculation models. Two days after hatching, the TLs of thirty

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randomly-selected larvae were measured to the nearest 0.01 mm under transmitted light. The left sagittal otolith of each larva was collected and measured under a high-resolution digital microscope camera Leica DFC 425 (Leica Microsystems Ltd. 2001) mounted on a high- performance dissecting microscope Leica MZ7.5 (7.9:1 zoom). The radii of these whole otoliths were measured to the nearest 0.1 µm under transmitted light using the computer imaging package Leica Application Suite version 3.6.0.

4.2.3 BIVARIATE GROWTH MODEL AND ASSOCIATED BACK-CALCULATION

APPROACH

The bivariate growth model employed in this study, comprising both somatic and otolith growth curves and a bivariate statistical distribution of deviations, was fitted using an objective function written for Template Model Builder (package ‘TMB’, Kristensen et al.

2015), in combination with the function ‘nlminb’, within R (R Development Core Team

2011) as described in Chapter 3. For both fish lengths and otolith radii, the expected size at age t, i.e., 푆(푡), is represented in this model by either a modified version of the von

Bertalanffy equation with an oblique linear asymptote or a form of the versatile growth curve described by Schnute (1981), using the approaches described in Chapter 2.

The modified von Bertalanffy equation is:

푆(푡) = 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆) = 푐{1 − exp[−푎푠(푡 − 푏푠)]} + 푑(푡 − 푏푠), where

푦 ⁄(휏 − 푏 ) − 푦 ⁄(휏 − 푏 ) 푐 = 2 2 푠 1 1 푠 , {1 − exp[−푎푠(휏2 − 푏푠)]}⁄(휏2 − 푏푠) − {1 − exp[−푎푠(휏1 − 푏푠)]}⁄(휏1 − 푏푠) and

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푦 − 푐{1 − exp[−푎 (휏 − 푏 )]} 푑 = 1 푠 1 푠 . 휏1 − 푏푠

The Schnute (1981) model, which comprises the following four equations, is:

푏−1 −푎(푡−휏1) if 푎푆 ≠ 0, 푏푆 ≠ 0 푏 푏 푏 1 − 푒 [푆휏 + (푆휏 − 푆휏 ) ] 1 2 1 1 − 푒−푎(휏2−휏1)

−푎(푡−휏1) if 푎 ≠ 0, 푏 = 0 푆휏2 1 − 푒 푆 푆 푆휏 exp [ln ( ) ( )] 1 −푎(휏2−휏1) 푆휏1 1 − 푒 푆(푡) = 푔푆(푡| 푆휏1, 푆휏2, 푎푆, 푏푆) = 푏−1 if 푎 = 0, 푏 ≠ 0 푏 푏 푏 푡 − 휏1 푆 푆 [푆휏1 + (푆휏2 − 푆휏1) ( )] 휏2 − 휏1

푆휏2 푡 − 휏1 if 푎푆 = 0, 푏푆 = 0 푆휏1 exp [ln ( ) ( )] 푆휏1 휏2 − 휏1

where 푆휏1 and 푆휏2 are the expected sizes at two specified reference ages 휏1 and 휏2, and 푎푆 and

푏푆 are parameters that determine the shape of the curve. The minimum and maximum ages at

capture of each species were used as the reference ages 휏1 and 휏2, respectively, in this study.

For each of the analyses undertaken in this study, the data were separated into two

subsets, the first of which was used when fitting the bivariate growth or regression models,

and the second for which expected lengths of the fish, given their observed values of otolith

radii at given ages, were predicted using the fitted models. The bivariate growth model was

fitted simultaneously to length and otolith size at capture, using age as an explanatory

variable, to obtain estimates of the parameters of the somatic and otolith growth curves

(Chapter 3). Deviations of observed fish lengths and associated otolith radii at capture from

those respective growth curves were assumed to possess a bivariate normal, normal-

lognormal, lognormal-normal, or bivariate lognormal distribution (Appendix S3.3 in Chapter

3).

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When predicting the expected length of each fish at age using the bivariate growth model, the estimate calculated using the somatic growth curve was adjusted using information from the deviation of the observed from the predicted radius of the otolith. For this, the length at age predicted using the somatic growth curve was adjusted to the mean of the conditional distribution of the lengths at that age given the deviation between the observed and predicted otolith radii and the bivariate distribution of the deviations of lengths and otolith radii

(equations for calculating the conditional distribution of each of the two variables of the various bivariate statistical distributions are presented in Chapter 2). Likewise, the estimates of expected otolith radii at age were obtained by adjusting the values predicted using the otolith growth curve to the values of the conditional means of the radii at age given the deviations between the observed lengths at age and the expected lengths at age predicted using the somatic growth curve. The resulting estimates of the lengths at age t given the

̂ ∗ observed otolith radii at those ages, i.e., 퐿푡|푅푡=푅 , and otolith radii at age given the observed

̂ ∗ fish lengths at those ages, i.e., 푅푡|퐿푡=퐿 , were then used in the subsequent cross-validation and back-calculation sections of this study. Observed and expected lengths and otolith radii at age t are denoted by 퐿푡, 퐿̂푡, 푅푡, and 푅̂푡, respectively, and particular observed values of length and radius by 퐿∗ and 푅∗, respectively.

A modified form of the bivariate growth model, in which somatic and otolith growth curves were constrained to pass through the biological intercept, was also fitted to the lengths and otolith radii at capture for A. butcheri. For this, 휏1 was set to the age of the fish used

when calculating the biological intercept, and 퐿휏1 and 푅휏1 were set to the total length and otolith radius at the biological intercept, thereby reducing the number of parameters to be estimated when fitting the bivariate model.

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For back-calculation of lengths of A. butcheri using the bivariate growth model, it was assumed that the proportional deviation of fish length at capture from the expected length given the observed otolith radius at that age remained constant throughout life. Accordingly,

an estimate of the length of fish j with age at zone 푡푘, i.e., 퐿푗,푡푘, was calculated as

̂ 퐿푡푘|푅푡 =푅푗,푡 퐿 = 푘 푘 퐿 , 푗,푡푘 퐿̂ 푗,푡푐 푡푐|푅푡푐=푅푗,푡푐

where 퐿푗,푡푐 and 푅푗,푡푐 are the length and otolith radius at age 푡푐 when the fish was caught, and

푅푗,푡푘 is the otolith radius at the edge of opaque zone k.

4.2.4 PROPORTIONALITY-BASED AND CONSTRAINT-BASED BACK-CALCULATION

APPROACHES

The accuracy and precision of lengths and otolith radii predicted using the above bivariate growth model were compared with those predicted using the regression models (or derived from or based on those models) of the proportionality-based back-calculation approaches described by Morita and Matsuishi (2001) and Finstad (2003) (Table 4.1). The three-parameter Morita and Matsuishi (2001) ‘age effect’ (AE) model employs age at capture in the relationship between otolith size and fish length, while the four-parameter ‘interaction term’ (IT) model of Finstad (2003) extends the Morita and Matsuishi (2001) model by incorporating an interaction between fish length and age at capture. Vigliola and Meekan

(2009) reported a modified three-parameter form of the ‘age effect’ model that employs fish length and otolith size as the dependent and independent variable, respectively, terming this the ‘age effect Body Proportional Hypothesis’ or AEBPH model. For this study, the four- parameter ‘interaction term’ model was re-arranged and an analogous four-parameter form of model developed to express fish length in terms of otolith size (Table 4.1). These have been

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termed a ‘re-arranged interaction term’ (RIT) model, and an ‘analogous interaction term’

(AIT) model, respectively. The regression equations of the proportionality-based back- calculation approaches of Morita and Matsuishi (2001) and Finstad (2003), and those derived from or based on those regression equations, were also re-written as equations constrained to pass through the biological intercept (Tables 4.1 and 4.2).

A further analysis compared back-calculated estimates of lengths produced for

A. butcheri using the proportionality-based back-calculation approach developed using the bivariate growth model with those calculated using the approaches described by Morita and

Matsuishi (2001) and Finstad (2003), and the constraint-based back-calculation model described by Vigliola et al. (2000). Only published versions of these traditional back- calculation formulae, and versions of these constrained to pass through the biological intercept, were used. Details of the various back-calculation approaches are presented below.

The back-calculation formula (with modified notation) of the ‘age effect’ model of

Morita and Matsuishi (2001), which employs the parameters (i.e., 훼, 훽 and 훾) estimated using the regression equation Eq. 1 (Table 4.1), is

(11) ∗    푅푗,푘  퐿푗,푘 = − + (퐿푐,푘 + + 푡푐,푘) − 푡푗,    푅푐,푘  while that of Finstad (2003), i.e., the ‘interaction term’ model, which uses the parameters

(i.e., 훼, 훽, 훾, and 훿) as estimated by fitting Eq. 2 (Table 4.1), is

(12) 푅푗,푘 [(+퐿푐,푘+푡푐,푘+훿퐿푐,푘푡푐,푘) −− 푡푗] ∗ 푅푐,푘 퐿푗,푘 = . +훿푡푗

∗ 퐿푗,푘 is the back-calculated length of fish k with age at zone 푡푗, 퐿푐,푘 is the observed length of fish k at capture, i.e., at age 푡푐,푘, 푅푗,푘 is the observed radius of the otolith of fish k at age 푡푗, and 푅푐,푘 is the observed radius of the otolith of fish k at capture.

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The regression models employed by Morita and Matsuishi (2001) and Finstad (2003) were constrained to pass through the biological intercept by rewriting 훼 as a function of the length, radius and age at that intercept (reducing the number of parameters by 1) using equations 1 and 2 (Table 4.1). The re-parameterised back-calculation formulae become

(13) ∗ 푅BI−훽퐿BI−훾푡BI 푅BI−훽퐿BI−훾푡BI 훾 푅푗,푘 훾 퐿푗,푘 = − + (퐿푐,푘 + + 푡푐,푘) − 푡푗 훽 훽 훽 푅푐,푘 훽 and

푅푗,푘 (14) [푅BI+(퐿푐,푘−퐿BI)+훾(푡푐,푘−푡BI)+훿(퐿푐,푘푡푐,푘−퐿BI푡BI)] { 푅푐,푘} ∗ −(푅BI−퐿BI−훾푡BI−훿퐿BI푡BI)−푡푗 퐿푗,푘 = , +훿푡푗

where 푅BI = otolith radius at biological intercept (µm), 퐿BI = total length (mm) at biological intercept, and 푡BI = age (years) associated with the biological intercept, and where the values of the parameters in Eq. 13 are obtained by fitting the regression model described by Eq. 6

(Table 4.2), and those in Eq. 14 by fitting the regression model of Eq. 7 (Table 4.2).

The back-calculation formula, i.e., modified Fry model, of Vigliola et al. (2000) is

(15) ∗ [푙푛(퐿푐,푘−)−푙푛(퐿퐵퐼−)]×[푙푛(푅푗,푘)−푙푛(푅퐵퐼)] 퐿푗,푘 =  + exp [푙푛(퐿퐵퐼 − ) + ] [푙푛(푅푐,푘)−푙푛(푅퐵퐼)]

 +   where  is the fish body length at otolith formation, with  = 1 2,  = 퐿 −  푅 1 , and 2 1 퐵퐼 1 퐵퐼

2 2 = 퐿퐵퐼 − 2푅퐵퐼, and where 1, 1, 2, and 2 are parameters estimated by fitting the following regression equations (Vigliola and Meekan 2009) to observed fish lengths and otolith radii at capture.

 (16) 1 1 퐿 = 퐿퐵퐼 − 1푅퐵퐼 + 1푅

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1 (17) 2 퐿− 퐿 +  푅 2 푅 = ( 퐵퐼 2 퐵퐼 ) 2

4.2.5 ANALYSES

All analyses were undertaken using R (R Development Core Team 2011).

Ten-fold cross-validation and hold-out validation

Two cross-validation methods, i.e., a ten-fold cross-validation and a hold-out cross- validation (Kohavi 1995; Then et al. 2015), were employed in this study to assess the predictive performance of the regression equations of the different proportionality-based back-calculation approaches. That is, they were used to assess the ability of those equations to produce predictions of length and otolith radius at capture that matched the measured values for fish, data for which had been excluded when fitting the regression equations. Such comparion differs from comparisons that employ likelihood ratios or AICs, which assess the extent to which predicted values match observed values of a dependent variable for data included when fitting the model. Note that cross-validation was employed only to assess the ability of the regression equations of the back-calculation approaches to produce accurate predictions of expected lengths and otolith radii, given known values of the independent variables of the regression equation, but not to assess the validity of the back-calculation approach in predicting the lengths of individual fish at ages prior to their capture.

For the ten-fold cross-validation for each species, i.e., A. butcheri, A. japonicus,

B. frenchii, E. armatus, E.coioides and G. hebraicum, the 50 fish in the sample used for the study described in Chapter 3 were assigned randomly to ten groups. For each of the proportionality-based back-calculation approaches (Tables 4.1 and 4.2), the following analysis was undertaken using firstly the equation relating fish length to otolith radius and age at capture, and subsequently the equation for the relationship between otolith radius and

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fish length and age at capture. Excluding each of the ten groups in turn, the regression model was fitted to the data from the other nine groups and used to calculate the expected value of the dependent variable for each fish in the excluded group. Following Then et al. (2015), the

Root Mean Square Error (RMSE) of the resulting predicted values of fish lengths and otolith radii from the corresponding observed values of those variables was employed as an overall measure of the predictive ability of the regression equation. This was calculated as

1 푅푀푆퐸 = √ ∑푛 푒2, 푛 푖=1 푖

where 푒푖 is the difference between the observed and predicted values of the dependent variable for observation i (푖 = 1, 2, … , 푛) and n is the number of observations (Dunn et al.

2002; Chai and Draxler 2014). A measure of the bias of the predicted values from the corresponding observed values for the excluded fish was obtained by calculating the overall mean of the differences between the observed and expected values, i.e., the Mean Error

1 푀퐸 = ∑푛 푒 (Walther and Moore 2005). 푛 푖=1 푖

The above cross-validation analysis was undertaken separately for the bivariate growth model and each of the alternative proportionality-based back-calculation approaches, i.e., the

‘age effect’, ‘interaction term’, AEBPH, ‘re-arranged interaction term’, and ‘analogous interaction term’ models (Tables 4.1 and 4.2). The difference between the RMSE calculated for each alternative model and that for the bivariate growth model, expressed as a percentage of the latter, was calculated as

%∆푅푀푆퐸 = 100 (푅푀푆퐸 − 푅푀푆퐸푏푖푣푎푟푖푎푡푒 푚표푑푒푙)⁄푅푀푆퐸푏푖푣푎푟푖푎푡푒 푚표푑푒푙.

Ten-fold cross-validation was also undertaken, as described above for the samples of 50 fish from each species, using lengths and otolith radii for 120 individuals of A. butcheri, which had been randomly selected from the sample of 128 fish collected for this species. This procedure was repeated while constraining the length-otolith radius relationships to pass

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through the biological intercept, noting that A. butcheri was the only one of the six studied species for which such an intercept was available.

Note that the holdout or ten-fold cross-validation approaches were applied only to the back-calculation based approaches as it was not possible to employ these with the modified

Fry back-calculation formula of Vigliola et al. (2000). This model assumes that, for an individual fish, the curve relating fish lengths and otolith radii passes through both the biological intercept and the length and otolith radius of the fish at capture.

For A. butcheri, for which fish lengths and otolith radii and ages at capture had been recorded for 128 fish, the above ten-fold cross-validation was complemented with a hold-out validation (Kohavi 1995). This was undertaken by fitting each regression equation of the different back-calculation models to the data for the 50 A. butcheri used in the first of the ten- fold cross-validations, i.e., the 50 fish that had been used in the study reported in Chapter 3, and using the resultant fitted equation to calculate the expected values of the dependent variables for the remaining 78 fish and, from these, the RMSE and ME of those deviations.

As in the ten-fold cross-validation for the 120 fish, the holdout validation analyses were repeated for A. butcheri using the version of the bivariate growth model for which the somatic and otolith growth curves had been constrained to pass through the biological intercept, together with the regression equations of the other proportionality-based back- calculation approaches that had been similarly modified.

Back-calculation

Estimates of lengths at ages at zones were calculated for A. butcheri using the various proportionality-based and the constraint-based back-calculation formulae for both the case when the biological intercept was not included and the case when the model was constrained to pass through the biological intercept. For this, the bivariate growth model and various

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regression models of the different back-calculation approaches were fitted to the fish lengths, otolith radii and ages at capture for the 50 fish that had been used for the study reported in

Chapter 3. Using the resulting parameters, the back-calculation formulae of the different approaches were used to produce estimates of the length at each age at zone for each of the

78 A. butcheri that had been excluded in the preceding step when fitting the models, which were then compared. Mean lengths of A. butcheri within each age class of the sample were also calculated and compared with the means of the back-calculated lengths at the ages at zones bounding the otolith radii for those age classes.

4.3 RESULTS

4.3.1 ACCURACY AND PRECISION OF LENGTH AND OTOLITH RADIUS ESTIMATES

FOR SIX SPECIES

Based on the results of the ten-fold cross-validation for 50 fish of each of the six species, the bivariate growth model produced lower estimates of RMSE for fish length than were obtained using the other models and thus improved prediction performance for all six species (Table 4.3a). In contrast, the re-arranged form of the interaction term model generally produced an estimate of RMSE (i.e., between 26 and 284) for fish length, which was greater

(i.e., predictions with greater error) than was produced by either the bivariate growth model

(RMSE = 17 to 87) or the Age Effect Body Proportional Hypothesis model (RMSE = 23 to

174) across all species. This was particularly the case for E. coioides and G. hebraicum for which the percentages by which the RMSEs of the re-arranged form of the interaction term model exceeded those of the bivariate growth model by as much as 241 and 353%, respectively. These large RMSEs appear to be due to occasional very small values of the denominator in Eq. 4 (Table 4.1), suggesting that this form of model may be sensitive to such values and, if employed in future studies, should be used with caution. Note that in the cases

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of A. butcheri and G. hebraicum, the analogous form of the interaction term model produced an estimate of RMSE for the length predictions similar (i.e., differing by only ~ 4%) to that obtained using the bivariate growth model (Table 4.3a). The bivariate growth model produced low values of positive and negative bias, and particularly in the cases of A. butcheri and E. coioides for which the model produced the lowest estimates of ME and observed lengths were either slightly overestimated or underestimated, respectively.

The bivariate growth model also produced better predictions of otolith radius for four of the six species, i.e., A. butcheri, A. japonicus, E. armatus and G. hebraicum, by providing values of RMSE lower than those calculated using the age effect and interaction term models

(Table 4.3b). The predictions from the age effect model, in the case of A. butcheri, were nearly equal to those of the new model (differing by only ~ 1%) and provided as good a fit.

The interaction term model, however, produced estimates of otolith radius for B. frenchii and

E. coioides with lower (by ~ 5% and ~ 4%, respectively) RMSEs than those of the bivariate growth model. Although estimates of the ME were always very low and marginally different, the bivariate growth model did not produce the most accurate observed otolith radii compared with the age effect and interaction term models.

4.3.2 ACCURACY AND PRECISION OF LENGTH AND OTOLITH RADIUS ESTIMATES

FOR ACANTHOPAGRUS BUTCHERI, WITH AND WITHOUT A BIOLOGICAL INTERCEPT

For the holdout validation for A. butcheri, the re-arranged form of the interaction term model produced length estimates with lower RMSEs (differing by ~ 2%) than were obtained using the bivariate growth model (Table 4.4a). In the case of the ten-fold cross-validation of

120 individuals of A. butcheri, however, the re-arranged form of the interaction term model was only marginally better (by < 1%) than the new bivariate growth model. When the models were constrained by the biological intercept, the bivariate growth model provided better

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predictions of lengths of A. butcheri for both the holdout (RMSE = 17) and the ten-fold cross- validation with 120 fish (RMSE = 17, Table 4.4a). Overall, when constraining the models to pass through the biological intercept, there was a slight decrease in the quality of the length predictions produced by the Age Effect Body Proportional Hypothesis, the re-arranged and the analogous forms of the interaction term models, i.e., for all cases other than that for the bivariate growth model with ten-fold cross-validation (Table 4.4a). The RMSE for predictions of length calculated using the bivariate growth model only improved slightly, i.e., by < 1%, when the biological intercept was imposed. In terms of accuracy, estimates of the

ME matched those of the RMSE, such that the re-arranged form of the interaction term model and the bivariate growth model, which both produced lower RMSEs than the other models, also produced the most accurate observed lengths.

Low values of the RMSEs for the predictions of the otolith radii, with or without the biological intercept, indicated that the bivariate growth model was best for both the holdout and for the ten-fold cross-validation using 128 and 120 fish, respectively (Table 4.4b).

Although inclusion of the biological intercept slightly increased (by ~ 11%) the RMSEs of predicted otolith radii for the age effect model, a much greater (~ 8-fold) increase was produced in the RMSEs calculated using the interaction term model, suggesting that this latter model may occasionally be less robust depending on the data (or species). In the case of the holdout validation, the age effect model constrained to pass through the biological intercept produced estimates of predicted otolith radii that were least biased compared with predicted values produced by the other models, whereas, in the case of the ten-fold cross- validation with 120 fish, the interaction term model without the constraint of the biological intercept produced the most accurate estimates.

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4.3.3 COMPARISON OF LENGTH PREDICTIONS FOR ACANTHOPAGRUS BUTCHERI

FROM DIFFERENT BACK-CALCULATION APPROACHES

Means of back-calculated lengths calculated using all of the proportionality-based approaches (without biological intercept) underestimated the mean observed lengths at age for the younger A. butcheri collected from the Wellstead Estuary (Table 4.6; Fig. 4.1). Thus, the means of the back-calculated total lengths of fish with age at zone = 3 years, which ranged from 161 (for the bivariate growth model) to 170 mm (for the interaction term model), were less than the mean observed length, i.e., 171 mm, for the fish with ages of 2.6 years at the date on which the sample was collected (May 2013). Although fish with ages between 3 and 6 years were not available in the sample, the bivariate growth model produced mean back-calculated estimates of length that were consistent with the means of the observed lengths for fish with ages > 6 years. The age effect model, however, only produced estimates of back-calculated length that were consistent with the observed lengths for fish with ages between 6 and 8 years, and > 10 years, the lengths for other ages were underestimated. The interaction term model only produced consistent estimates for fish aged from 6 to 8 years, underestimating the lengths of older fish. The modified Fry model produced estimates of back-calculated lengths that underestimated the means of the observed lengths for fish of all ages within the sample (Table 4.6).

The consistency of estimates of back-calculated lengths at the ages at each of the zones with means of observed lengths for fish with ages lying between the ages at those zones was improved by imposing the biological intercept as a constraint (Table 4.6; Fig. 4.1).

In the case of the bivariate growth model, the means of the back-calculated estimates were consistent with the means of the observed lengths for A. butcheri of all observed ages. The estimates of the lengths calculated using the age effect back-calculation model were

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consistent with observed lengths for fish for all ages except 9 to 10 years, which were underestimated. Consistent length estimates were produced by the interaction term model only for fish with ages up to 7 years, but, for older fish, lengths were underestimated.

The age effect model produced estimates of lengths which were generally more similar to those calculated using the bivariate growth model than those obtained using the interaction term and modified Fry models, particularly at older ages (Fig. 4.1a and b). At young ages, however, back-calculated lengths estimated using the bivariate growth model

(with biological intercept) were more comparable to those calculated using the modified Fry model (Table 4.6; Fig. 4.1e).

4.4 DISCUSSION

4.4.1 CROSS-VALIDATION OF ESTIMATES DERIVED FROM RELATIONSHIP BETWEEN

LENGTH, OTOLITH RADIUS AND AGE

Although the bivariate growth model is likely to produce more reliable predictions of length or otolith radius than the regression equations of the proportionality-based back- calculation approaches considered in this study, results are species-dependent. Evidence for this is provided by the finding that, although lengths at capture predicted by the bivariate growth model from otolith sizes and ages at capture for individuals of six fish species were more reliable than those produced using the regression equations of the back-calculation approaches, the bivariate growth model produced the most reliable predictions of otolith radii from fish lengths and ages at capture for only four of the six species. For the other two species, the regression equation of the interaction term back-calculation produced the lowest estimates of RMSE.

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Cross-validation using 50 individuals, holdout validation using 128 individuals, and ten- fold cross-validation using 120 individuals of A. butcheri from the Wellstead Estuary demonstrated that, given fish lengths and ages, the most reliable estimates of otolith radii for this species were produced by the bivariate growth model, regardless of whether or not that model was constrained to pass through the biological intercept. Using the bivariate growth model with the biological intercept constraint, a similar result was obtained for estimates of fish length when ten-fold cross-validation was applied to the 50 randomly-selected individuals from the sample of 128 fish and when the larger datasets of 128 and 120 fish were analysed using holdout and ten-fold cross-validation, respectively. When not constrained by the biological intercept, however, the regression equation of the re-arranged interaction term model produced the most reliable estimates of length for the holdout validation using 128 fish and the ten-fold cross-validation using 120 fish. Thus, without the constraint of the biological intercept, the results for the smaller sample were influenced by sample size. Given such inconsistency, it would therefore be appropriate to base predictions for A. butcheri on analyses employing models fitted to the larger of these datasets, i.e., the holdout cross- validation employing 128 fish or the ten-fold cross-validation using 120 individuals of this species. For this cross-validation, the most reliable estimates of fish lengths of A. butcheri given ages and otolith radii were obtained using the version of bivariate growth model that employed the biological intercept, while equally reliable estimates of otolith radii given ages and fish lengths were produced using the bivariate growth model with and without the biological intercept.

It was concluded above that, when constrained to pass through the biological intercept and samples are of sufficient size to adequately represent the population from which they are drawn, the bivariate growth model is likely to provide the most reliable estimates of both fish lengths and otolith radii. This may be due to the considerable flexibility of the bivariate

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growth model, whereby the forms of the curves used to represent somatic and otolith growth are selected from the wide range of alternative forms described by the Schnute (1981) or modified von Bertalanffy growth models, and from alternative forms of bivariate statistical distributions for the distribution of deviations from those growth curves. This flexibility allows it to account for the particular characteristics of the relationships exhibited between observed fish lengths, otolith sizes and ages at capture of the individuals in the samples of the different species. In contrast, the linear regression equations, and the nonlinear regression equations produced by incorporating an interaction term or by re-arranging the linear equations are far more prescriptive. Although those latter models provide good descriptions of the allometric relationships between fish lengths, otolith sizes, and take into account the ages at capture for the sampled fish of the different species, their ability to adjust to the characteristics of the data are constrained by their fixed functional forms. While the flexibility of the forms of the growth curves used by the bivariate growth model allow it to produce a better description of the available data, there is the risk that, with small sample sizes, noise in those data may influence the forms of growth curve that are selected. In this context, the alternative forms of curves that the Schnute (1981) growth model encompasses include a number of curves that, when describing somatic or otolith growth, are biologically infeasible.

Prior to adopting a particular form of somatic growth curve, Katsanevakis and

Maravelias (2008) recommended that a broad range of alternative growth curves should be explored when fitting to lengths at ages. The different approaches described in various back- calculation studies (e.g., Campana 1990; Francis 1990; Sirois et al. 1998; Vigliola et al. 2000;

Morita and Matsuishi 2001), each employing slightly different equations to describe the relationships between length and otolith size (and, in some approaches, age at capture), reflect an awareness that the extent to which the different regression equations describe the

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data varies among samples from different species. The report by Vigliola and Meekan (2009) that as many as 22 different back-calculation approaches have been proposed suggests that a flexible approach, which identifies the most suitable form of regression equation, such as that used when fitting the bivariate growth model, is required to ensure that exploration of alternative model forms is undertaken using a systematic, well-defined procedure with explicit criteria for model selection.

The current study is apparently the first to employ cross-validation to explore the reliability of the values of fish length or otolith size predicted by the regression equations used in the different back-calculation approaches. Most earlier studies have only explored the extent to which predicted fish lengths associated with the various growth zones matched the mean of the observed lengths of fish from the different age classes (e.g., Pierce et al. 1996;

Sirois et al. 1998; Pajuelo and Lorenzo 2003; Zengin et al. 2006) or, in the few cases when recaptured tagged and otolith-marked individuals were available, the extent to which back- calculated estimates of length matched the lengths of the fish at release (e.g., Panfili and

Tomás 2001; Roemer and Oliveira 2007; Li et al. 2008; Michaletz et al. 2009). These latter approaches are of considerable value, particularly those that validate back-calculated length estimates by comparison with lengths at tagging of recaptured, otolith-marked fish. Although cross-validation is unable to test the reliability of the final back-calculated estimates of lengths of individual fish, it offers the benefit to proportionality-based back-calculation approaches of elucidating the reliability of the expected lengths predicted for fish by the regression equations, on which the estimates of back-calculated lengths of individual fish rely.

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4.4.2 COMPARISON OF LENGTH PREDICTIONS BETWEEN BACK-CALCULATION

APPROACHES

Back-calculated lengths at ages estimated for individuals of A. butcheri using the proportionality-based approach developed for the bivariate growth model were found to be very similar to those calculated using the three alternative traditional back-calculation formulae, i.e., the age effect, interaction term and modified Fry models. Overall, however, and particularly when using a biological intercept, length estimates produced using the bivariate growth model were more consistent with the observed mean lengths at age at capture than those based on the other approaches. These results suggest that, for some species other than A. butcheri, the proportionality-based bivariate growth approach developed in the current study will produce estimates of length at age that are more accurate than those produced by back-calculated traditional approaches.

To address potential bias resulting from continued increase in otolith size despite the reducing rate of somatic growth as fish age, the back-calculation approach developed by

Morita and Matsuishi (2001) was the first to incorporate age as a predictor variable. Although this model produces less biased length estimates than earlier proportionality-based approaches for which growth rates of slow-growing fish were overestimated, it is also the least precise model (Morita and Matsuishi 2001). Indeed, large errors in predicted size produced by the age effect model were reported by Wilson et al. (2009), confirming the age effect model’s sensitivity to growth effects and to the accuracy and precision of the regression fitted to the relationship between fish length, otolith radius and age. Finstad (2003) found that incorporation of the interaction term into the age effect model contributed significantly to the quality of the fit of the length and otolith radius to age relationship, with difference between the two models being most pronounced in the youngest age classes. The

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results of the current study demonstrate that inclusion of the biological intercept improved the consistency of back-calculated length estimates with mean observed lengths for younger fish for both the age effect and interaction term models, thus overcoming to some extent the bias in predicted size introduced by these models (e.g., Vigliola and Meekan 2009; Wilson et al.

2009).

Wilson et al. (2009), who were the first to validate modern back-calculation models using longitudinal data collected and analysed at the individual level from multiple internal and external tagging trials, showed that, for two marine cleaning gobies Elacatinus evelynae and Elacatinus prochilos, the modified Fry model provided the most accurate (and least biased) size at age estimates despite the presence of age, growth and time-varying growth effects in the dataset. The better performance of the modified Fry model was explained by the allometric nature of the relationship between fish length and otolith size at the individual level, and that the model is constrained to biological intercepts (Vigliola et al. 2000; Wilson et al. 2009). Despite its complex form, the approach preferred by Vigliola and Meekan (2009) for use in routine back-calculation was the modified Fry model (Vigliola et al. 2000; Wilson et al. 2009). In the present study, however, back-calculated estimates of length at age from the bivariate growth model with the biological intercept constraint were closer to mean observed lengths at age than those produced by the modified Fry model, suggesting that, at least for A. butcheri from the Wellstead Estuary, the new approach may produce more accurate back-calculated estimates of fish length than the modified Fry approach.

The allometric relationship between fish length and otolith radius is formed by the changes in the sizes of these variables with age, throughout the life of the fish (Xiao 1996).

That is, the allometric relationship used in traditional back-calculation approaches integrates the effects of the growth of both variables. The bivariate growth model proposed in this study, on the other hand, explicitly describes the growth of each of these variables, offering

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greater opportunity to investigate the factors affecting that growth and thus, indirectly, the form of the relationship between length and otolith radius. The bivariate growth model may thus represent a valuable alternative to the modified Fry model as it provides a useful link between somatic and otolith growth models and back-calculation approaches, and a more realistic representation of the relationship between these variables, and age, through the life of fish.

For this study, as with the majority of other back-calculation studies, data for tagged and marked fish were not available. There would be value in comparing the performance of the proportionality-based bivariate growth approach against that of other back-calculation approaches using recaptures from an appropriate tagging study in which otoliths of individually-tagged fish have been chemically marked prior to their release, similarly to the methods carried out by Panfili and Tomás (2001) and Li et al. (2008). It has been suggested that such tagging studies provide the most suitable data to validate the performance of a back- calculation formula (Casselman 1983; Vigliola and Meekan 2009).

4.4.3 SUMMARY AND CONCLUSIONS

Based on cross-validation results across a range of fish species, the RMSEs of predictions of expected fish length and otolith size produced by the new bivariate growth model were found typically to be equal to or better than those produced using the regression equations employed by the selected traditional back-calculation approaches considered in this study. The results of the analyses suggest that, for A. butcheri from the Wellstead Estuary, the expected length predicted for an individual fish based on its age and otolith size using the bivariate growth model (particularly when fitted to samples of greater size) is likely to be more reliable than those estimates produced using those other regression models, and thus likely to lead to more reliable estimates of back-calculated length. The proportionality-based

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back-calculation approach developed using the bivariate growth model, when constrained to pass through the biological intercept for this species, produced mean back-calculated length estimates that were more consistent with the mean observed lengths at age than those of other traditional back-calculation approaches. The results of this study strongly support the conclusion that back-calculated lengths calculated for A. butcheri in the Wellstead Estuary using the proportionality-based bivariate growth approach, and employing the biological intercept, are more reliable than those produced by the alterative back-calculation approaches that were considered. The approach is likely to be of value for other back-calculation studies, and may, as in the case of A. butcheri, provide estimates of back-calculated length that improve on those produced by traditional approaches.

Acknowledgements

This research was sponsored by Murdoch University and Recfishwest. Gratitude is expressed to Murdoch University for logistical support whilst in the field, P. Coulson for his skilled assistance in the laboratory and as second reader of otolith zones counts,

G. Thompson for his microscopy expertise and for helping detect the otoliths of A. butcheri larvae, and J. Williams and D. Yeoh for assisting in the collection of A. butcheri for this study. I also thank Michaël Manca for his advice and Bruce Ginboy at the Australian Centre for Applied Aquaculture Research, Challenger, Institute of Technology (Fremantle), for providing black bream eggs for the purpose of determining the biological intercept used in this study.

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Table 4.1. Regression equations of the back-calculation approaches described by Morita and

Matsuishi (2001), Finstad (2003) and Vigliola and Meekan (2009), or derived from those

equations.

Eq. Regression equation Source

(1) 푅 =  + 퐿 + 푡 Morita and Matsuishi (2001)

(2) 푅 =  + 퐿 + 푡 + 훿퐿푡 Finstad (2003)

(3) 퐿 = 훼 + 훽푅 + 훾푡 Re-arranged version of eq. 1 (AEBPH in Vigliola and Meekan 2009)

(4) 퐿 = (푅 + 훼 + 훾푡)⁄(훽 + 훿푡) Re-arranged from eq. 2

(5) 퐿 = 훼 + 훽푅 + 훾푡 + 훿푅푡 Analogous linear form of eq. 2

Note. R = otolith radius (µm); L = total length (mm); t = age (years);

훼, 훽, 훾, and 훿 = parameters of regression equation. AEBPH = Age Effect Body Proportional

Hypothesis.

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Table 4.2. Regression equations, which pass through the biological intercept, modified (or developed) from the equations of Morita and Matsuishi (2001), Finstad (2003) and Vigliola and Meekan (2009).

Eq. Regression equation Source

(6) 푅 = 푅BI + 훽(퐿 − 퐿BI) + 훾(푡 − 푡BI) Modified from eq. 1

(7) 푅 = 푅BI +  (퐿 − 퐿BI) +  (푡 − 푡BI) + 훿 (퐿푡 − 퐿BI푡BI) Modified from eq. 2

(8) 퐿 = 퐿BI + 훽(푅 − 푅BI) + 훾(푡 − 푡BI) Modified from eq. 3

(9) 퐿 = ([퐿BI( + 훿푡BI) − 푅BI − 푡BI] + 푅 + 푡)/(푏 + 훿푡) Modified from eq. 4

(10) 퐿 = 퐿BI +  (푅 − 푅BI) +  (푡 − 푡BI) + 훿 (푅푡 − 푅BI푡BI) Modified from eq. 5

Note. 푅BI = otolith radius at biological intercept (µm); 퐿BI = total length (mm) at biological intercept; 푡BI = age (years) associated with the biological intercept; R = otolith radius (µm);

L = total length (mm); t = age (years); 훼, 훽, 훾, and 훿 = parameters of regression equation.

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Table 4.3a. Root Mean Square Error (RMSE) and mean error (ME) of total length estimates

for the bivariate growth model and models derived from the Morita and Matshuishi (2001)

and Finstad (2003) models calculated using ten-fold cross-validations for samples of 50 fish

for each of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. The percentages

by which the RMSE of the alternative models differ from that of the bivariate growth model

are presented in parentheses.

Species Accuracy and BG AEBPH RIT AIT precision measures

Acanthopagrus butcheri RMSE 16.97 22.90 (34.96) 29.53 (74.04) 17.69 (4.26)

ME 0.08 -0.22 -2.98 0.25

Argyrosomus japonicus RMSE 87.36 173.62 (98.74) 144.62 (65.54) 123.60 (41.48)

ME -2.35 -3.30 28.35 -3.28

Bodianus frenchii RMSE 22.49 37.97 (68.80) 26.53 (17.94) 32.21 (43.22)

ME 0.50 0.14 0.06 1.51

Epinephelides armatus RMSE 38.66 44.29 (14.57) 62.06 (60.54) 42.90 (10.97)

ME 2.26 0.11 5.84 0.37

Epinephelus coioides RMSE 55.97 60.20 (7.57) 190.64 (240.61) 58.82 (5.09)

ME -0.37 -0.35 -23.09 -0.25

Glaucosoma hebraicum RMSE 62.67 68.52 (9.30) 284.27 (353.47) 65.26 (4.11)

ME 2.23 -0.83 -40.61 -0.85

Note. Bold font identifies the minimum RMSE for each species. BG = Bivariate Growth

model; AEBPH = Age Effect Body Proportional Hypothesis; RIT = the re-arranged form of

the regression equation used by Finstad (2003); AIT = the analogous form of the regression

equation used by Finstad (2003).

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Table 4.3b. Root Mean Square Error (RMSE) and mean error (ME) of otolith radius estimates for the bivariate growth model and for the Morita and Matshuishi (2001) and

Finstad (2003) models calculated using ten-fold cross-validations for samples of 50 fish for each of Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. The percentages by which the

RMSE of the alternative models differ from that of the bivariate growth model are presented in parentheses.

Species Accuracy and BG AE IT precision measures

Acanthopagrus butcheri RMSE 0.069 0.070 (1.10) 0.072 (3.78)

ME 0.003 < -0.001 0.001

Argyrosomus japonicus RMSE 0.484 0.519 (7.28) 0.548 (13.19) ME 0.025 0.009 -0.035

Bodianus frenchii RMSE 0.108 0.105 (-2.72) 0.103 (-4.72)

ME 0.001 -0.001 < 0.001

Epinephelides armatus RMSE 0.095 0.101 (6.61) 0.100 (4.99)

ME 0.006 < 0.001 0.001

Epinephelus coioides RMSE 0.097 0.106 (8.98) 0.094 (-3.55)

ME 0.001 -0.001 0.001

Glaucosoma hebraicum RMSE 0.176 0.199 (13.14) 0.191 (8.54)

ME 0.004 -0.002 < 0.001

Note. Bold font identifies the minimum RMSE for each species. BG = Bivariate Growth model; AE = Age Effect model; IT = Interaction Term model.

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Table 4.4a. Root Mean Square Error (RMSE) and mean error (ME) of total length estimates

for the bivariate growth model and for models derived from the Morita and Matshuishi

(2001) and Finstad (2003) models calculated using holdout and ten-fold cross-validations

with and without the biological intercept for samples of Acanthopagrus butcheri. The

percentages by which the RMSE of the alternative models differ from that of the bivariate

growth model are presented in parentheses.

Without the Biological Intercept With the Biological Intercept

Method Accuracy and Number BG AEBPH RIT AIT BG AEBPH RIT AIT precision of fish measures

RMSE N = 128 17.14 21.52 16.84 17.36 17.34 22.58 21.34 18.08 Holdout validation (25.56) (-1.72) (1.28) (30.22) (23.05) (4.27) ME 2.44 3.59 2.23 2.34 2.76 5.63 3.58 3.18

RMSE N = 120 16.90 21.74 16.84 17.46 16.79 22.23 17.89 18.11 Ten-fold cross- (28.64) (-0.37) (3.34) (32.43) (6.52) (7.85) validation ME 0.02 -0.39 -0.01 -0.03 0.03 1.03 1.31 1.08

Note. Bold font identifies the minimum RMSE. BG = Bivariate Growth model; AEBPH =

Age Effect Body Proportional Hypothesis; RIT = the re-arranged form of the regression

equation used by Finstad (2003); AIT = the analogous form of the regression equation used

by Finstad (2003). The holdout validation approach used 128 fish, 50 of which were

employed when calculating the parameters of the models and 78 of which were held outside

the fitting process for use in testing the accuracy and precision of model predictions. The ten-

fold cross-validation involved the use of 120 fish, with each prediction based on a model

fitted to data for 108 fish and predictions calculated for the other 12 fish on each pass through

of the approach.

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Table 4.4b. Root Mean Square Error (RMSE) and mean error (ME) of otolith radius

estimates for the bivariate growth model and the Morita and Matshuishi (2001) and the

Finstad (2003) models calculated using holdout and ten-fold cross-validations with and

without the biological intercept for samples of Acanthopagrus butcheri. The percentages by

which the RMSE of the alternative models differ from that of the bivariate growth model are

presented in parentheses.

Without the Biological Intercept With the Biological Intercept

Method Accuracy and Number BG AE IT BG AE IT precision of fish measures

RMSE N = 128 0.070 0.076 0.072 0.071 0.084 0.570 Holdout validation (8.46) (3.28) (18.75) (702.21) ME -0.016 -0.019 -0.018 -0.016 -0.006 0.559

RMSE N = 120 0.070 0.074 0.072 0.070 0.082 0.563 Ten-fold cross- (6.28) (2.95) (16.13) (700.32) validation ME 0.002 -0.001 < -0.001 0.002 0.007 0.550

Note. Bold font identifies the minimum RMSE. BG = Bivariate Growth model; AE = Age

Effect model; IT = Interaction Term model. The holdout validation approach used 128 fish,

50 of which were employed when calculating the parameters of the models and 78 of which

were held outside the fitting process for use in testing the accuracy and precision of model

predictions. The ten-fold cross-validation involved the use of 120 fish, with each prediction

based on a model fitted to data for 108 fish and predictions calculated for the other 12 fish on

each pass through of the approach.

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Table 4.5. Mean observed total length (mm) and mean age of fish within each age class in the 2013 sample of Acanthopagrus butcheri, together with mean back-calculated total lengths (mm) for fish at ages at zones bounding the age classes, calculated using different back-calculation approaches.

Mean age of fish in age class at capture* and at zone (years) 2 2.62* 3 6 6.62* 7 7.62* 8 9 9.62* 10 10.62* 11

Observed TL 170.80 236.00 253.15 279.50 286.50 (mm) (5.27) (2.58) (0.94) (3.52) (3.26)

Mean of back-calculated lengths for fish of different ages at zones

BG model 136.43 161.32 226.79 244.16 259.60 272.01 282.97 295.01 (2.57) (2.11) (1.24) (0.83) (1.40) (1.78) (3.01) BG model 144.67 173.23 231.37 246.77 260.64 271.65 281.50 291.84 with BI (1.93) (2.78) (2.05) (1.29) (0.84) (1.08) (1.51)

AE model 138.44 169.12 227.00 241.68 256.83 267.60 278.64 289.41 (3.51) (5.95) (3.79) (2.91) (2.91) (2.86) (3.13)

AE model 156.25 181.27 229.76 243.43 256.93 267.61 277.52 286.54 with BI (3.03) (4.29) (3.00) (2.65) (1.56) (1.72) (0.46)

IT model 140.20 169.57 224.58 238.43 253.29 264.15 275.16 284.67 (3.62) (6.43) (4.14) (4.13) (2.04) (2.86) (1.77) IT model 158.77 182.38 226.97 239.44 252.23 262.99 272.46 279.06 with BI (3.40) (4.90) (3.54) (4.35) (3.16) (2.90) (2.18)

MF model 145.36 168.54 221.81 235.98 248.97 262.00 271.55 277.82 (2.03) (3.87) (3.15) (4.71) ( 4.65) (4.41) (4.87)

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Note. Standard errors are indicated between parentheses. Bold font indicates the range of back-calculated mean lengths at ages at zone between which mean observed total length (TL) falls for the corresponding age at capture (*). BI = biological intercept; BG = Bivariate Growth model;

AE = Age Effect model described by Morita and Matsuishi (2001); IT = Interaction Term model described by Finstad (2003); MF = Modified

Fry model described by Vigliola et al. (2000).

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136

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CHAPTER 5 – THE RATIO OF INDIVIDUAL TO AVERAGE OTOLITH

SIZE AT AGE DOES NOT REMAIN CONSTANT THROUGHOUT LIFE OF

ACANTHOPAGRUS BUTCHERI

Abstract

This study explored whether, throughout the life of an individual of a selected year class from an estuarine population of the sparid, Acanthopagrus butcheri, from south-western Australia, the size of its otolith remained in constant proportion to the average size of the otoliths of fish of the same age. For individuals of this species, the hypothesis of constant proportionality with age is invalid as the relative deviations, i.e., natural logarithms of the ratios of otolith size for individual fish to average otolith size at age, vary markedly among different periods of life. The variance of the relative deviations was significantly greater for the first than subsequent opaque zones. Annual increments between the first and second opaque zones in the otoliths exhibited compensation for faster or slower growth between hatching and the formation of the first opaque zone. The relative deviations of otoliths for individual fish exhibited less consistency among the opaque zones formed at younger than older ages but, as the age increased, the relative deviations became increasingly constant, particularly for adjacent zones. Such increasing consistency for older fish may have been due to the annual increment in otolith size becoming small in comparison with the size of the otolith at the opaque zone prior to that growth. Annual growth increments in otoliths for these individuals of A. butcheri were proportional to their initial size prior to that growth, however, suggesting that growth rates of otoliths at these ages are correlated.

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5.1 INTRODUCTION

Back-calculation, the estimation of fish lengths at ages prior to capture based on the widths of growth zones in their otoliths or other hard body structures, is frequently used in fisheries and fish ecology studies (e.g., Campana 2005; Vigliola and Meekan 2009). The results from this invaluable approach are often employed to assess the effect of length- dependent selectivity and fishing mortality on the length and age compositions of a population, and the influence of factors affecting otolith and fish growth (e.g., Sinclair et al.

2002a; Vigliola and Meekan 2002; Takasuka et al. 2007). Each back-calculation approach relies on an assumption that, throughout life, there is a specific relationship between the length of a fish and the size of its otolith or between the sizes of the annual increments in those measurements. Differences between the assumed forms of those relationships and the ways in which these relate to the lengths and otolith radii of individual fish at different ages throughout life have resulted in the development of a wide-range of alternative back- calculation formulae (e.g., Vigliola and Meekan (2009) listed 22 alternative approaches), with the accuracy of the lengths predicted by each formula determined by the reliability of its associated assumptions. These different assumptions have resulted in back-calculation approaches that fall into three categories, i.e., regression, proportionality and constraint-based approaches (Vigliola and Meekan 2009).

In the case of proportionality-based back-calculation approaches, predictions of lengths for an individual at different ages throughout its life rely on (often untested) hypotheses that length remains proportional to expected length given otolith size, or otolith size remains proportional to expected otolith size given length (Whitney and Carlander 1956;

Francis 1990). Otolith size and fish lengths are positively related for fish of the same age, with faster growing fish possessing otoliths that are larger than the average size for that age

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(Francis and Campana 2004). It is a common assumption of mixed effects models of otolith growth that, through inclusion of a random effect for different fish, the growth rate of otolith of an individual fish relative to that of other fish will persist throughout life (e.g., Weisberg et al. 2010). That is, the rates of otolith growth of fish with faster growing otoliths will exceed those of fish with slower-growing otoliths throughout life. This study explores the hypothesis that, throughout the life of an individual from a selected population of fish, the ratio of the size of its otolith relative to the average size of the otoliths of fish of the same age remains constant throughout life. This is subsequently referred to as the hypothesis of ‘proportionality of otolith radius with age’ (PORA).

As a case study, the above hypothesis, i.e., PORA, was explored using data for Black

Bream Acanthopagrus butcheri (Munro 1949) collected from the Wellstead Estuary in May,

2013. This sparid completes its life cycle within its natal estuary and can live for up to

~ 31 years (e.g., Sarre and Potter 2000; Jenkins et al. 2006). Measurements of the otolith size of individuals of this species were considered likely to be particularly well suited for this study as such otoliths are relatively large and their sections exhibit clearly-defined opaque zones (Sarre and Potter 2000). The confounding influence of inter-annual environmental differences in estuarine conditions experienced by different year classes within different estuaries has been avoided by using data for individuals of a single year class in a sample from the selected population of A. butcheri.

The overall objective of this study was to explore whether the data for A. butcheri support the PORA hypothesis more strongly than the alternative hypothesis that, for an individual from a selected year-class, the ratio of the size of its otolith to the average size of the otoliths of fish of the same age differs markedly (according to the criteria specified below) among ages at different periods of its life. Thus, initially, for each A. butcheri, the relative deviation of the otolith radius at each opaque zone from the corresponding mean size

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was calculated as the natural logarithm of the ratio of its otolith size at that zone to the average otolith size for fish of the same age. If PORA is true for A. butcheri, it would be expected that: (1) the variances of these relative deviations at successive opaque zones would not exhibit statistically-significant differences (assuming a level of significance of

α = 0.05); (2) the correlation among fish between the relative deviations for the individual fish at each opaque zone in each pair of zones would be positive and statistically significant;

(3) of a set of alternative linear equations describing the relationship between the relative deviation of otolith size from the mean for the individual fish at each opaque zone and those for the corresponding fish at each earlier age, the intercept and slope of the equation that best describes the data would not differ significantly from 0 and 1, respectively (i.e., the relative deviation at one opaque zone would be approximately equal to that at the previous zone); and

(4) of a set of alternative linear equations describing the relationship between the annual increments in sizes of otoliths between each pair of consecutive opaque zones and their initial sizes at the first of those zones, the intercept of the equation that best describes the data would not differ significantly from 0 (i.e., the increment would be approximately proportional to the initial radius). Recognising that it is not possible to test a null hypothesis and noting also the limitations imposed by sample size, the above expectations were explored for individuals of A. butcheri and the results assessed to determine the extent to which the data supported the hypothesis of PORA rather than lack of constant proportionality. Thus, for objectives 3 and 4, model comparison to determine support for PORA, rather than falsification of (i.e., testing to disprove) the alternative hypotheses, was employed.

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5.2 MATERIAL AND METHODS

5.2.1 SAMPLING, FISH PROCESSING AND OTOLITH MEASUREMENTS

A sample of 128 A. butcheri was collected in May 2013 by seine and gill netting in the

Wellstead Estuary at 34°50’S and 118°60’E on the south-western coast of Australia. A

21.5 m long seine net, comprising two wings, each 10 m long, and a 1.5 m wide bunt, fished to a depth of ~ 1.5 m and swept an area of ~ 116 m2. Seining was undertaken along the shoreline during daylight, while the sunken composite multifilament gill nets were set, parallel to the shore at dusk, and then retrieved ~ 2 to 3 h later. Each gill net contained seven panels, each panel being 20 m long with a height of 2 m with netting of different stretched mesh size, i.e., 51, 63, 76, 89, 102, 115 or 127 mm. Fish were euthanized in an ice slurry immediately after retrieval of the net. The total length (TL) of each A. butcheri was measured to the nearest 1 mm and its two sagittal otoliths removed and stored dry.

The left sagittal otolith of each A. butcheri was embedded in clear epoxy resin and cut transversely into ~ 0.3 mm sections through the primordium and perpendicular to the sulcus acusticus, using an Isomet® low-speed saw (Buehler Ltd., Lake Bluff, Illinois). The sections were cleaned, polished with wet and dry carborundum paper (grade 1 200) with tap water and mounted on microscope slides under coverslips using DePX mounting adhesive. The otolith sections were examined under reflected light against a black background using a high- resolution digital microscope camera (Leica DFC 425) mounted on a dissecting microscope

(Leica MZ7.5). High-contrast digital images of the sectioned otoliths were analysed using the computer imaging package Leica Application Suite version 3.6.0 (Leica Microsystems Ltd.

2001). As shown in earlier studies, the annually-formed opaque zones are clearly defined in sectioned otoliths of A. butcheri (e.g., Sarre and Potter 2000; Potter et al. 2008). The central opaque zones (primordia) of these otoliths are relatively small and distinct, which enables the

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distance from the centre of that zone to the outside edge of the first opaque zone within each otolith to be measured precisely.

The number of opaque zones in each sectioned otolith was counted independently and on different dates by E. Ashworth and P. Coulson (Centre for Fish and Fisheries Research,

Murdoch University, Western Australia). The counts of these two readers differed only in the cases of five otoliths, for each of which, following re-examination, a count was mutually- agreed by the readers. Each fish was then assigned an age based on the number of opaque zones in its sectioned otolith, its date of capture, a mean birth date (i.e., the approximate mid- point of the spawning season for A. butcheri in the Wellstead Estuary, 1 October) and knowledge of the time of the year (i.e., late spring) when new opaque zones become delineated from the otolith periphery (Chapter 2; for details of the ageing approach, see also

Sarre and Potter 2000).

The distance from the primordium of each sectioned otolith to the outside edge of the first opaque zone and the increments between the outside edges of all successive opaque zones were measured to the nearest 1 µm under reflected light. All distances were measured along an axis perpendicular to the opaque zones adjacent to the posterior edge of the sulcus.

The radii from the primordium to the outside edges of successive opaque zones were calculated as the sums of the associated increments. Thus, for fish j,

푅푗,푡=푎+1 = 푅푗,푡=푎 + ∆푅푗,푡=푎+1 where 푅푗,푡=푎 is the radius associated with the age t at which opaque zone a became delineated (subsequently referred to as the ‘age’ of the fish at that zone) and ∆푅푗,푡=푎+1 is the increment in otolith size between opaque zones 푎 and 푎 + 1, where, at the primordium, 푎 = 0 and 푅푗,푡=0 = 0 µm.

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5.2.2 ANALYSES

Transformation of data

The analyses were restricted to data for otoliths of fish from the 7+ age class as preliminary exploration had revealed that this strong year class was the only one with sufficient fish (n = 53) for detailed exploration. This restriction ensured that, at each age, all fish used in analyses had potentially been exposed to the same range of annual environmental conditions within each year of life.

The radius of an otolith 푅푗,푡 of fish j (1 ≤ 푗 ≤ 푛) at age t years is assumed to be related to the average (i.e., arithmetic mean) of the radii 푅̅푡 of the otoliths for fish of the same age by

푅푗,푡 = 푅̅푡 exp(휀푗,푡),

where exp(휀푗,푡) reflects the relative deviation of the size of the otolith of fish j at age t from

5 its expected value. It was assumed that, at this age, 푅푗,푡 was lognormally distributed with a

2 2 2 variance of 휎 , i.e., 푅푗,푡~LN(ln(푅̅푡), 휎 ), and thus exp(휀푗,푡)~LN(0, 휎 ).

At age t = a years, i.e., the age at which opaque zone a (a = 1, 2, …, 푛푎, where

푛푎 = 7 is the number of opaque zones in otoliths of fish from the selected age class) became delineated, the radius of the otolith of fish j is denoted by 푅푗,푡=푎 and the arithmetic mean of the radii of the fish of this age class by 푅̅푡=푎. For fish j, an estimate of the natural logarithm of the ratio 푅푗,푡=푎⁄푅̅푡=푎, i.e., the relative deviation 푆푗,푡=푎 (subsequently termed the

‘deviation’) of its otolith size from the mean size for fish of that age was calculated as

5 Ashworth, E. C., N. G. Hall, S. A. Hesp, P. G. Coulson, and I. C. Potter. Age and growth rate variation influence the functional relationship between somatic and otolith size. Can. J. Fish. Aquat. Sci. In press.

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푆푗,푡=푎 = 휀푗,푡=푎 = ln(푅푗,푡=푎) − ln(푅̅푡=푎), where 푆푗,푡=푎 is normally distributed,

2 푆푗,푡=푎~푁(0, 휎 ) and ‘ln’ denotes the natural logarithm.

Homogeneity of variance

If, as hypothesised, the relative deviation, 푆푗,푡=푎, remains constant through the life of fish j, then 푆푗,푡=푎, = 푆푗 for all values of a. Furthermore, the variance of 푆푗,푡=푎 over all n fish will be constant for each age a, i.e., Var(푆푗,푡=푎) = Var(푆푗). The variations among the values of 푆푗,푡=푎 at different opaque zones, a, were therefore compared using box-whisker plots to determine whether any systematic trend was present.

Homogeneity of variance of the deviations 푆푗,푡=푎 among the different opaque zones a

(a = 1, 2, …, 푛푎) was tested using both Levene’s test and the non-parametric Fligner-Killeen test.

Correlation analysis

A strong and statistically-significant, positive correlation between the values of the deviations for the different fish at each of the ages associated with the formation of the different opaque zones would exist if, as hypothesised, the relative deviation of the size of each otolith from its average size remains constant through the life of the fish. The value of the Pearson correlation coefficient relating the deviations for the individual fish at each pair of opaque zones, 푎1 and 푎2, where 1 ≤ 푎1 < 푛푎 and 푎1 < 푎2 ≤ 푛푎, was calculated and tested using a one-tailed t-test to determine whether it exceeded 0. The extent to which the otolith radius retained constant proportionality as age increased was assessed by considering the trend in the magnitudes (and P-values) of the correlations among the relative deviations at the different pairs of opaque zones.

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For an individual fish, it is recognised that the relative deviations of the otolith radii from the mean at successive ages are repeated measures and thus highly likely to be positively correlated. In this context, a finding of a significant negative correlation for a pair of opaque zones would be considered a demonstration that PORA is invalid for those opaque zones. A positive correlation, however, does not demonstrate that PORA is true, but simply indicates that the data are not inconsistent with the hypothesis of constant proportionality. A highly significant positive correlation may be considered to reflect strong support but not evidence for PORA, i.e., it would provide corroborative evidence for, but not validation of,

PORA.

It is also recognised that the correlations between deviations at different opaque zones are not independent as they relate to data from the same set of otoliths. Results of the statistical comparisons of the correlations, and those of the linear regression analyses, reflect examination of different aspects of the same data and, if in agreement with PORA, should not be considered as providing additional, independent, supporting evidence for PORA. If the results of different analyses suggest that the data fail to support PORA, those analyses simply identify different aspects of deviation from PORA, but do not provide additional evidence that PORA is invalid.

Linear relationships between deviations at different ages

The following equations were fitted by minimizing the negative log-likelihood using

R (R Development Core Team 2011) to describe the relationship between the values of 푆푗,푡=푎 at each pair of ages 푡 = 푎1 and 푎2 (defined as above).

(1) 푆푗,푎2 = 훼1 + 휂푗

(2) 푆푗,푎2 = 푆푗,푎1 + 휂푗

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(3) 푆푗,푎2 = 훼3 + 푆푗,푎1 + 휂푗

(4) 푆푗,푎2 = 훽4푆푗,푎1 + 휂푗

(5) 푆푗,푎2 = 훼5 + 훽5푆푗,푎1 + 휂푗 where α and β are parameters of these five models and η푗 is assumed to be a random variate

2 drawn from a normal distribution, i.e., 휂푗~푁(0, 휎휂 ). The first of these models (Eq. 1) assumes that the relative deviations of the otoliths of fish at age 푡 = 푎2 bear no relationship to the relative deviations for the same fish at age 푡 = 푎1, whereas the second equation (Eq. 2) assumes that, allowing for some random variation, the relative deviations of the otoliths of individual fish at age 푡 = 푎2 are essentially the same as those of the same fish at age 푡 = 푎1.

The third model (Eq. 3) assumes that, again allowing for some random variation, the relative deviations of the otoliths of fish at age 푡 = 푎2 differ by a constant amount to those of the same fish at age 푡 = 푎1, while the fourth model (Eq. 4) assumes that the relative deviations of otoliths of the fish at age 푡 = 푎2 are proportional to, but not equal to, those of the same fish at age 푡 = 푎1. Finally, the fifth model (Eq. 5) is a linear function which assumes that, after accounting for a constant difference and allowing for some random variation, the relative deviations of the otoliths of fish at age 푡 = 푎2 are proportional to those of the same fish at age 푡 = 푎1.

Note again that, for fish j, the radius of the otolith at the age when opaque zone 푎 + 1 becomes delineated, 푅푗,푡=푎+1, is the sum of the radius associated with the delineation of opaque zone a, i.e., 푅푗,푡=푎, and the increment in otolith size between those two opaque zones,

∆푅푗,푡=푎+1. Thus,

푅̅푎 ∆푅푗,푡=푎+1 푆푗,푡=푎+1 = ( ) 푆푗,푡=푎 + . 푅̅푎+1 푅̅푎+1

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For the deviation, 푆푗,푎, to be constant through the life of the fish, it is therefore necessary that

푅̅푎 ∆푅푗,푡=푎+1 = 푅̅푎+1 (1 − ) 푆푗,푡=푎. 푅̅푎+1

Recognising that both 푆푗,푡=푎 and ∆푅푗,푡=푎+1 are subject to error, equations 1–5 explore the extent to which

∆푅푗,푡=푎+1 ∝ 푆푗,푡=푎,

and thus whether, for all a, it is reasonable to conclude that 푆푗,푡=푎+1 = 푆푗,푡=푎 .

For each fitted model, values of the negative log-likelihood and the corrected Akaike

Information Criterion (AICc) were calculated (Hurvich and Tsai 1989). The equation used to calculate AICc was 퐴퐼퐶푐 = 2휆 + 2퐾 + [2퐾 (퐾 + 1)]⁄(푛 − 퐾 − 1), with Akaike

Information Criterion 퐴퐼퐶 = 2휆 + 2퐾 and where λ is the negative log-likelihood, K is the number of parameters and n represents the sample size (Hurvich and Tsai 1989; Anderson and Burnham 2002). For each pair of ages 푡 = 푎1 and 푡 = 푎2, the difference in AICc of model i with respect to the AICc of the model that is best supported by the data, i.e., that with the lowest value of AICc, was calculated as 훥푖(AICc) = AICci − min(퐴퐼퐶푐). To determine the level of support provided by the data for model 푖, the Akaike Weights (AWs) were computed (and subsequently expressed as percentages) as,

푀 훥 (퐴퐼퐶푐) 훥 (퐴퐼퐶푐) 퐴푊 = exp [− 푖 ]⁄∑ exp [− 푚 ] , 푖 2 2 푚=1 where M is the number of candidate models (Burnham and Anderson 2002; Katsanevakis and

Maravelias 2008). The most appropriate of the five models (Eqs 1–5), given model complexity (i.e., number of parameters), was identified as the one with the smallest value of

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AICc, and thus the greatest value of AW. A finding that the relative deviations of otolith radii from their associated means are best described by Eq. 2, Eq. 4 (provided that the 95% confidence interval for 훽4 includes the value 1), or Eq. 5 (provided that the confidence intervals for 훼5 and 훽5 include the values 0 and 1, respectively) has been taken as an indication that the otolith data support PORA, otherwise they support one of the alternative hypotheses. Note again that, for an individual fish, the deviations at successive zones are repeated measures from the same fish, and thus comparisons of the models describing the relationships between deviations for different pairs of opaque zones are not independent.

Linear relationships between annual increment and initial otolith size

The parameters α and β of the following equations were estimated using R (R

Development Core Team 2011) to produce three alternative relationships between the increment in otolith radius ∆푅푗,푡=푎+1 between opaque zones a and a + 1 (where 1 ≤ 푎 ≤ 6) and the radius of the otolith at the first of these zones, 푅푗,푡=푎, i.e.,

∆푅푗,푡=푎+1 = 훼6 + 휂푗 (6)

∆푅푗,푡=푎+1 = 훽7푅푗,푡=푎 + 휂푗 (7)

∆푅푗,푡=푎+1 = 훼8 + 훽8푅푗,푡=푎 + 휂푗 (8)

As with the linear relationships between deviations at different ages, values of the

AICc and AW were calculated for each fitted equation and the alternative models compared.

If PORA is valid, it would be expected that the data would support Eq. 7 more strongly than the other equations. PORA was also considered to be valid if Eq. 8 provided the best description of the relationship, but only if the 95% confidence interval for 훼8 overlapped the value 0. Again, it is noted that the results for different ages are not independent, due to the issue of ‘repeated measures’.

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5.3 RESULTS

5.3.1 HOMOGENEITY OF VARIANCE

The interquartile distances of the deviations 푆푗,푡=푎 at opaque zones a = 2 to 7 were of similar magnitude, but that at opaque zone 1 was considerably greater than at each of the other zones (Fig. 5.1). The variance of the deviations was greatest for the first opaque zone

(i.e., 0.0113) than for subsequent zones, which ranged between 0.0038 and 0.0044 (Fig. 5.1).

Although the variances of the deviations at ages associated with opaque zones 1 to 7 differed significantly (Levene: P < 0.01; Fligner-Killeen: P < 0.01), no such difference was detected between those for opaque zones 2 to 7 (Levene: P = 0.996; Fligner-Killeen: P = 0.997).

5.3.2 CORRELATION ANALYSIS

The correlations between deviations, 푆푗,푡=푎 , for different pairs of opaque zones, i.e., different ages, ranged from 0.34, when comparing deviations for the first and seventh opaque zones, to 0.99, for those between the fifth and sixth opaque zones (Fig. 5.2). All correlations were significantly greater than 0, typically with P < 0.001, but P < 0.01 for comparisons of deviations for opaque zone 1 with those for zones 5 and 6, and P < 0.05 for those for zones 1 and 7 (Fig. 5.2). The magnitude of Pearson’s correlation coefficient increased (and the value of P decreased) as the age associated with the smaller opaque zone of the pair, 푎1, increased, and particularly with decreasing age separation between the zones. This feature was further exemplified for the different pairs of opaque zones by the approximately linear relationship for the plots of deviations for the different fish for the different pairs of opaque zones and by the declining scatter of points about those relationships as the age of the earlier-formed opaque zone of the pair increased (Fig. 5.2), and the number of years separating the compared zones decreased.

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5.3.3 LINEAR RELATIONSHIPS BETWEEN DEVIATIONS AT DIFFERENT AGES

For each of the 21 different pairs of opaque zones, linear models of the forms specified by Eqs 1–5 were fitted to describe the relationship between the relative deviations of the otolith radii at the two zones. Based on the values of AICc and taking model complexity into account, 76% of these relationships were best described by the fourth

equation, i.e., 푆푗,푎2 = 훽4푆푗,푎1 + 휂푗, with AWs > 40% (Tables 5.1a and b; Table S5.1). The values of 훽4 (i.e., the coefficient representing the slope in Eq. 4) were greatest for the relationship between deviations at adjacent opaque zones, e.g., the greatest value of 훽4 was

0.98 for the relationships of the deviations at opaque zones 4 and 6 with those at zones 3 and

5, respectively, but decreased as the separation between opaque zones increased, e.g., the smallest value of 훽4 was 0.20 for the relationship between the deviations at opaque zone 7 with those at zone 1 (Table 5.1b). In those cases for which Eq. 4 provided the best description of the relationships, the 95% confidence intervals for 훽4 encompassed the value 1 only for the comparisons of the relative deviations at opaque zone 6 with those at both zones 3 and 4, and those at zone 7 with those at both zones 4 and 5.

In the remaining 5 of the 21 cases, i.e., those where the relative deviations at zones 4 through 7 were compared with the deviations at the preceding zone and, in the case of zone 5, also with deviations at zone 3, the relationships were best described by the second equation,

i.e., 푆푗,푎2 = 푆푗,푎1 + 휂푗, with AWs > 42% (Tables 5.1a and b; Table S5.1). In these cases, where the data were best described using Eq. 2 rather than Eq. 4, the estimated value of 훽4 for

Eq. 4 exceeded 0.96, i.e., 훽4 ≅ 1, and the resulting relationship is thus essentially of the same form as that of Eq. 2. The increase in model complexity associated with the inclusion of a value of 훽4 that differed from 1 (i.e., Eq.4 rather than Eq. 2) was not justified by the resultant improvement in model fit.

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Thus, based on those cases for which the relative deviations of otolith radii from their associated means were best described by Eq. 2 or Eq. 4 where the 95% confidence interval for the slope encompassed the value 1, the PORA hypothesis appeared to be supported by the otolith data for opaque zones 3 through 6 and 4 through 7. This was not true for the other zones, for which the deviations at the outermost of the pairs of opaque zones appeared to be proportional to those at the younger age, where the constant of proportionality was less than

5.3.4 LINEAR RELATIONSHIPS BETWEEN ANNUAL INCREMENT AND INITIAL

OTOLITH SIZE

For each of the six different pairs of opaque zones, i.e., from ages 푎 to 푎 + 1, where

1 ≤ 푎 ≤ 6, linear models of the forms specified by Eqs 6–8 were fitted to determine which best described the relationship between the increments in otolith radii and the radii of the otoliths prior to those annual growth increments. The relationship between the growth increment between opaque zones 1 and 2 and the radii of the otoliths at zone 1 was best described by Eq. 8, where ∆푅푗,푡=푎+1 = 0.28 − 0.23푅푗,푡=푎 + 휂푗 with AW = 98% (Table 5.2;

Table S5.2). That is, fish with larger otolith radii at opaque zone 1 have smaller otolith increments between opaque zones 1 and 2 than fish with smaller radii, and vice versa.

In the case of the increment between opaque zones 2 and 3 and the radius of the otolith at zone 2, the relationship was best described by Eq. 6, i.e., ∆푅푗,푡=푎+1 = 0.09 + 휂푗 with AW = 72%, indicating that otolith increments between these two opaque zones were essentially random and unrelated to initial otolith size.

The values of AICc for the relationships between increments from opaque zones 3 through 6 (to the subsequent opaque zones) and the radii of the otoliths at the opaque zones

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prior to those growth increments were best described by the seventh equation, i.e.,

∆푅푗,푡=푎+1 = 훽7푅푗,푡=푎 + 휂푗, with AWs > 51% (Table 5.2; Table S5.2). The values of the

AICcs indicated that the improvement in fit obtained by using Eq. 8 rather than Eq. 7 did not justify the inclusion of the additional parameter.

Although the annual increments for opaque zones 3 through 6 (to the subsequent opaque zones) were found to be proportional to the initial radius, thus supporting PORA, the increment data between zones 1 and 2 and between 2 and 3 provided little support for the hypothesis of constant proportionality.

5.4 DISCUSSION

This study explored the hypothesis that, for individuals of a selected year class (i.e., age 7 fish) in a sample collected from an estuarine population of A. butcheri, the ratio of the size of the otolith relative to the average size of the otoliths for fish of the same age remains constant throughout life. The results indicate that, for A. butcheri collected in the Wellstead

Estuary in May 2013, the hypothesis of constant proportionality of these ratios with age for individual fish was invalid as the ratio varied markedly throughout the life of individual fish, particularly when comparing the ratios at the ages associated with the first and subsequent opaque zones. As age increased (from 3 years onward) and the separation between opaque zones decreased, however, the ratios of otolith radius for each fish relative to mean radius did become approximately constant, and thus consistent with PORA for the ages associated with opaque zones 3 through 6 and 5 through 7. These findings are discussed in greater detail below.

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5.4.1 OTOLITH GROWTH AT FIRST OPAQUE ZONE

The finding that the variance of the relative deviations of otolith size from the mean size at the first opaque zone in the otoliths of A. butcheri significantly exceeded the variances of those deviations at subsequent zones is not consistent with the hypothesis of constant proportionality of otolith radius with age throughout the entire life of the individuals of this species. There is, however, no evidence that the variances at subsequent opaque zones, i.e., 2 to 7, differed. The greater variation among relative deviations of otolith sizes of fish at the first opaque zone than at other zones may be related to differences, among individuals, in the timing of hatching (females of A. butcheri are multiple spawners and their spawning period in the Wellstead Estuary extends from early to late spring) relative to the time of year at which the first zones became delineated (Sarre and Potter 1999) or variation among growth rates of otoliths of individuals in their first year of life. Such differences among growth rates of otoliths within the first year of life may reflect variation in the water temperature, the main environmental factor found to affect the regulation of calcification rate of otoliths, to which different individuals are exposed (e.g., Mosegaard et al. 1988; Lombarte and Lleonart 1993;

Folkvord et al. 2004).

Further evidence of inconsistency between the relative deviations of otolith sizes of fish at the first and subsequent opaque zones was provided by the findings of the regression analyses. Although the relative deviations of the otolith radii at opaque zone 1 were proportional to those at subsequent opaque zones, the constant of proportionality invariably differed from 1, and the best description of the relationship between the increment in otolith size between opaque zones 1 and 2 and the otolith radius at the first opaque zone indicated that increments had a negative linear relationship with initial otolith sizes. The data for 7- year-old A. butcheri from the Wellstead Estuary thus provide little support for the hypothesis of constant proportionality with age between age 1 and subsequent ages.

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5.4.2 OTOLITH GROWTH AT SECOND OPAQUE ZONE

The finding of a negative linear relationship between the increment between opaque zones 1 and 2 and the initial sizes of the associated opaque zones at zone 1 suggests the existence of a compensatory growth process for otoliths of individuals of A. butcheri of this age. That is, the annual increments in size exhibited by otoliths of larger than average size are less than those exhibited by smaller otoliths, thus providing, at least to some extent, regression towards mean otolith size at opaque zone 2 and reducing the variance of relative deviations exhibited at opaque zone 1 to that exhibited at subsequent opaque zones. It is possibly pertinent to note that structural growth (i.e., body size) compensation, i.e., where accelerated growth arises to attain partial (or full) recovery from a period of poor growth, has been found to occur in individuals of several species following a period of resource deprivation (e.g., Sæther and Jobling 1999; Morgan and Metcalfe 2001; Zhu et al. 2001;

Johnsson and Bohlin 2006) or exposure to predation (e.g., McCormick and Hoey 2004).

Because otolith growth is closely related to metabolic rate (e.g., Mosegaard et al. 1988;

Wright 1991; Fey 2005), these same factors may also induce elevated compensatory growth in the otoliths following the first year of growth for individuals of A. butcheri in the population of the Wellstead Estuary.

Although the variance of the relative deviations of the 7-year-old A. butcheri at opaque zone 2 did not differ from those at subsequent opaque zones and the correlation between these deviations and those at subsequent opaque zones were positive and statistically significant, the results of the regression analyses provided little further support for the hypothesis that relative deviations at this zone maintained constant proportionality with those at subsequent opaque zones. The finding that the increment between opaque zones 2 and 3 increased by a random and constant amount, unrelated to initial otolith size at opaque zone 2

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is possibly consistent with the suggestion of a compensatory process associated with otolith growth for individuals in the first few years of life. This last result, however, suggests that these increments might best be described as random variates from a normal distribution, unrelated to previous otolith sizes, thus reflecting the processes of continuous calcification and organic matrix deposition involved in the formation of opaque zones within otoliths and entrained by photoperiod and diurnal rhythmicity (Mugiya 1987; Wright et al. 1992). Thus, although possibly consistent with a compensatory mechanism, the appearance of compensatory otolith growth of young A. butcheri may simply be a consequence of random chance and regression towards the mean.

5.4.3 OTOLITH GROWTH AT THIRD AND SUBSEQUENT OPAQUE ZONES

Statistically significant, positive correlations between the relative deviations of the radii of otoliths of 7-year-old individuals of A. butcheri from their means at different opaque zones confirmed that, although not necessarily constant, the expected relationships between these deviations persisted throughout their lives. As the age at which the opaque zones were formed increased and as the number of years separating the ages at which those zones formed decreased, the positive correlation between the relative deviations strengthened. This finding was confirmed by the results of fitting the linear models relating the relative deviations at different pairs of opaque zones and those relating the annual increments to initial otolith size.

Such a trend of increasing correlation is expected as the radius of an otolith at an opaque zone is determined by its radius at the preceding opaque zone and the (relatively smaller) increment in radius between those two zones, whether or not that annual increment is random or proportional to the initial size at the first of those zones. The latter distance becomes an increasingly smaller proportion of otolith radius as the number of opaque zones increases, thereby increasing the consistency between relative deviations at successive opaque zones at

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older ages. This provides the probable explanation for the observed increase in correlation between relative deviations for the 7-year-old A. butcheri from the Wellstead Estuary as the number of opaque zones increased and the number of years separating the opaque zones declined.

The results of the linear regression analysis of the relationships between annual increment in otolith radius and initial radius for 7-year-old A. butcheri from the Wellstead

Estuary indicated that, from the third and subsequent opaque zones, the annual increments were not random but appeared to be proportional to the initial size. This suggests that, regardless of whether otolith size relative to the mean for fish of the same age remains constant, faster (or slower) growing otoliths of individual A. butcheri are likely to remain faster (or slower) growing, at least through ages 3 through 6 years. Mixed effects models, such as that developed by Weisberg et al. (2010), assume, through inclusion of a random effect for different fish, that the growth rate of otoliths of individual fish relative to that of other fish will persist throughout life, e.g., faster growing otoliths continue to grow faster throughout life. Although such an assumption may be valid for individuals of A. butcheri older than 3 years, the results of the current study suggest that this assumption would be invalid for younger fish.

5.4.4 IMPLICATIONS

Although the characteristics of the sectioned otoliths of A. butcheri made them well suited for use in this study, it should be recognised that each population of this species is confined to its natal estuary for life (Chaplin et al. 1998). Environmental conditions within an estuary are typically more variable than those in adjacent marine waters, and vary seasonally along a gradient with distance from the estuary mouth (e.g., Loneragan et al. 1989; Young et al. 1997; Chaplin et al. 1998). Although constraining the data employed in this study to a

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single year class removed the effect of inter-annual variation in the growth conditions experienced by the fish, annual growth of individuals is still likely to have been affected by movements of those fish within the estuary and non-random differences in their spatial distribution throughout the estuary, coupled with variation of that distribution at different ages (Elsdon and Gillanders 2005; Smith 2006; Hindell et al. 2008). Such differences would have increased the variability of the relative increments in otolith size exhibited between successive growth zones by individual fish (particularly for the first year of life) and influenced the results of the analyses. It would be informative to repeat this study using data for marine species with different life history patterns and which exhibit little migratory behaviour. For individuals of such species, the relative deviations of otolith radii from the mean radius for fish of the same age are likely to be more similar throughout life than those for A. butcheri.

Interestingly, A. butcheri was one of the three (of the six) species studied for which deviations from the somatic and otolith growth curves fitted to lengths and otolith sizes at capture were found to exhibit a significant positive correlation (Chapter 3). If deviations of otolith size from expected values at age are proportional for older individuals (i.e., > 3 years) of this species, it might then be expected that deviations of fish length from expected values at age would also be proportional later in life, i.e., faster growing fish remain faster growing at least for those older ages. Furthermore, and similarly to otolith growth where increasing consistency at consecutive ages followed greater variability at the first opaque zone, it is also possible that a lack of proportionality throughout life in the deviations of fish length from expected values at age may occur due to greater length variations at younger ages and subsequent compensatory growth. This may consequently raise issues for back-calculation of lengths at young ages for A. butcheri, as proportionality-based approaches generally rely on constant proportionality and are based on the hypothesis that, throughout life, the length of an

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individual fish remains proportional to expected length given otolith size, or vice versa

(Whitney and Carlander 1956; Francis 1990). The insertion of a biological intercept into the back-calculation approach may be one way to mitigate this potential problem (Campana

1990). Growth compensation coupled with greater variation at the first opaque zone may, however, lessen the strength of the fish length-otolith size relationship at younger ages. It would thus be appropriate in future studies to extend the validation of back-calculation approaches to the youngest ages.

The possibility that growth compensation has occurred in the otoliths of younger individuals from a single year class of A. butcheri is of considerable interest as such compensation is of major importance for life trait studies. Studies involving a greater number of year classes for a broader range of species should be undertaken to determine whether their otoliths exhibit similar patterns of apparent growth compensation.

Proportionality of otolith radius at age

In combination, the findings relating to the homogeneity of the variances of the relative deviations at opaque zones 2 to 7, the statistically significant positive correlations between relative deviations at different opaque zones, and the results of the comparisons of the linear models relating both the relative deviations at different opaque zones and the annual increments with initial otolith size failed to demonstrate that, for 7-year-old

A. butcheri from the Wellstead Estuary, the relative deviations of otolith radii from their mean remained approximately constant through successive ages. Although the data supported the PORA hypothesis for the relative deviations of the otoliths at some groups of opaque zones, the deviations for the opaque zones formed, particularly at younger ages, departed from that hypothesis. Following an extensive search of the literature, it appears that the current study is one of the first attempts to explore in detail how, for a selected species, the

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deviations of the sizes of otoliths of individual fish relative to the mean size of otoliths for fish of the same age vary throughout life.

Acknowledgements

This research was funded by Murdoch University and Recfishwest, and conducted in accordance with conditions in permit R2561/13 issued by the Murdoch University Animal

Ethics Committee. Gratitude is expressed to P. Coulson for his skilled assistance in the laboratory and as second reader of otolith zone counts, and to J. Williams and D. Yeoh for assisting in the collection of A. butcheri.

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Table 5.1a. Corrected Akaike Information Criteria (AICc) for the fitted models (Eqs 1-5) describing the relationships between the

deviations, 푆푗,푎, at each pair of ages 푎1 and 푎2, and, where present in the equations, the slopes (β) of those models.

Age 풂ퟐ of Age 풂ퟏ of AICc 1 AICc 2 AICc 3 AICc 4 AICc 5 dependent independent variable, 푺 variable, 푺 풋,풂ퟐ 풋,풂ퟏ 푺풋,풂ퟐ = 휶ퟏ + 휼풋 푺풋,풂ퟐ = 푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휶ퟑ + 푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휷ퟒ푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휶ퟓ + 휷ퟓ푺풋,풂ퟏ + 휼풋 휷4 휷5

0.52 0.52 2 1 -133.87 -140.53 -138.52 -194.58 -192.35 (0.42-0.61) (0.42-0.62) 0.40 0.40 3 1 -140.55 -118.10 -116.06 -173.33 -171.08 (0.28-0.52) (0.28-0.52) 0.88 0.88 3 2 -140.55 -250.18 -248.03 -254.71 -252.46 (0.79-0.97) (0.79-0.97) 0.32 0.32 4 1 -138.23 -103.69 -101.61 -156.06 -153.81 (0.18-0.46) (0.18-0.46) 0.81 0.81 4 2 -138.23 -201.36 -199.20 -205.83 -203.59 (0.67-0.96) (0.67-0.96) 0.98 0.98 4 3 -138.23 -276.72 -274.56 -274.76 -272.51 (0.90-1.06) (0.90-1.06) 0.26 0.26 5 1 -139.63 -95.88 -93.79 -150.87 -148.62 (0.11-0.41) (0.11-0.41) 0.74 0.74 5 2 -139.63 -182.33 -180.17 -189.55 -187.30 (0.58-0.90) (0.57-0.91) 0.93 0.93 5 3 -139.63 -239.80 -237.63 -239.34 -237.10 (0.82-1.04) (0.82-1.04) 0.97 0.97 5 4 -139.63 -319.62 -317.46 -318.88 -316.63 (0.92-1.02) (0.92-1.02) Note. Bold font indicates the model that best describes the data. 95% confidence intervals of the slopes (β) are indicated between parentheses.

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Table 5.1b. Corrected Akaike Information Criteria (AICc) for the fitted models (Eqs 1-5) describing the relationships between the

deviations, 푆푗,푎, at each pair of ages 푎1 and 푎2, and, where present in the equations, the slopes (β) of those models.

Age 풂 of Age 풂 of ퟐ ퟏ AICc 1 AICc 2 AICc 3 AICc 4 AICc 5 dependent independent variable, 푺풋,풂ퟐ variable, 푺풋,풂ퟏ 휷4 휷5 푺풋,풂ퟐ = 휶ퟏ + 휼풋 푺풋,풂ퟐ = 푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휶ퟑ + 푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휷ퟒ푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휶ퟓ + 휷ퟓ푺풋,풂ퟏ + 휼풋

0.23 0.23 6 1 -141.11 -92.88 -90.79 -150.02 -147.77 (0.08-0.38) (0.08-0.38) 0.69 0.69 6 2 -141.11 -174.62 -172.46 -183.82 -181.57 (0.52-0.87) (0.52-0.87) 0.89 0.89 6 3 -141.11 -223.76 -221.60 -224.85 -222.60 (0.76-1.01) (0.76-1.01) 0.94 0.94 6 4 -141.11 -283.28 -281.12 -284.04 -281.79 (0.87-1.01) (0.87-1.01) 0.98 0.98 6 5 -141.11 -367.56 -365.40 -367.04 -364.79 (0.95-1.01) (0.95-1.01) 0.20 0.20 7 1 -141.87 -89.34 -87.25 -148.36 -146.12 (0.05-0.35) (0.04-0.35) 0.64 0.64 7 2 -141.87 -164.44 -162.29 -175.77 -173.53 (0.45-0.82) (0.45-0.83) 0.83 0.82 7 3 -141.87 -202.18 -200.02 -205.26 -203.02 (0.67-0.98) (0.67-0.98) 0.89 0.89 7 4 -141.87 -241.28 -239.12 -243.42 -241.17 (0.79-1.00) (0.79-1.00) 0.94 0.94 7 5 -141.87 -281.72 -279.56 -281.97 -279.72 (0.87-1.02) (0.87-1.02) 0.97 0.97 7 6 -141.87 -313.90 -311.74 -312.73 -310.48 (0.92-1.03) (0.92-1.03)

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Note. Bold font indicates the model that best describes the data. 95% confidence intervals of the slopes (β) are indicated between parentheses.

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Table 5.2. Corrected Akaike Information Criteria (AICc) for the fitted models (Eqs 6-8) describing the relationships between the increment in

otolith radius ∆푅푗,푡=푎+1 between opaque zones a and a + 1 and the radius of the otolith at the first of these zones 푅푗,푡=푎, and, where present in the equations, the intercepts (α) and slopes (β) of those relationships.

Age 풂 + ퟏ of Age 풂 of

dependent independent AICc 6 AICc 7 AICc 8 ∝6 ∝8 휷7 휷8 variable, ∆푹풋,풕=풂+ퟏ variable, 푹풋,풕=풂 ∆푹풋,풕=풂+ퟏ = 휶ퟔ + 휼풋 ∆푹풋,풕=풂+ퟏ = 휷ퟕ푹풋,풕=풂 + 휼풋 ∆푹풋,풕=풂+ퟏ = 휶ퟖ + 휷ퟖ푹풋,풕=풂 + 휼풋 0.18 0.28 0.44 -0.23 2 1 -240.96 -196.13 -248.97 (0.18-0.19) (0.22-0.34) (0.42-0.47) (-0.37-0.09) 0.09 0.08 0.16 0.02 3 2 -294.31 -287.66 -292.22 (0.09-0.10) (0.02-0.14) (0.15-0.16) (-0.08-0.12) 0.08 0.02 0.12 0.09 4 3 -298.61 -303.05 -301.09 (0.08-0.08) (-0.04-0.08) (0.11-0.12) (0.01-0.18) 0.07 0.03 0.10 0.06 5 4 -331.53 -334.43 -333.82 (0.07-0.08) (-0.02-0.07) (0.09-0.10) (0.00-0.12) 0.06 0.02 0.07 0.05 6 5 -366.51 -373.85 -372.91 (0.06-0.06) (-0.01-0.05) (0.07-0.08) (0.02-0.09) 0.08 0.03 0.09 0.06 7 6 -310.52 -313.62 -312.45 (0.08-0.09) (-0.03-0.08) (0.09-0.10) (0.00-0.12)

Note. Bold font indicates the model that best describes the data. 95% confidence intervals of the intercepts (α) and slopes (β) are indicated

between parentheses.

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Figure 5.2. Scattergrams of the deviations for individual fish at each pair of opaque zones (lower left triangle), histograms of deviations at each opaque zone (diagonal), and the correlations (and associated P-values) for the different pairs of opaque zones of individual fish (upper right triangle).

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Supplemental materials of Chapter 5

Age 풂ퟐ of Age 풂ퟏ of dependent independent AW1 AW2 AW3 AW4 AW5 variable, 푺풋,풂ퟐ variable, 푺풋,풂ퟏ 푺풋,풂ퟐ = 휶ퟏ + 휼풋 푺풋,풂ퟐ = 푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휶ퟑ + 푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휷ퟒ푺풋,풂ퟏ + 휼풋 푺풋,풂ퟐ = 휶ퟓ + 휷ퟓ푺풋,풂ퟏ + 휼풋 2 1 0.00 0.00 0.00 0.75 0.25 3 1 0.00 0.00 0.00 0.75 0.25 3 2 0.00 0.07 0.02 0.68 0.22 4 1 0.00 0.00 0.00 0.75 0.25 4 2 0.00 0.07 0.02 0.68 0.22 4 3 0.00 0.54 0.18 0.20 0.07 5 1 0.00 0.00 0.00 0.75 0.24 5 2 0.00 0.02 0.01 0.73 0.24 5 3 0.00 0.42 0.14 0.33 0.11 5 4 0.00 0.44 0.15 0.31 0.10 6 1 0.01 0.00 0.00 0.75 0.24 6 2 0.00 0.01 0.00 0.75 0.24 6 3 0.00 0.28 0.09 0.48 0.15 6 4 0.00 0.31 0.10 0.45 0.14 6 5 0.00 0.42 0.14 0.33 0.11 7 1 0.03 0.00 0.00 0.73 0.24 7 2 0.00 0.00 0.00 0.75 0.25 7 3 0.00 0.13 0.05 0.62 0.20 7 4 0.00 0.19 0.07 0.56 0.18 7 5 0.00 0.35 0.12 0.40 0.13 7 6 0.00 0.48 0.16 0.27 0.09

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Note. Bold font identifies the model that best describes the data. The intercepts (훼1, 훼3 and 훼5) and the slopes (훽4 and 훽5) are the parameters of the associated models, and 휂푗 is the random normal variate for fish j.

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AW6 AW7 AW8 Age 풂 + ퟏ of Age 풂 of dependent independent ∆푹풋,풕=풂+ퟏ = 휶ퟔ + 휼풋 ∆푹풋,풕=풂+ퟏ = 휷ퟕ푹풋,풕=풂 + 휼풋 ∆푹풋,풕=풂+ퟏ = 휶ퟖ + 휷ퟖ푹풋,풕=풂 + 휼풋 variable, ∆푹풋,풕=풂+ퟏ variable, 푹풋,풕=풂 2 1 0.02 0.00 0.98 3 2 0.72 0.03 0.25 4 3 0.07 0.67 0.25 5 4 0.12 0.51 0.37 6 5 0.02 0.61 0.38 7 6 0.12 0.57 0.31

Note. Bold font identifies the model that best describes the data. The intercepts (훼6 and 훼8) and the slopes (훽7 and 훽8) are the parameters of the associated models, and 휂푗 is the random normal variate for fish j.

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171

CHAPTER 6 – THE EFFECTS OF LENGTH-DEPENDENT FISHING

MORTALITY ON RELATIONSHIPS BETWEEN FISH LENGTH, OTOLITH

SIZE AND AGE AT CAPTURE

Abstract

In fished populations, it is recognised that size-selective fishing mortality influences fish length at age, such that fast growing young fish and slow growing old fish are overrepresented in catch samples. This study employed simulated samples drawn from populations with different patterns of length-dependent selectivity and fishing mortalities to explore the effects of length-dependent fishing mortality on the relationship between fish length, otolith size and age at capture for six species with different biological characteristics.

The findings demonstrate that, due to the cumulative effect of fishing mortality on survival, the mean age at which a given fish length or otolith size is attained decreases with increasing length-dependent fishing mortality, with the effect becoming greater as fish lengths or otolith sizes increase. Similarly, mean length for fish with otoliths of a given size decreases with increasing fishing mortality with the effect increasing as otolith sizes of fish become larger.

Likewise, but to a lesser extent, mean otolith sizes for a given fish length decrease with increasing fishing mortality, with the effect becoming greater as fish became larger. Mean otolith sizes at age of younger fish of the studied species appeared little affected by the increase in the size at which fish became selected by the fishing gear. Although otolith size at age predicted by fitted growth curves for older fish appeared little affected by increasing fishing mortality, predicted fish lengths at age of older fish and fish lengths at otolith size of fish with larger otoliths decreased with increasing fishing mortality. Lastly, deviations from otolith growth curves of individual fish supported the hypothesis that fish with faster growing otoliths are caught at younger ages and, as a consequence of the cumulative effects of length-

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dependent fishing mortality, fish with slower growing otoliths are caught at older ages. As noted in earlier studies, the effect of length-dependent fishing mortality on the catches taken from a population and its cumulative effect on the composition of growth characteristics of fish that survive to older ages need to be recognised when fitting somatic growth curves to lengths at age of a random sample drawn from the catch. The current study has demonstrated that the effect of such mortality on otolith growth models and on relationships between fish length and otolith size should also be considered when undertaking back-calculation analyses.

Improved knowledge of the relationship between fish length and otolith size, in combination with growth curves relating fish lengths and otolith sizes with ages at capture and data on annual increments of the sizes of otoliths of individual fish, may facilitate the development of approaches to account for the various effects of changes in length-dependent fishing mortality.

6.1 INTRODUCTION

The age and length compositions of fish collected from catches made by size-selective fishing gear are typically not representative of the corresponding compositions of the fish populations from which the catches were taken due to the limited age and size ranges of the fish that are caught (Smale and Taylor 1987; Berkeley et al. 2004; Hsieh et al. 2010). Thus, for example, Lucena and O’Brien (2001) demonstrated that, in southern Brazil, the gill nets used by commercial fishers caught faster-growing young fish and slower-growing old

Pomatomus saltatrix from the stock. Although selectivity of fishing gear is a function of fish length, not otolith size (Francis 1990), the marginal distributions of both fish lengths and otolith sizes of fish caught by the gear will be influenced by length-dependent fishing mortality due to the relationship between fish length, otolith size, and age. As, for fish of the same age, otolith sizes are likely to be positively correlated with fish sizes (e.g., Hare and

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Cowen 1995), the effect of length-dependent fishing mortality on otolith size may be similar to its effect on fish length.

In addition to the direct effect of size-dependent fishing mortality on the sizes of the fish that are caught, with over-representation of fast-growing young fish in catches, the cumulative effects of such mortality include both a reduction in the numbers of older fish in the population and over-representation of slow-growing individuals among those older fish

(e.g., Kristiansen and Svåsand 1998; Taylor et al. 2005; Kendall and Quinn 2012). Taylor et al. 2005 have demonstrated that, as a result of such bias, the von Bertalanffy growth curve fitted to lengths at age of individuals collected from an exploited population has a greater growth coefficient (K) and lower asymptotic length (퐿∞) than would be obtained if the same growth curve had been fitted to data from the same population when it had experienced less size-dependent fishing mortality. Because, for some species, deviations of fish length from the somatic growth curve and otolith size from the otolith growth curve are correlated

(Chapter 36), it is reasonable to postulate that individuals of these species with faster growing otoliths will be over-represented in catches of young fish and, because of the cumulative effects of size-dependent fishing mortality on otolith size, slower growing otoliths will be over-represented in catches of older fish. Thus, both length and otolith size at age of older fish are expected to decrease in response to increased levels of length-dependent fishing mortality. Furthermore, as gear selectivity is length- but not otolith size-dependent, it might be expected that, for older fish, the cumulative effect of increasing fishing mortality on otolith size at age is less than the effect on fish length at age (Francis 1990). Consequently, the length-otolith size relationship would show a reduction in the lengths of those fish with

6 Ashworth, E. C., N. G. Hall, S. A. Hesp, P. G. Coulson, and I. C. Potter. Age and growth rate variation influence the functional relationship between somatic and otolith size. Can. J. Fish. Aquat. Sci. In press.

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larger otolith sizes. For older fish, as the mean age for fish of a given length or given otolith size would be reduced as fishing mortality increased, it would therefore also be expected that the mean length of fish associated with a given otolith radius and mean otolith radius at a given fish size would also decrease.

The overall objective of this study was to explore the effect of length-dependent fishing mortality on the relationship between fish length and otolith size. To achieve this objective, simulated samples were drawn from catches taken with different patterns of length- dependent selectivity and different levels of fishing mortality from populations with different growth characteristics. These samples were analysed to explore the following hypotheses: (1) because of the cumulative effect of fishing mortality on survival, the mean age at which a given fish length or otolith size is attained decreases with increasing levels of length- dependent fishing mortality; (2) for the same level of fishing mortality, as otolith sizes at age of fish are correlated with their lengths, the mean otolith sizes at age of younger fish increase as their selectivity is reduced, i.e., as the mean length at which they are selected by the fishing gear increases; (3) for fish with larger otoliths, the mean length at a given otolith size decreases with increasing fishing mortality; (4) the mean otolith size at a given length decreases with increasing fishing mortality with the effect increasing as fish become larger;

(5) predicted fish lengths and otolith sizes at age of older fish, and predicted fish lengths at otolith size for fish with larger otoliths decrease with increasing fishing mortality; (6) deviations of the otolith sizes from the otolith growth curves used in generating the simulated samples are consistent with the hypothesis that, as a consequence of length-dependent selectivity and the cumulative effects of fishing mortality, fish with faster growing otoliths are caught at younger ages and those with slower growing otoliths at older ages. The effects of fishing mortality and length-dependent selectivity on the relationships between predicted

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lengths and otolith radii were explored for six species with different biological characteristics and the implications for back-calculation were considered.

6.2 MATERIAL AND METHODS

6.2.1 SELECTED SPECIES FOR THE SIMULATION

The species employed in this simulation study were Black Bream, Acanthopagrus butcheri (Munro 1949); Mulloway, Argyrosomus japonicus (Temminck and Schlegel 1843);

Foxfish, Bodianus frenchii (Klunzinger 1880); Breaksea Cod, Epinephelides armatus

(Castelnau 1875); Goldspotted Rockcod, Epinephelus coioides (Hamilton-Buchanan 1822) and West Australian Dhufish, Glaucosoma hebraicum Richardson, 1845. Maximum ages and total lengths of these species ranged from ~ 19 to 78 years and from ~ 480 to 2 000 mm, respectively (Table 6.1).

6.2.2 SIMULATION

The samples used in this study were drawn from simulated catches taken from populations of the above six species, thereby exploring the effects of fishing mortality and selectivity for species with different somatic and otolith growth characteristics and different instantaneous rates of natural mortality. To simplify the analyses, females and males of each population were assumed to possess common somatic and otolith growth curves (and bivariate distribution of deviations from those curves), and common instantaneous rate of natural mortality, M, and to exhibit the same selectivity when exposed to fishing mortality.

The somatic and otolith growth of individuals from each population were assumed to be described by a bivariate growth model of the form developed in Chapter 3.

The parameters of the growth curves, an estimate of instantaneous rate of natural mortality, and arbitrarily-selected values spanning biologically-feasible ranges of the

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parameters of the two logistic selectivity functions for each species, which were used as input to the simulations, are presented in Table 6.2. The lower and upper reference levels of age, i.e., 휏1 and 휏2, and the parameters and forms of the fitted somatic and otolith growth curves, and bivariate distribution of deviations of observed lengths and otolith radii at age from those fitted growth curves, were those that were determined and reported in Chapter 3 (Table 6.2).

The simulated population of each species was assumed to have attained a stable state, with constant annual recruitment and stable age and length compositions while experiencing one of three specified levels (0.1, 0.4, and 0.8 year-1) of constant fishing mortality F (for fully-selected fish of each species), with one of two arbitrarily-specified length-dependent selectivity patterns, specific to the species. The larger values of fishing mortality were chosen to reflect the effects of overfishing (i.e., F > M) and extreme overfishing (i.e., F >> M), thus producing greater contrast and allowing the effects of increased levels of fishing mortality to be determined more readily. Note that the values of fishing mortality and length-dependent selectivity were arbitrarily selected to provide contrast in the simulation, and are not those experienced by the populations from which the samples studied in Chapter 3 were drawn.

The logistic curves describing the two selectivity scenarios for each species had a common slope but differed in the length (퐿50) at which selectivity attained a value of 0.5. That is, a common length difference separated 퐿50 from the length 퐿95, at which selectivity for individuals of that species was 0.95, e.g., for A butcheri, the values of 퐿50 for the two selectivity patterns, i.e., 200 and 300 mm, were separated from the values of their respective

퐿95s by 125 mm (Table 6.2).

The following procedure, which was undertaken using R (R Development Core Team

2011), was employed to simulate the drawing of a random fish from the catch of each species

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for the selected value of fully-selected fishing mortality and pattern of length-dependent selectivity.

1) A pair of deviations from the somatic and otolith growth curves, which was

assumed to characterise the growth of an individual fish and to remain constant

throughout the life of that fish, was randomly drawn from the bivariate

distribution of such deviations, where the form of that bivariate distribution, i.e.,

either bivariate normal, bivariate normal-lognormal, or bivariate lognormal, was

that used in the bivariate growth model for that population (Chapter 3).

2) The current age of the fish 푡0 was set to 휏1, the lower of the two reference ages

used when fitting the bivariate growth model (Table 6.2).

3) Following Hampton and Majkowski (1987), the age 푡1 at which the simulated

individual experienced natural mortality or encountered the fishing gear was

calculated as 푡1 = 푡0 − ln(1 − 푟1)⁄푍, where 푟1 is a uniformly-distributed random

number in the range from 0 to 1, i.e., 푟1~U(0,1), and Z is the instantaneous rate of

total mortality for fully-selected fish. The latter variable was calculated as

Z = M + F. If 푡1 − 푡0 exceeded an arbitrarily small time step (in this study,

1/52 year-1), it was assumed that the fish did not experience natural mortality or

encounter the fishing gear during the time step, and thus the current age 푡0 was

incremented by the time step, and step 3 was repeated.

4) A uniformly-distributed random number 푟2 in the range from 0 to 1, i.e.,

푟2~U(0,1), is then drawn, and compared with the ratio 푀⁄푍. If 푟2 ≤ 푀⁄푍, it is

assumed that the fish died of natural causes at age 푡1, otherwise it is assumed that

it encountered the fishing gear. If the fish died from natural causes, the simulation

procedure returns to step 1 to draw a new fish from the population.

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5) If it was concluded at step 4 that the fish encountered the fishing gear, the

expected length and otolith radius for a fish of age 푡1 were calculated using the

somatic and otolith growth curves of the bivariate growth model (for details of the

model and calculation of expected length and otolith radius, refer to Chapter 3).

6) The length and otolith radius of the simulated individual at age 푡1 were then

calculated by adjusting the expected length and otolith radius using the pair of

deviations determined at step 1.

7) The selectivity 푆퐿 of a fish with a total length 퐿푇 (mm) equal to that of the

simulated individual at age 푡1 (calculated at step 6) was then calculated from the

logistic equation (e.g., Punt and Kennedy 1997)

1 푆퐿 = 1 + exp[− log(19) (퐿푇 − 퐿50)/(퐿95 − 퐿50)]

where 푆퐿 is the probability that a fish of total length 퐿푇 is caught if it encounters

the fishing gear, and the total lengths at which 50 and 95% of fish are expected to

be retained are denoted by 퐿50 and 퐿95, respectively.

8) A uniformly-distributed random number 푟3 in the range from 0 to 1, i.e.,

푟3~U(0,1), was then drawn and compared with 푆퐿. If 푟3 > 푆퐿, it was assumed that

the fish evaded the gear. In this case, the current age 푡0 was reset to 푡1, and the

simulation procedure in steps 3 to 8 repeated.

9) If 푟3 ≤ 푆퐿 at step 8, it was assumed that the fish was caught, the count of fish in

the sample was incremented and the details of the age at capture, 푡1, the current

length and otolith size of the fish, and the deviations from the somatic and otolith

growth curves, which were assumed to characterise the growth of that fish, were

recorded. If the number of fish in the sample had not yet reached the required

sample size, the simulation described in steps 1 to 9 was repeated.

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10) When the required sample size was attained, the simulation ceased and the

sample was subjected to the required analysis.

6.2.3 SIMULATION SCENARIOS

The simulation procedure described above was employed to generate samples of

50 000 fish for each species for each of six scenarios. The scenarios investigated for each species comprised each of the three fully-selected fishing mortalities, i.e., F = 0.1, 0.4 and

-1 0.8 year , coupled with each of the two selectivity curves for that species (parameters a, b, 푦1 and 푦2 of the somatic and otolith growth curves for each of the three fishing mortalities for each species are listed in Tables 6.3, 6.4 and 6.5). Values of 퐿50 and 퐿95 for each of the two selectivity scenarios for each species are listed in Table 6.2, where the pair of smaller values represent the values of 퐿50 and 퐿95 for one selectivity scenario, and the pair of larger values represent the parameters for the second selectivity pattern, i.e., that with lower selectivity for smaller fish.

Plots of age compositions, length compositions, and distributions of otolith radius of the fish in the simulated samples for the different species were examined to confirm (and thus provide ‘face validity’) that, consistent with expectation, the distributions became increasingly skewed towards smaller values, with increasing truncation of larger values as fishing mortality increased, and that the relative numbers of younger or smaller fish present in the samples were reduced when the 퐿50 of the selectivity curve was increased.

6.2.4 ANALYSES

The mean age, and associated 95% confidence limits, of the fish within each of 1 000 length slices, i.e., length classes, were calculated for each of the samples generated for each species using each of the three fishing mortalities in combination with the selectivity curve

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with the lower value of 퐿50 for that species. The interval of the length classes was determined by dividing the maximum length of the fish in the sample by 1 000. The cumulative effect of fishing mortality on mean age at a given length for each species was then compared visually among the different mortalities by plotting the mean ages against the associated length class midpoints. The large size of the simulated samples (50 000 individuals), was necessary to ensure that sufficient individuals were contained in each slice such that, in the main range of data for this and subsequent analyses, the signal in plots of the resulting estimates of means was not overwhelmed by noise resulting from a paucity of individuals in the various slices.

To examine the effects of fishing mortality on mean age at a given length in greater detail, the length ranges of the samples for the three different fishing mortalities were divided into length classes, with common length midpoints and a length interval calculated by dividing the length range of the sample for the greatest fishing mortality by 200. For the resulting length classes within each sample ≥ 10 fish, the mean and associated standard error

(SE) were calculated. Differences between the mean ages of fish in corresponding length classes within the samples for fishing mortalities of both 0.4 and 0.8 year-1 and the mean ages of the fish from the sample produced using the fishing mortality of 0.1 year-1 were calculated and tested to determine whether the mean age for the greater of the two mortalities was less than that for the smaller mortality. Thus, for fish in corresponding length classes, the number of fish, mean age, and standard deviation of ages within the length slice of the sample produced using fishing mortality F, denoted by 푛퐹, 푥̅퐹, and 푠퐹, respectively, were calculated.

The difference, 푑퐹1,퐹2, between the means of the samples for fishing mortalities 퐹1 and 퐹2, the

SE of that difference SE퐹1,퐹2, the degrees of freedom df퐹1,퐹2 and the t-statistic for the Aspin-

Welch unequal-variance t-test (Murphy 1967) was then calculated as follows.

푑퐹1,퐹2 = 푥̅퐹2 − 푥̅퐹1

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SE = 푠2 ⁄푛 + 푠2 ⁄푛 퐹1,퐹2 √ 퐹1 퐹1 퐹2 퐹2

2 2 2 2 2 2 2 df퐹1,퐹2 = (푠퐹1⁄푛퐹1 + 푠퐹2⁄푛퐹2) ⁄{[(푠퐹1⁄푛퐹1) ⁄(푛퐹1 − 1)] + [(푠퐹2⁄푛퐹2) ⁄(푛퐹2 − 1)]}

푡퐹1,퐹2 = 푑퐹1,퐹2⁄SE퐹1,퐹2

Using the above equations, which assume that the SEs for the sample means differ, the differences between the mean ages for the corresponding length slices were tested to determine whether, for each pair, the mean age for the greater of the two mortalities was less than that for the smaller mortality. The resulting differences were plotted against the midpoints of the length slices, using different symbols to discriminate between significant

(푃 ≤ 0.05) and insignificant (푃 > 0.05) differences.

The same approaches, as described above, were employed to compare the effect of fishing mortality on mean ages at given otolith sizes for samples generated using the value of

퐿50 for the alternative selectivity scenario for each species. The analysis was also repeated, using the same methods, to compare the effect of the two different values of 퐿50 for each species on mean ages at given otolith sizes for samples of each species generated using each of the three different levels of fishing mortality. Likewise, the same approaches were used to explore the effects of fishing mortality on the mean lengths of fish with given values of otolith radius and on the mean otolith radii of fish with given values of length.

The bivariate growth model was fitted to the lengths and otolith radii at the ages of capture of the fish in the different samples generated using the different fishing mortalities and selectivity curves for the six species (for a detailed description of the model and the method used to fit it, refer to Chapters 2 and 3). The predicted fish lengths and otolith radii at given ages were calculated using the resulting fitted somatic and otolith growth curves. These

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predicted lengths and otolith radii were plotted against age, and against each other, together with the lengths and otolith radii at age calculated using the ‘true’ growth curves that had been employed when generating the simulated samples. For each species, the curves of both predicted lengths at age and radii at age and fish length versus otolith radius were compared visually to explore the effect of the different fishing mortalities.

Deviations of fish lengths and otolith radii for individual fish from the somatic and otolith growth curves for each species, which were used when generating the simulated samples for both the lowest and the highest values of fishing mortality, were plotted. Each of the resulting figures was overlayed with a plot of a smoothing spline fitted to the deviations of length or otolith size (i.e., the dependant variables) at age at capture (i.e., the independent variable) using the smooth.spline function in R. The resulting plots were examined to assess the effect of fishing mortality on the distributions of deviations exhibited at different ages by the different species.

6.3 RESULTS

6.3.1 DISTRIBUTIONS OF AGE, LENGTH AND OTOLITH SIZE

For all species, the numbers of older and larger individuals, and individuals with larger otolith sizes, declined with increasing levels of fishing mortality, providing confirmation that the data produced by the simulation procedure were consistent with the expectation of increased truncation of these variables due to the cumulative effects of fishing mortality (Figs S6.1-S6.3). Plots of the distributions produced for each species using the two selectivity patterns displayed changes consistent with the different values of 퐿50 employed when simulating the data (Figs S6.4-S6.6).

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6.3.2 EFFECT OF FISHING MORTALITY ON MEAN AGE FOR A GIVEN LENGTH AND

OTOLITH SIZE

The mean ages of fish within the length slices for the larger fish of each species decreased as fishing mortality increased (Fig. 6.1). In contrast, the mean ages of fish in the length slices of the smaller fish were little affected by increases in fishing mortality. The mean ages of the larger fish of each species displayed increasing variability with length, consistent with the smaller numbers of individuals in the length slices for the larger fish (Fig.

6.1).

One-tailed Aspin-Welch t-tests demonstrated that, throughout those ranges of lengths for which samples generated with the different fishing mortalities overlapped, mean ages of the fish of each species in the length slices for samples generated using fishing mortalities of

0.4 and 0.8 year-1 were typically significantly less than those for the fish in the samples generated using a fishing mortality of 0.1 year-1 (Fig. 6.2). Note, however, that the individual samples on which this analysis (and subsequent analyses) was based included simulated data for 50 000 individuals, and thus, although differences may be found to be statistically significant (e.g., Fig. 6.2), biological significance should also be considered as the magnitude of the differences may be too small to be of concern. The differences in mean ages ranged from 0 to 6 years (Fig. 6.2), and the larger differences therefore would be of biological significance, noting that ages of simulated data ranged from 15 to 40 years (Fig. 6.1) and that the larger fishing mortalities reflect levels of exploitation that far exceed the F-based limit reference points set by the managers of these Western Australian fish stocks. On occasion, although the mean ages for length slices of both smaller and larger fish of each species were typically less for fishing mortalities of 0.4 and 0.8 year-1 than those for the slices generated using 퐹 = 0.1 year-1, the differences were not statistically significant (푃 > 0.05). For smaller fish, the lack of significance was due to small differences in mean age, but for larger fish, it

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appeared to result from the small numbers of fish within the different length slices and the large SEs of the differences between the mean ages of the fish in the slices created using the different fishing mortalities. For smaller fish, the mean ages of fish within the length slices for the fishing mortality of 0.8 year-1 were similar to those produced for the lower values of fishing mortality (F = 0.1 year-1). For length slices in the middle and upper length ranges, however, the mean ages of fish for the fishing mortality of 0.8 year-1 were typically significantly less than those associated with the fishing mortality of 0.4 year-1 (Fig. 6.2).

As with mean ages at lengths, the point estimates of the mean ages at given otolith radii for larger otoliths of each of the six species decreased as fishing mortality increased

(Fig. 6.3), particularly for E. armatus (Fig. 6.3d). At smaller otolith sizes, however, the point estimates of the mean age were unaffected by the different values of fishing mortality. The mean ages for the different radius slices were typically significantly less (푃 < 0.05) for the samples generated using fishing mortalities of 0.4 and 0.8 year-1 than 0.1 year-1 (Fig. 6.4).

Differences ranged from 0 to 5 years and, given the lifespans of the different species, were considered of biological significance, noting again that the larger fishing mortalities represent excessively high exploitation. In contrast, however, the mean ages for the greater fishing mortalities were, on occasion, not significantly less (푃 > 0.05) for radius slices at the lower and upper ends of the range. As for mean ages at given lengths, the mean ages at given values of otolith radius were typically significantly less (푃 < 0.05) for F = 0.8 year-1 than for

F = 0.4 year-1 (Fig. 6.4).

6.3.3 EFFECT OF INCREASE IN MEAN LENGTH AT FIRST CAPTURE (L50) ON MEAN

OTOLITH SIZE FOR A GIVEN AGE

For a fishing mortality of F = 0.1 year-1, the point estimates of the mean otolith radius

(of the fish within the different age slices) at a given age for the older fish of each of the six

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species were little affected by the increase in the mean length at first capture (Fig. 6.5). The same was the case with a fishing mortality of F = 0.8 year-1 for A. butcheri, A. japonicus,

B. frenchii and E. armatus, which appeared little affected by the increased values of 퐿50 (Figs

6.6a, b, c and d). With this greater level of fishing mortality, the mean otolith radii at age for older individuals of E. coioides exhibited a relatively consistent but small (and probably biologically insignificant) increase (Fig. 6.6e). In contrast, for a fishing mortality of

F = 0.8 year-1, the mean otolith radii in the age slices of the older individuals of G. hebraicum appeared to increase to a level that might be considered of biological significance when the value of 퐿50 was increased (Fig. 6.6f).

The one-tailed Aspin-Welch t-test demonstrated that, throughout those ranges of ages for which samples generated with the different values of 퐿50 overlapped, mean otolith radii in the age slices for samples generated using larger values of 퐿50were not significantly greater

(푃 > 0.05) than those for the fish in the samples generated using a smaller value of 퐿50 for the highest value of fishing mortality (F = 0.8 year-1) (Fig. 6.7). On occasion, and particularly for E. coioides and G. hebraicum, the mean otolith radii of the age slices for the larger values of 퐿50were significantly greater (푃 < 0.05) than those associated with the smaller values of

퐿50 in the middle and upper age ranges (Fig. 6.7).

6.3.4 EFFECT OF FISHING MORTALITY ON MEAN FISH LENGTH FOR A GIVEN

OTOLITH SIZE

The mean lengths of the fish within the slices of otolith radius for the larger otoliths of each species decreased (to an extent that was considered of biological significance) as fishing mortality increased (Fig. 6.8), particularly for both B. frenchii and E. armatus (Figs 6.8c and d). The mean fish lengths for the otolith radius slices of smaller otoliths appeared to be little affected by increased levels of fishing mortality, while the larger otoliths of each species

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displayed increasing variability of mean lengths within the radius slices. Within the middle of the range of otolith radii, the mean lengths of fish within the different radius class intervals for the samples of each species generated using fishing mortalities of 0.4 and 0.8 year-1 were typically significantly less (푃 < 0.05) than those for the samples of fish created using a fishing mortality of 0.1 year-1 (Fig. 6.9). Within the lower and upper regions of the ranges of otolith radii for the different species, however, the mean lengths for samples generated using the greater levels of fishing mortality were not always significantly less (푃 > 0.05) than those for samples generated with F = 0.1 year-1 (Fig. 6.9). Occasionally for A. butcheri and

A. japonicus, the differences between the different levels of fishing mortality of the mean lengths for otolith radius slices scattered throughout the entire range of values of otolith radii were not significant (푃 > 0.05) (Figs 6.9a and b).

6.3.5 EFFECT OF FISHING MORTALITY ON MEAN OTOLITH SIZE FOR A GIVEN FISH

LENGTH

The mean otolith radii of the fish in the length slices for the larger fish of each species decreased slightly with increasing fishing mortality (Fig. 6.10), but to a much lesser extent than with the mean fish length given otolith size. For A. butcheri, A. japonicus, and

B. frenchii, these differences appeared likely to be of biological significance. The mean otolith radii in the length slices for the smaller fish were not affected to any marked extent by increasing fishing mortality. The values of mean otolith radius within the length slices at the middle and upper ends of the length ranges for A. butcheri, A. japonicus, B. frenchii and

G. hebraicum samples, created using F = 0.4 and 0.8 year-1, were usually (but much less so for G. hebraicum) significantly lower (푃 < 0.05) than those for samples generated using

F = 0.1 year-1 (Figs 6.11a, b, c, and f). Differences between mean otolith radii for length

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slices of E. armatus and E. coioides were typically not significant (푃 > 0.05) throughout their length ranges (Figs 6.11d and e).

6.3.6 EFFECT OF FISHING MORTALITY ON PREDICTED FISH LENGTH AND OTOLITH

SIZE AT AGE, AND ON PREDICTED FISH LENGTH AT OTOLITH SIZE

At the upper end of their respective age ranges, as fishing mortality increased, the lengths at age predicted using the bivariate growth model decreased for five of the six species, i.e., A. butcheri, A. japonicus, B. frenchii, E. armatus and G. hebraicum (Figs 6.12a, b, c, d and f). This was particularly apparent for A. japonicus, B. frenchii and G. hebraicum, for which the asymptotic lengths (퐿∞) fell below those of the ‘true’ growth curves used when generating the simulated samples (Figs 6.12b, c and f). Increases in fishing mortality appeared to have little effect on the predicted lengths at age for E. coioides (Fig. 6.12e). Note that a plot of the mean lengths of E. coioides within age slices (not shown) exhibited a decline in length of older fish with increasing fishing mortality similar to that of the other five species. It thus appears that, for E. coioides, the finding that predicted lengths for the growth curves fitted to the data for F = 0.4 and 0.8 year-1 at the upper ends of their respective age ranges exceed those for F = 0.1 year-1 is an artefact resulting from the form of the fitted growth curves and the relative paucity of length data at the upper end of the age range.

Increasing fishing mortality did not affect predicted otolith radius at age for

A. butcheri, A. japonicus, B. frenchii, E. armatus and E. coioides (Figs 6.13a, b, c, d and e).

Predicted values of otolith radius at age for older G. hebraicum decreased markedly (Fig.

6.13f), however, although values for younger fish were unaffected.

Predicted lengths associated with larger otolith radii decreased as fishing mortality increased for four of the six species, i.e., A. butcheri, A. japonicus, B. frenchii, and

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G. hebraicum (Figs 6.14a, b, c and f). This was particularly apparent, and considered to be of biological significance, for A. japonicus and B. frenchii for which, the asymptotic lengths

(퐿∞) fell below that of the ‘true’ growth curves (cf. Figs 6.12b and c; Figs 6.14b and c). For

A. butcheri, the decrease was slight and likely to be of little biological significance (Fig.

6.14a). In the case of G. hebraicum, a similar slight decrease in predicted lengths was evident for fish with larger otoliths when fishing mortalities of F = 0.1 and 0.4 year-1 were employed

(Fig. 6.14f). In contrast, for F = 0.8 year-1, predicted lengths at otolith size for individuals of

G. hebraicum with larger otoliths appeared to increase. Although the youngest individuals of all species other than E. armatus exhibited a slight increase in predicted length at otolith radius (Fig. 6.14d), the statistical significance of the difference was not tested and, other than in the cases of A. japonicus and E. coioides (Figs 6.14b and e), is likely to be of little biological significance.

6.3.7 CHANGES IN DISTRIBUTION OF DEVIATIONS OF FISH LENGTH AND OTOLITH

SIZE WITH AGE

For F = 0.1 year-1 and with A. butcheri, A. japonicus, and B. frenchii, for which deviations from the ‘true’ somatic growth curve had been assumed (when simulating) to be normally distributed, fish included in the samples from the simulated catches exhibited high frequencies of positive length deviations at younger ages, i.e., < 2 years (Figs 6.15a, b and c).

The distribution of length deviations declined with increasing age, with the average deviation approaching zero and, for A. butcheri and B. frenchii, becoming slightly negative at older ages. For E. armatus, for which the deviations of length at age were assumed to be lognormally distributed about the ‘true’ somatic growth curve (Fig. 6.15d), a high frequency of positive length deviations was evident in the samples for fish of younger ages, but the mean of the distribution declined towards one as age increased. The distribution of the

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deviations of length for E. coioides (normally distributed deviations from ‘true’ somatic growth curve) followed a similar trend to those of A. butcheri and B. frenchii (Fig. 6.15e).

For G. hebraicum (lognormally distributed deviations from ‘true’ curve), the positive bias in the mean of the distribution of deviations of the youngest fish initially increased as age increased to ~ 4 years, then progressively declined to become negatively biased for fish with ages greater than ~ 11 years (Fig. 6.15f).

The distributions of otolith radius deviations for the fish sampled with F = 0.1 year-1 from populations of both A. butcheri and A. japonicus (lognormally distributed deviations from

‘true’ otolith growth curve), were positively biased, i.e., their means exceeded 1, for younger fish, but little bias was apparent for older fish (Figs 6.16a and b). No trend with age was evident in the deviations of radius at age within the fish in the sample of B. frenchii

(lognormally distributed deviations) (Fig. 6.16c). For E. armatus (lognormally distributed deviations) (Fig. 6.16d), a slight positive bias, i.e., mean > 1, was evident in the distribution of otolith radius deviations for fish of younger ages, but the bias decreased as fish became older. Deviations of otolith radius for E. coioides (normally distributed deviations) were positively biased at younger ages, with the bias decreasing with age and becoming slightly negative at older ages (Fig. 6.16e). In the case of G. hebraicum, for which deviations of both length and otolith radius had been assumed to be lognormally distributed about their respective ‘true’ growth curves, the distribution of otolith radius deviations followed a similar trend to that of the length deviations, with an initial positive bias that increased to age

~ 3 years (Fig. 6.16f). Deviations for both length and otolith radius of the fish in the sample, became increasingly biased towards positive values as age increased to ~ 3 years, before declining to become negatively biased after age reached ~ 11 years.

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When the fishing mortality used to generate the samples was increased to F = 0.8 year-1, the effect on both deviations of length and otolith radius was more pronounced than that observed for samples generated using F = 0.1 year-1 (Figs 6.17 and 6.18). For all species, the distribution of both length and otolith radius deviations declined markedly with age to become increasingly negatively biased (or, in the case of lognormally distributed deviations, less than 1) at older ages (Figs 6.17 and 6.18).

6.4 DISCUSSION

Based on an extensive literature search, this study appears to be one of the first to explore the effects of length-dependent fishing mortality on otolith size at age and, consequently, the effect of this on the fish length-otolith size-age relationship.

6.4.1 EFFECT OF FISHING MORTALITY ON MEAN AGE FOR FISH OF A GIVEN FISH

LENGTH OR OTOLITH SIZE

For the larger fish in samples of each of the six species considered in the present study, the mean age of fish of a given length was found to decrease as fishing mortality increases, while the effect for smaller fish tended to be insignificant. This result stems from the cumulative effect of fishing mortality with age, the declining rate of increase in length at age with increasing age, and the distribution of lengths about the somatic growth curve.

Faster growing fish of a given length are caught at a younger age than slower growing fish of that length, but increasing levels of fishing mortality reduce the survival of fish, such that there are fewer older fish of the given length that are caught. The smaller fish are growing more rapidly, and thus the age range over which fish are of a given small length is far smaller than the age range over which fish are of a given large length, noting also that selectivity typically reduces the effect of length-dependent fishing mortality on the smaller fish (e.g.,

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Machiels and Wijsman 1996; Jennings et al. 1998; Law 2000). The effect of fishing mortality on the numbers of small fish of different ages is thus much less obvious than its effect on the larger fish.

The effect of increasing the fishing mortality on the mean age of fish of a given otolith radius was similar to that for the mean age of fish of a given length. That is, an increased level of fishing mortality reduces the mean age given that otolith size because of the cumulative effect of fishing mortality on survival, the shape of the otolith growth curve, and the distribution of otolith radii at age about the otolith growth curve. Due to the curvature of the otolith curve with respect to age, combined with the form of the distribution of deviations about the otolith growth curve (shown in Chapter 3 to be typically lognormal), the effect is greater for older fish with larger otoliths than for the younger fish that possess smaller otoliths. This explains why the differences between mean ages at given otolith sizes were typically of statistical significance for fish with larger otoliths, but not always for those with smaller otoliths. It should be noted, however, that these differences (even if small) may be found to be statistically significant because of the large numbers of simulated fish in the samples for the different scenarios, where those sample sizes were necessary to produce the numbers of fish required within the different slices to calculate means of sufficient precision.

6.4.2 EFFECT OF SELECTIVITY ON MEAN OTOLITH SIZE FOR FISH OF A GIVEN AGE

For the same level of fishing mortality, only the fastest growing of the younger fish, i.e., those with the greatest size, continue to be caught when fishing gear that selects for larger rather than smaller fish replaces the fishing gear with lesser 퐿50, thus producing an increase in length at age for the very young fish within the sample. Greater numbers of faster- growing fish, which would have been caught by the fishing gear with the lesser 퐿50, survive

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to attain ages and sizes at which they become susceptible to capture by the gear with the greater value of 퐿50.

As, for some species, otolith sizes of fish of a given age are correlated with their lengths (e.g., Hare and Cowen 1995), it had been expected that the effect of size-dependent fishing mortality on otolith size at age would be similar to that on fish length at age. Although the effect on mean otolith radii at ages of older A. butcheri, A. japonicus, B. frenchii and

E. armatus, of employing the selectivity curves with the greater rather than smaller value of

퐿50, appeared inconsequential, mean otolith radii at ages of older E. coioides and, particularly, G. hebraicum exhibited small increases consistent with expectation. The extent to which the mean otolith radius at age increases depends upon the somatic and otolith growth curves, the magnitude and form of the statistical distribution of deviations about those growth curves, the shape of the two selection curves, the mean lengths at which fish are selected by those curves, and the levels of fishing mortality. The precise combination of these factors that produced the small increases (or apparent lack of increase) in mean otolith size at age exhibited at older ages by the six different species could not be determined in the current study. This arose from the fact that, due to the different biological characteristics of the six species, it was not possible to specify parameters of the two selectivity curves for each species that would produce results that would be directly comparable among species.

For all species, however, there appeared to be little effect of selectivity on mean otolith sizes at age of younger fish. This is possibly due to the low selectivity of such fish, such that the capture of such fish is essentially random with respect to the range of lengths

(and otolith sizes) at those young ages.

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6.4.3 EFFECT OF FISHING MORTALITY ON MEAN FISH LENGTH FOR FISH OF A

GIVEN OTOLITH SIZE

The results of this study demonstrated that, for individuals of a given fish species, the mean length at a given otolith size decreased with increasing fishing mortality with the effect increasing as otolith sizes became larger. This effect was not apparent, however, for fish with smaller otolith sizes. Note that, because the number of fish within the otolith size slices was small and the SEs of the differences for the slices were large, the decrease in mean length at larger otolith sizes associated with increased fishing mortality was not always statistically significant. That a significant reduction in mean lengths at a given otolith size for fish with larger otoliths resulted from an increase in fishing mortality is due to a combination of the variability of lengths at ages, the variability of ages at which a given otolith size is attained, and the reduced survival of fish at older ages resulting from the cumulative effect of fishing mortality (e.g., Sinclair et al. 2002a; Taylor et al. 2005). The reduction of mean lengths at a given otolith size for fish with smaller otolith sizes, however, was not significant and is probably a consequence of the curvature of the somatic and otolith growth curves and thus the limited age range over which young fish are of a given otolith size.

In the cases of A. japonicus, and to some extent also A. butcheri, differences between the mean lengths for otolith radius slices at the different levels of fishing mortality lacked statistical significance throughout the entire range of values of otolith radii. This finding may result from the fact that for these species, while otolith size at age continued to increase at a slower rate, lengths at age had approached their asymptotic lengths more closely at earlier ages than the other species in this study (Chapter 3).

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6.4.4 Effect of fishing mortality on mean otolith size for fish of a given fish length

The finding that, for larger individuals, mean otolith radius at a given fish length declined with increasing levels of fishing mortality, and was particularly marked for

A. butcheri, A. japonicus, and B. frenchii, appears to be related to the impact of increasing fishing mortality on the mean ages of the fish of a given length that are caught (Fig. 6.1), the shape of the relationship between otolith size and age (Fig. 6.5) and the relatively small impact of increased levels of fishing mortality on otolith size given age (Fig. 6.5).

6.4.5 EFFECT OF FISHING MORTALITY ON PREDICTED FISH LENGTH AND OTOLITH

SIZE AT AGE, AND ON PREDICTED FISH LENGTH AT OTOLITH SIZE

Increase in fishing mortality reduces the age and size ranges of the fish within exploited populations. When combined with gear selectivity, the resulting length-dependent fishing mortality produces bias in length-at-age samples due to over-representing of faster- growing young fish and, through the cumulative effect of such mortality, the predominance of slower-growing old fish in the catches (e.g., Hanson and Chouinard 1992; Kristiansen and

Svåsand 1998; Sinclair et al. 2002b; Taylor et al. 2005). Thus, for example, estimates of asymptotic length of a von Bertalanffy growth curve fitted to simulated lengths at ages of capture using a least squares approach have been shown to underestimate the true, i.e., intrinsic, asymptote as a result of length-dependent fishing mortality (Taylor et al. 2005). The results for five of the six species considered in the current study were consistent with this finding as lengths at age of older fish predicted using the somatic growth curves of the bivariate growth models fitted to the simulated data declined with increasing fishing mortality. The decline was particularly evident in the cases of A. japonicus, B. frenchii, and especially G. hebraicum, but less so in the cases of A. butcheri and E. armatus.

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Contrary to expectation, lengths at age of E. coioides between ~ 15 to 28 years old predicted using the somatic growth curves of the bivariate growth models fitted to samples generated with fishing mortalities of F = 0.4 and 0.8 year-1 were greater than values calculated using with the ‘true’ somatic growth curve employed when generating the simulated sample. Examination of mean lengths of sampled E. coioides within successive age slices demonstrated, however, that expected lengths at age declined with increasing fishing mortality, and that values predicted using growth curves fitted to the simulated samples had over-estimated the expected values within the age slices in this section of the age range.

In contrast to the results for predicted lengths at age, predicted otolith radii at age for five species of the six species, i.e., A. butcheri, A. japonicus, B. frenchii, E. armatus and

E. coioides, appeared little affected by increasing fishing mortality. This was not the case for

G. hebraicum, however, for which the otolith sizes at age of older fish predicted using the otolith growth curves of the bivariate growth models fitted to the simulated data declined with increasing fishing mortality. This appears to result from the effect on the age compositions of the samples for the different fishing mortalities of the relatively high 퐿50, the lognormal distribution of deviations of lengths at age from the somatic growth curve, and the form of that somatic growth curve. Accordingly, as in the case of the predictions for length at age, the predicted values of otolith radius at age for older fish of G. hebraicum are more sensitive to the different levels of fishing mortality than those of the other species.

For all species, increases in fishing mortality had little effect on the lower end of the fitted lengths at otolith radii curves, i.e., predicted lengths for fish with smaller otolith. As with predicted lengths at age, the predicted lengths at otolith size for fish with larger otoliths, calculated using the somatic and otolith growth curves of the bivariate growth models fitted to the simulated data, declined with increasing fishing mortality for A. japonicus and

B. frenchii. Although predicted lengths at age of older G. hebraicum had declined, this was

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offset in the relationship between predicted lengths and otolith sizes by the decline in predicted otolith sizes at age for these older fish. It thus appears that the extent to which the relationship between fish length and otolith size for older fish is affected by increased length- dependent mortality is determined by the relative magnitudes of the cumulative species- dependent effects of that mortality on fish lengths and otolith sizes at age.

6.4.6 CHANGES IN DISTRIBUTION OF DEVIATIONS OF FISH LENGTH AND OTOLITH

SIZE WITH AGE

High frequencies of positive deviations from the somatic growth curves of the bivariate growth model for sample fish of younger ages supported the expectation that fish which are faster growing were selected at younger ages. In the case of A. butcheri,

B. frenchii, E. coioides and G. hebraicum, high frequencies of negative deviations at older ages suggest that slower growing fish were selected at older ages for these species, as a consequence of the cumulative effects of length-dependent fishing mortality (e.g., Sinclair et al. 2002b; Taylor et al. 2005).

To my knowledge, following an extensive search of the literature, this is the first study in which the effect of mortality on the distribution of deviations of otolith size at age has been explored. As with deviations of lengths at age, deviations from the otolith growth curve of the bivariate growth model over age revealed that fish with faster growing otoliths were selected at younger ages, while for the same four species, i.e., A. butcheri, B. frenchii,

E. coioides and G. hebraicum, fish with slower growing otoliths were selected at older ages.

When the level of size-selective fishing mortality was increased to the highest level used in this study, the bias observed at a lower level of fishing mortality was accentuated. In the case of G. hebraicum, the observed rapid initial increase in the log-normally distributed deviations

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of length and otolith radius at age appears to be an artefact resulting from the influence of a small number of fish on the form of the smoothing spline fitted to the data.

6.4.7 IMPLICATIONS

Length-dependent fishing mortality produces bias in length-at-age samples by over- representing faster-growing young fish and, through the cumulative effect of such mortality, the predominance of slower-growing old fish in the catches (e.g., Hanson and Chouinard

1992; Kristiansen and Svåsand 1998; Sinclair et al. 2002b; Taylor et al. 2005). The effect of such fishing mortality on the biological characteristics of surviving individuals within fish populations extends to the characteristics of the otoliths of individuals in samples collected from those populations. Thus, the younger fish in those samples are typically those with faster growing otoliths, while older fish tend to be those with slower growing otoliths.

Although increases in fishing mortality appeared to have little effect on the relationship between predicted otolith size and fish age for five of the six species studied, there was little impact on the form of the predicted length-otolith size relationship for

G. hebraicum, the sixth species, as predicted lengths at age of older fish had been similarly affected by that mortality. In the absence of an effect of increased levels of fishing mortality on predicted otolith size at age, the full cumulative impact of the effect of that mortality on the predicted lengths at age of older fish was reflected in the effect of the different levels of fishing mortality on the relationship between predicted lengths and otolith sizes at age.

The effect of length-dependent fishing mortality on the biological characteristics of individuals can also affect the reliability of estimates of the parameters of the true, intrinsic growth curve, a key biological characteristic of a fish population (e.g., Ricker 1969; Hamley

1975; Vaughan and Burton 1994; Potts et al. 1998; Lucena and O’Brien 2001). Such bias may result in the apparent relationship between lengths at age and age having a form that differs

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from that of the true relationship. As with somatic growth (Taylor et al. 2005), there is need to account for, and remove the bias of, the effect of length-dependent fishing mortality when fitting otolith growth curves. The construction of a reliable curve that represents the true, intrinsic relationship between length and otolith size for individuals of a fish population requires that the bias associated with length-dependent fishing mortality is removed.

The simulation study demonstrated that the lengths, otolith sizes and ages that characterise sampled fish, and, for some species, affect the relationship between fish length and otolith size, are affected by length-dependent fishing mortality. Back-calculation that employs formulae derived from the apparent relationship between fish length and otolith size and using length and otolith data from sampled fish would be expected to produce predictions that are consistent with the characteristics of the observed fish lengths, i.e., have similar bias.

Use of a fishing mortality-corrected back-calculation formula would produce length predictions inconsistent with observed lengths at age. To obtain estimates of lengths at age that are unbiased with respect to fishing mortality, it would be best to combine back- calculation (using the bi-variate residuals) with adjustment for length selectivity in a single comprehensive analysis (e.g., as done in Taylor et al. 2005).

This study has demonstrated that, not only does length-dependent fishing mortality affect the parameters of fitted somatic growth curves, but so also is it likely to affect the parameters of otolith growth curves and of relationships between fish length and otolith size.

Through this, sufficient bias may be introduced into results of back-calculation and stock assessment to affect management decisions. By coupling lengths and ages at capture with otolith growth, and with annual increments in the sizes of otoliths of individual fish, together with improved understanding of the growth of otoliths and the relationship between fish lengths and otolith sizes, it may be possible to develop approaches that can assist in accounting for the effects of length-dependent fishing mortality on growth estimates.

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Acknowledgements

This research was sponsored by Murdoch University and Recfishwest. Gratitude is also expressed to Fisheries Research and Development Corp. for the financial support it provided for several of the earlier studies.

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Table 6.1. Maximum ages and total lengths (TL), sexuality, and habitats of Acanthopagrus

butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus

coioides, and Glaucosoma hebraicum, the biological characteristics of which were employed

in the simulation study.

Species Max. age Max. TL Sexuality Habitat References (years) (mm)

Acanthopagrus butcheri 31 530 Gonochorist Temperate estuaries Sarre and Potter (2000) Jenkins et al. (2006) Potter et al. (2008)

Argyrosomus japonicus 31 2 000 Gonochorist Coastal marine Farmer et al. (2005) waters, seasonally Gomon et al. (2008) entering estuaries

Bodianus frenchii 78 480 Protogynous Over and around Gomon et al. (2008) hermaphrodite coastal temperate Cossington et al. (2010) reefs Platell et al. (2010)

Epinephelides armatus 19 510 Gonochorist Over and around Moore et al. (2007) coastal temperate Gomon et al. (2008) reefs Platell et al. (2010)

Epinephelus coioides 22 1 110 Protogynous Subtropical/tropical Heemstra and Randall (1993) hermaphrodite mangrove nursery Heemstra (1995) habitats and coastal Pember et al. (2005) reefs

Glaucosoma hebraicum 41 1120 Gonochorist Temperate coastal Hesp et al. (2002) marine waters, Lenanton et al. (2009) around reefs Platell et al. (2010)

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Table 6.2. Parameters from the ‘true’ somatic and otolith growth curves used to generate simulated data for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

Species A. butcheri A. japonicus B. frenchii E. armatus E. coioides G. hebraicum 흉ퟏ 0.62 0.22 1.13 1.71 0.15 0.71

흉ퟐ 19.62 31.03 61.91 13.04 14.70 24.20

 0.31 0.13 0.06 0.22 0.50 0.63

Max. age 31 31 78 19 22 41

M 0.21 0.24 0.062 0.25 0.10 0.20

푳ퟓퟎ 200 200 150 150 500 450

300 400 250 200 800 700

푳ퟗퟓ 325 250 200 300 700 700

425 450 300 350 1 000 950

Growth type Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith 풂 0.19 0 0.24 0 0.28 0 0.38 0.33 0.12 0.15 0 1.31 풃 0.33 2.03 1 1.64 -0.38 1.78 -1 1 1 1 1.90 0.22

풚ퟏ 98 0.41 154 0.57 90 0.36 149 0.52 70 0.35 105 0.74 풚ퟐ 327 1.65 1 214 10.21 420 2.09 482 1.06 1 059 1.87 992 3.42 푺푫 17 0.06 79 0.09 21 0.09 0.11 0.10 61 0.11 0.12 0.09 Error type normal lognormal normal lognormal normal lognormal lognormal lognormal normal normal lognormal lognormal

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Growth curve Schnute Schnute Schnute Schnute Modified Schnute Schnute Schnute Schnute Schnute Schnute Modified (1981) (1981) (1981) (1981) von (1981) (1981) (1981) (1981) (1981) (1981) von Bertalanffy Bertalanffy References Current Chapter; Current Chapter; Current Chapter; Current Chapter; Current Chapter; Current Chapter; Chapter 3; Chapter 3; Chapter 3; Chapter 3; Chapter 3; Chapter 3; Sarre and Potter (2000); Farmer et al. (2005); Cossington et al. (2010) Moore et al. (2007) Pember et al. (2005) Hesp et al. (2002); Jenkins et al. (2006) Farmer (2008) Lenanton et al. (2009)

Note. 휏1= first reference age (youngest fish in the sample); 휏2= second reference age (oldest fish in the sample); 휌 = correlation of the bivariate

distribution of deviations from the expected length and radius at age; Max. age = maximum age; 푀푁 = natural mortality; 퐿50 and 퐿95 are sizes at

which 50% and 95%, respectively, of those fish sizes are retained; 푎 and 푏 = parameters of growth model; 푦1 = size at age 휏1; 푦2= size at age 휏2;

SD = estimated standard deviations for Schnute (1981) and modified von Bertalanffy growth curves of the bivariate growth models fitted to the

total lengths (mm) and otolith radii (mm) at their ages of capture. Error type = marginal distribution used for the deviations from the somatic and

otolith growth curves.

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Table 6.3. Parameters for the somatic and otolith growth curves of the bivariate growth model fitted to samples of length and otolith size at age

obtained through simulations employing a level of fishing mortality of F = 0.1 year-1 for Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

Species A. butcheri A. japonicus B. frenchii E. armatus E. coioides G. hebraicum

Growth type Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith

풂 0.19 0 0.29 0 0.06 0 0.39 0.31 0.12 0.15 0.07 0

풃 0.25 2.04 0.53 1.64 -2.75 1.77 -1.10 1.16 0.80 0.86 1.26 1.95

풚ퟏ 104.72 0.41 211.69 0.57 98.90 0.36 151.77 0.52 171.07 0.45 158.63 1.39

풚ퟐ 325.48 1.65 1 191.91 10.23 344.89 2.09 481.31 1.06 1 053.65 1.87 947.74 3.32

Note. 푎 and 푏 = parameters of growth model; 푦1 = size at age 휏1; 푦2= size at age 휏2 for Schnute (1981) and modified von Bertalanffy growth

curves of the bivariate growth models fitted to the total lengths (mm) and otolith radii (mm) at their ages of capture.

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Table 6.4. Parameters for the somatic and otolith growth curves of the bivariate growth model fitted to samples of length and otolith size at age

obtained through simulations employing a level of fishing mortality of F = 0.4 year-1 for Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

Species A. butcheri A. japonicus B. frenchii E. armatus E. coioides G. hebraicum

Growth type Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith

풂 0.19 < -0.01 0.36 0 0.15 0 0.37 0.34 0.09 0.13 0.14 0

풃 0.33 2.09 0.21 1.64 1.27 1.77 -1 1 1.06 1.06 0.96 3.75

풚ퟏ 104.15 0.41 217.91 0.57 114.42 0.36 151.37 0.52 156.62 0.43 125.28 0.87

풚ퟐ 321.41 1.64 1 135.61 10.21 301.96 2.10 476.72 1.06 1 048.32 1.86 814.63 2.98

Note. 푎 and 푏 = parameters of growth model; 푦1 = size at age 휏1; 푦2= size at age 휏2 for Schnute (1981) and modified von Bertalanffy growth

curves of the bivariate growth models fitted to the total lengths (mm) and otolith radii (mm) at their ages of capture.

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Table 6.5. Parameters for the somatic and otolith growth curves of the bivariate growth model fitted to samples of length and otolith size at age

obtained through simulations employing a level of fishing mortality of F = 0.8 year-1 for Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

Species A. butcheri A. japonicus B. frenchii E. armatus E. coioides G. hebraicum

Growth type Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith Somatic Otolith

풂 0.17 0 0.56 0 0.14 0 0.37 0.33 0.01 0.09 0.15 -0.06

풃 0.50 2.10 -0.54 1.63 1.46 1.77 -1 1 1.66 1.50 1.07 5.09

풚ퟏ 103.61 0.41 223.98 0.57 114.58 0.36 151.39 0.52 117.04 0.39 125.08 0.62

풚ퟐ 317.10 1.62 1 001.71 10.21 298.65 2.08 467.37 1.05 1 070.71 1.88 775.53 3.02

Note. 푎 and 푏 = parameters of growth model; 푦1 = size at age 휏1; 푦2= size at age 휏2 for Schnute (1981) and modified von Bertalanffy growth

curves of the bivariate growth models fitted to the total lengths (mm) and otolith radii (mm) at their ages of capture.

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Figure 6.1. Mean ages (years) of fish within length slices (mm) at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus,

Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

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Figure 6.2. Differences between the mean ages (years) of fish within corresponding length

(mm) classes for fishing mortalities, F = 0.4 and 0.8 year-1, and the mean age of the fish from the sample produced for the fishing mortality F = 0.1 year-1, using a one-tailed Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. Solid black line represents mean ages at length for fish in the samples generated using F = 0.1 year-1, blue circles represent the comparison between mean ages at length for fish in the samples generated using F = 0.1 and

0.4 year-1, and red circles represent the comparison between mean ages at length for fish in

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the samples generated using F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance (푃 < 0.05) of the comparisons.

209

210

Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

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between mean ages at otolith radius for fish in the samples generated using F = 0.1 and

0.8 year-1. Large blue or red circles indicate the statistical significance (푃 < 0.05) of the comparisons.

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Figure 6.5. Mean otolith radius (mm) of fish within age (years) slices at increasing levels of

-1 mean length at first capture (퐿50) for a fishing mortality of F = 0.1 year for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

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Figure 6.6. Mean otolith radius (mm) of fish within age (years) slices at increasing levels of

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Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum at the highest fishing mortality,

215

values of 퐿50. Large red circles indicate the statistical significance (푃 < 0.05) of the comparisons.

216

217

Aspin-Welch t-test for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum. Solid black line represents mean lengths at otolith radius for fish in the samples generated using

F = 0.1 year-1, blue circles represent the comparison between mean lengths at otolith radius for fish in the samples generated using F = 0.1 and 0.4 year-1, and red circles represent the

218

comparison between mean lengths at otolith radius for fish in the samples generated using

F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance

(푃 < 0.05) of the comparisons.

219

220

F = 0.1 year-1, blue circles represent the comparison between mean otolith radii at length for fish in the samples generated using F = 0.1 and 0.4 year-1, and red circles represent the

221

comparison between mean otolith radii at length for fish in the samples generated using

F = 0.1 and 0.8 year-1. Large blue or red circles indicate the statistical significance

(푃 < 0.05) of the comparisons.

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Figure 6.12. Length (mm) at age (years) calculated using the ‘true’ growth curves and predicted using the bivariate growth model at different levels of fishing mortality (F = 0.1,

0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

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Figure 6.13. Otolith radius (mm) at age (years) calculated using the ‘true’ growth curves and predicted using the bivariate growth model at different levels of fishing mortality (F = 0.1,

0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii,

Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

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Figure 6.14. Length (mm) at otolith radius (mm) calculated using the ‘true’ growth curves and predicted using the bivariate growth model at different levels of fishing mortality

(F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

225

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Figure 6.16. Distribution of otolith radius (mm) deviations with age (years) from the ‘true’ otolith growth curve at the lowest level of fishing mortality, F = 0.1 year-1, for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum. Red line = smoothing spline.

227

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Figure 6.18. Distribution of otolith radius (mm) deviations with age (years) from the ‘true’ otolith growth curve at the highest level of fishing mortality, F = 0.8 year-1, for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum. Red line = smoothing spline.

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Supplemental materials for Chapter 6

Figure S6.1. Age (years) compositions of the fish in the simulated samples at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

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Figure S6.2. Fish length (mm) compositions of the fish in simulated samples at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

231

Figure S6.3. Otolith radius (mm) compositions of the fish in the simulated samples at different levels of fishing mortality (F = 0.1, 0.4 and 0.8 year-1) for Acanthopagrus butcheri,

Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus, Epinephelus coioides, and Glaucosoma hebraicum.

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Figure S6.4. Age (years) compositions of the fish in the simulated samples at two levels of

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Figure S6.5. Fish length (mm) compositions of the fish in the simulated samples at two

-1 levels of mean length at first capture (퐿50) for a fishing mortality of F = 0.8 year for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum.

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Figure S6.6. Otolith radius (mm) compositions of the fish in the simulated samples at two

-1 levels of mean length at first capture (퐿50) for a fishing mortality of F = 0.8 year for

Acanthopagrus butcheri, Argyrosomus japonicus, Bodianus frenchii, Epinephelides armatus,

Epinephelus coioides, and Glaucosoma hebraicum.

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CHAPTER 7 – GENERAL DISCUSSION AND PERSPECTIVES

The overall objective of the thesis was to explore the relationship between lengths and otolith sizes of fish with respect to age, and to extend the range of current back-calculation approaches by developing a proportionality-based back-calculation method based on traditional growth equations. Cross-validation demonstrated that, for the majority of the six fish species employed in this study, the accuracy and precision of lengths at age given otolith size, and expected otolith sizes at age given length, predicted using the new approach exceeded those obtained using the regression equations for the length-otolith size-age relationships of earlier published, back-calculation methods. This was particularly true for

A. butcheri when growth curves were constrained to pass through the biological intercept calculated using data for recently-hatched larvae. Estimates of back-calculated length at ages produced by the new back-calculation approach, particularly when constrained to pass through the biological intercept for A. butcheri, were typically more consistent with mean observed lengths at corresponding ages than those of alternative approaches considered in this study. By assuming a bivariate distribution of deviations of length and otolith size at age from somatic and otolith growth curves selected from a suite of alternative equations of commonly-used forms, the new model provides a useful link between somatic growth models and traditional back-calculation approaches. Investigation of a ‘proportionality of otolith radius with age’ hypothesis (PORA) demonstrated that otolith size of individual A. butcheri, particularly for younger fish, did not remain a constant proportion of expected otolith size throughout life, but rather the proportion became increasingly constant with increasing age, particularly for adjacent opaque zones. Simulation demonstrated that the cumulative effects of length-dependent fishing mortality influence the relationships between fish lengths, otolith

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sizes and ages by reducing the relative number of faster-growing old fish in the population.

The implications of these findings are discussed below.

Somatic growth is a fundamental property of living organisms and thus one of the most important measurable life-history parameters for individuals and species, and particularly in the case of indeterminate growth (Austin et al. 2011; Einum et al. 2012; Paine et al. 2012; Pardo et al. 2013). An accurate description of somatic growth is essential to understand life histories, ecosystem dynamics, comparative demography, and productivity, and to assess sustainability of exploited populations and species (Beddington and Kirkwood

2005; Frisk et al. 2005). Back-calculation and sclerochronological studies of the allometric relationships between fish length and otolith size at capture (e.g., Xiao 1996) have enhanced our understanding of somatic growth of juvenile and adult fish. The intervals between successive opaque zones within otoliths, and the effect of environmental factors on these intervals, have provided information on somatic growth and on annual changes in length of fish throughout life.

Based on an extensive search of the literature, the current study is one of the first to describe the lengths and otolith sizes at ages of capture of both juveniles and adults by simultaneously fitting somatic and otolith growth curves, and allowing for a bivariate relationship between the deviations from those growth curves. The traditional forms of the size-at-age curves used in this model are essentially mechanistic, particularly in the case of somatic growth, which has a basis in the underlying growth process. In previous studies, simple linear or curvilinear equations have typically been used to describe the fish body- otolith size relationship (e.g., Campana and Jones 1992; Dulčić 1998; Harvey et al. 2000;

Jawad and Al-Mamry 2012; Zan et al. 2015). Such forms of empirical relationship, derived from the observed form of the relationship between length and otolith radius, have also been used in traditional back-calculation approaches, although, within the last two decades, age

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has been introduced as an additional explanatory variable (Morita and Matsuishi 2001;

Finstad 2003). Cross-validation analysis, however, demonstrated that, for most of the species examined in this study, the bivariate growth model produced more reliable predictions of these variables than the alternative linear and curvilinear models that related length and otolith size to age. Due to the flexibility of the curves it explored, which allowed for a wide range of patterns of growth and from which it selected the forms of the curves it used to represent somatic and otolith growth, this model provided better descriptions of the relationships between fish length and age and between otolith size and age exhibited by the six species than those produced by the more rigid linear and curvilinear alternatives. This finding suggests that the use of more appropriate descriptions of somatic (e.g., Katsanevakis and Maravelias 2008) and otolith growth processes, based on mechanistic models, might improve our understanding of the relationship between fish length, otolith size and age, and assist in understanding the factors likely to affect this relationship. In this context it is noted that, while the Schnute (1981) growth model offers considerable flexibility of model form, not all of its curves are biologically feasible. Thus, when fitted to limited data, results may be affected by the variability of size (fish length or otolith radius) at age and the assumptions made regarding the nature of the variation of sizes at age about the growth curve.

Although somatic growth is indeterminate (i.e., continues throughout life) for most fish species, metabolic theory suggests that fish length must approach an asymptotic size as age increases, i.e., the rate of somatic growth is expected to decrease with age (e.g., von

Bertalanffy 1938; West et al. 2001). For one of the species examined in this study

(B. frenchii), however, somatic growth was better described by a modified von Bertalanffy model, which assumes that lengths approach an oblique linear asymptote with increasing age.

Although, in the context of metabolic theory, this finding is unusual, Knight (1968) maintains that the assumption that lengths approach a [horizontal] asymptote is a “mathematical artefact

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of the data analysis” rather than a “fact of nature”. Indeed, as found in the simulation component of this study, samples which contain few old fish in the catch may provide insufficient information on the curvature associated with declining somatic growth rates at older ages, and data for younger fish are likely to dominate in their influence on the type of growth curve that is fitted, particularly when the range of alternative curves is not constrained to those that are feasible (e.g., some forms of the Schnute (1981) model allow for infeasible curves that, if employed to describe growth of adult fish, are linear or increase exponentially with age). Cases in which the fitted bivariate growth model selects a modified von

Bertalanffy curve to best describe somatic growth may thus result from a paucity of older fish in the samples. It would be appropriate, therefore, to constrain the somatic curves fitted in the bivariate growth model to mechanistic forms of curves which are consistent with metabolic theory.

Otolith growth, which results from the deposition of organic matrix fibres and carbonate crystals (Campana 1999; Morales-Nin 2000), is rapid in younger fish but declines with age, as was demonstrated in the data for the six species studied. Accretion of crystals continues throughout life (Campana 1999), suggesting that, although otolith growth slows, otolith size is likely to exhibit continued increase with age, such as was determined for four of the six species that were studied (i.e., a modified von Bertalanffy curve with an oblique asymptote for G. hebraicum or, for A. butcheri, A. japonicus, and B. frenchii, a particular form of the Schnute (1981) curve with similar pattern of growth). In the case of E. armatus, however, otolith size at age exhibited a greater decline with age than that of the other five species, possibly explaining the selection of a traditional von Bertalanffy curve as best describing otolith growth. Based on the results from this study, it is suggested that otolith size is best described by a growth curve that, for young fish, exhibits a rapid monotonic increase that declines with increasing age and, for older fish, approaches a constant linear rate of

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increase. Further investigation is required to elucidate whether the curves used in this study might be improved to provide better, biologically-plausible descriptions of the processes involved in otolith growth.

Although somatic growth curves are typically fitted by assuming that deviations from expected lengths are normally distributed with constant variance (Bowker 1995), the variance of length at age typically increases with age (Taylor et al. 2005). In this study, however, length deviations were found to have a lognormal marginal distribution about the somatic growth curve for only two of the six species, i.e., E. armatus and G. hebraicum. In contrast, otolith size deviations for five of the six species had lognormal marginal distributions. These findings suggest that, when fitting somatic or otolith growth curves, the form of statistical distribution employed to describe the scatter of deviations about the growth curve should be explored.

Since Templeman and Squires’ (1956) study of the relationship between otolith lengths and weights, and the rate of growth of Melanogrammus aeglefinus (L.), several morphometric variables relating to whole otoliths have been employed in exploration of fish length-otolith size relationships (Jawad et al. 2012; Zan et al. 2015). These have included otolith length, width, thickness and mass (e.g., Steward et al. 2009; Williams et al. 2015).

Such measurements have also been used to estimate age using multiple regression models

(e.g., Steward et al. 2009) or, for example, through application of a random forest approach

(e.g., Williams et al. 2015). In addition, morphometric variables relating to sagittal otoliths have been subjected to shape analyses, the results being used to discriminate between stocks of a species (e.g., Agüera and Brophy 2011).

In this study, as is typical in studies involving back-calculation, the radius of the otolith along a particular axis of measurement was employed as the measure of otolith size.

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For further studies exploring relationships between somatic and otolith growth and the effects on these of environmental factors, however, there would be value in investigating which of the distances along various alternative axes of measurement might provide the most reliable description of otolith growth. In this context, it is noted that otolith growth is related to metabolic rate (influenced by temperature, food intake, dominance status and other activity cycles) (e.g., Mosegaard et al. 1988; Metcalfe et al. 1992; Yamamoto et al. 1998; Fey 2005), which can lead to morphological differences in otoliths through altered metabolism and changes in shape with age (e.g., Wright 1991; Bang and Grønkjær 2005) under different environmental processes (e.g., food availability, Gagliano and McCormick 2004) and among different stocks with differing growth rates (e.g., Campana and Casselman 1993; DeVries et al. 2002). It would therefore be useful to determine whether incorporation of other dimensions of the otolith, such as width or thickness, might provide a better description of the overall fish length, otolith size and age relationship by including changes in otolith growth and shape through life. Combinations of different otolith morphometrics, e.g., length and width, could be explored to improve upon the description of otolith growth for species with differing otolith morphologies. A principal component analysis (PCA), for example, might be employed to determine the contributions of the various morphometrics to the principal component reflecting overall ‘otolith size’ for a given species and employing the scores for that component as measures of otolith size in subsequent analyses. Note that back-calculation estimates requires measurements of otolith size along selected axes of sectioned otoliths, however, and, based on the otoliths of the six species considered in this study, the opaque zones along other axes are likely to be ill defined in certain areas of the otoliths of numerous fish.

Otolith mass, which is easy to measure (e.g., Worthington et al. 1995; Pilling et al.

2003), and is based on whole rather than sectioned otoliths, is now increasingly employed,

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together with counts of opaque zones, to facilitate more accurate estimation of age. Otolith mass is also less subjective than measures of otolith size along a particular axis of measurement (e.g., Pawson 1990; Hunt 1992; Worthington et al. 1995; Cardinale et al. 2000;

Pilling et al. 2003, Lepak et al. 2012). The use of mass as a measure of otolith size, or its inclusion in an appropriate way with other measures of otolith size, e.g., PCA, would be likely to improve the precision of the resulting description of the length-otolith size-age relationship. For studies of the ways and extents to which somatic and otolith growth are influenced by variation in environmental factors, measurements of otolith size as measures of mass rather than distance (in either whole or sectioned otoliths) are likely to be far more rapid and less costly. Again, it is noted that, for back-calculation, measurements of otolith size as distances along selected axes will still be required. For fish aged using methods that employ otolith mass, otolith mass is already recorded and available for analysis. If ages are based on counts of opaque zones in broken or sectioned otoliths, the other of the pair of otoliths remains whole and is typically available for measurement and recording of its mass. Studies of otolith growth based on mass are therefore likely to be more rapid and economic than if based on otolith distance measurements, allowing increased sample sizes, and represent possibly ancillary information to ageing methods that improve accuracy of age estimates.

In this study, to improve precision, multiple measurements of otolith radius along the same axis were averaged to fit otolith growth curves. Such distance measurements are subjective, particularly when identifying the axis of measurement for each otolith section and, on that axis, the centre of the primordium, or when determining the edge of that otolith or the edges of its successive opaque zones. Otolith size measurements are thus likely to be affected by errors. As with age determination (Campana and Jones 1992; Campana 2001), there would be value in determining the precision of independent measurements of the same set of otoliths made on separate occasions by different readers, e.g., by calculating the coefficient of

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variation (CV). The establishment for each species of a reference set of otoliths, for use in training and refreshing skills in otolith measurement, would prove valuable in ensuring consistency of measurements.

Back-calculation of lengths at ages prior to capture is used to reconstruct individual growth histories of teleosts from the microstructure present within otoliths, and represents an important tool in fisheries science and fish ecology (e.g., Francis 1990; Campana 2005;

Vigliola and Meekan 2009). A central aspect for traditional back-calculation studies is its reliance on the relationship between fish length and measures of growth zones formed at validated, regular intervals in otoliths of fish. Although there have been many studies that use this relationship, by either using allometric relationships which integrate the effects of the growth of body and otolith size over the entire life history (e.g., Francis 1990; Tremblay and

Giguère 1992; Sirois et al. 1998; Vigliola et al. 2000) or linear relationships between these variables over a relatively narrow age range (e.g., Ozawa and Peñaflor 1990; Wright and

Bailey 1996), few have attempted to validate the resulting estimates of back-calculated lengths at age and even fewer have compared back-calculated lengths at age with recorded lengths at age for individual fish (Francis 1990; Vigliola and Meekan 2009).

Back-calculation validation studies based on individual fish involve tagging of fish, chemical marking of otoliths (e.g., through injection of tetracycline, release, subsequent recapture after a period at liberty), back-calculation to estimate the length of the fish at the time when its otolith was chemically marked, and comparison of that back-calculated length with the recorded length of the fish at release (e.g., Panfili and Tomás 2001; Li et al. 2008).

Although tagging experiments have been suggested as the best validation method by which to evaluate the performance of back-calculation formulae (Casselman 1983), only a small number of such studies have been undertaken (Vigliola and Meekan 2009). In this study, such a validation approach was precluded by cost and time constraints. In the latter context, it is

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pertinent to note that the period between the release of tagged, otolith-marked fish and their subsequent recapture in such validation studies is typically short (i.e., ranging from a month to a year) (e.g., Morita and Matsuishi 2001; Roemer and Oliveira 2007; Li et al. 2008;

Michaletz et al. 2009), and it is thus likely that the length of the fish at capture will be highly correlated with the length at which its otolith was marked. Results from the evaluation of

PORA in the current study suggest that, for such a short period, it would be highly likely that back-calculated lengths would prove to be reliable estimates of recorded lengths, particularly in the case of larger, older fish. Back-calculation studies, however, typically use otoliths of the older fish to estimate lengths at which the fish were much younger, i.e., age differences that extend over a number of years. Tagging studies to validate the accuracy of back- calculation over such extended periods would require survival of sufficient fish over similar periods at liberty. Moreover, to validate the accuracy of back-calculated lengths at younger ages when fish are possibly difficult to catch, a tagging study would need to release a considerable number of tagged, otolith-marked individuals of those younger age classes.

Despite these limitations, studies involving the release of tagged, otolith-marked fish, or maintenance of such fish in a laboratory, provide the only available direct method of assessing the accuracy of back-calculated estimates of length. It is recommended, however, that the ranges of ages of released tagged fish and associated times at liberty are recognised when considering the validity of the back-calculated length estimates.

Many contemporary back-calculation approaches assume constant proportionality of fish length and otolith size at capture to the expected sizes of those variables throughout the life of the individual fish (Whitney and Carlander 1956; Francis 1990; Vigliola and Meekan

2009). The finding of this study when investigating PORA that, in the case of A. butcheri from the Wellstead Estuary, proportionality of otolith radius to expected radius at different ages did not remain constant through the youngest ages raises a note of caution for such

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proportionality-based back-calculation studies. It is suggested that, to the extent possible, the age range of the fish employed in determining the length-otolith size relationship should extend to the youngest fish for which back-calculated estimates of length are to be determined. The finding also supports use of a biological intercept (Campana 1990) in the back-calculation approach.

A key premise when considering the implications of the results of sclerochronological studies of fish is that the effects of environmental factors on somatic growth are similar to those exhibited in otolith growth. Certainly, if this premise is true, valuable information on the factors affecting somatic growth could be elucidated. Somatic and otolith growth processes differ, however, with the former being associated with metabolism and energy allocation among activity, reproduction, cell maintenance and growth (e.g., Charnov et al.

2001; West et al. 2001). In contrast, however, while regulated by the biology of the fish and partially controlled by the deposition of organic matrix fibres, otolith growth is primarily determined by accretion of carbonate crystals (e.g., Wright 1991; Payan et al. 1998; Campana

1999; Morales-Nin 2000). As a consequence, determination of the effect of environmental variables on the relationship between annual increments in fish and otolith size continues to challenge fishery scientists. A possible direction for future studies may be to explore the relationships between otolith chronologies developed in sclerochronological studies (e.g.,

Matta et al. 2010; Gillanders et al. 2012; Black et al. 2013), or year-dependent growth derived from growth curves fitted to otolith increment data using mixed-effects models

(Weisberg et al. 2010), and parameters of time-varying growth derived from lengths at age of different year classes over the same extended periods of time (Szalai et al. 2003; He and

Bence 2007; Cottingham et al. 2016). To provide sufficient contrast in environmental conditions, such studies might benefit from application to populations in different estuaries of an estuarine-dependent species, such as A. butcheri (Chaplin et al. 1998; Potter and Hyndes

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1999; Sarre and Potter 1999; Cottingham et al. 2016), noting that, as demonstrated in the simulation component of this study, differing levels of exploitation would need to be taken into account.

Although the current study has focused on back-calculation and factors that affect this approach, its longer-term intent was to identify research strategies to determine the forms of the underlying relationship between somatic and otolith growth and how these relationships are affected by different environmental factors. The study has provided a promising new back-calculation approach, and has afforded an improved understanding of the growth of the otoliths of A. butcheri, and the factors affecting this growth and its relationship with somatic growth. Future field and experimental studies should be undertaken to explore in greater detail the effect of temperature and other environmental factors on both somatic and otolith growth, and on the relationship between these two processes. Such studies would provide valuable information of use to managers to account for the effect of environmental change and global climate change on fish stocks.

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APPENDICES

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Figure SA. Sectioned otoliths of the six study species, showing the location of the radius (red line) from248 the primordium to the outer edge of the otolith, to the right of the sulcus and perpendicular to the opaque zones. Scale bar = 1 mm.

Figure S3.3. Relationships

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GLOSSARY OF ACRONYMS

AE - Age Effect model (Morita and Matsuishi 2001)

BCF - Back-calculation formula

BI - Biological Intercept (Campana 1990)

BPH - Body Proportional Hypothesis (Whitney and Carlander 1956; Francis 1990)

MF - Modified-Fry model (Vigliola et al. 2000)

SPH - Scale Proportional Hypothesis (Whitney and Carlander 1956; Francis 1990)

TVG - Time-Varying Growth model (Sirois et al. 1998)

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FIN

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