Growth Models and Analysis of Crustacean Growth Data
Chuan Hui Foo B.Sc Edu (Hon), M.Sc (Hon)
A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2017
School of Mathematics and Physics Abstract
Crustaceans represent one of the most important fishery species in the world, both ecologically and economically. Understanding the growth pattern of these species is fundamental to their stock assessment and sustainability management. Unlike most other fishery species, crustaceans must shed their exoskeleton periodically in order to grow, a process known as ‘moulting’. As a result, their growth trajectories do not follow a linear pattern. Traditional growth models, how- ever, are based on continuous growth trajectories and thus are not appropriate for modellling stepwise growth in crustaceans.
This thesis develops novel methodology for modelling discrete growth pattern in crustacean species. We introduce new stochastic growth models that incorporate discontinuous jumps, taking into account individual heterogeneity and environmental variability. There are two different settings, data from an artificial condition (tank data) and data from the natural environment (tag-recapture data). We propose new approaches for modelling the growth of crustaceans from each data type, respectively. A likelihood approach is constructed to estimate the parameters of our growth models.
Our methodology addresses four major challenges in modelling crustacean growth. Firstly, as previously mentioned, crustacean growth is a discrete stepwise process and hence traditional models, which assumes continuous growth over time, is not appropriate for use.
Secondly, growth patterns are significantly affected by individual variability. The process of moulting involves interaction between two major stochastic processes of growth, namely the intermoult period (the time interval between two successive moults) and the moult increment (the increase in size between moults). The former varies significantly according to individual factors, such as sexual maturity. In particular, adult females are required to moult more often than adult males in order to produce juveniles and thus have a shorter intermoult period. Moreover, in general, the intermoult period increases with size, whereas the moult increment decreases over time. Thirdly, intrinsic variations and environmental conditions can also influence growth patterns. Biologically, all individuals possess a different terminal size that is partly determined by their genetics. Such phenomena are commonly referred to as forms of intrinsic variation. Apart from these variations, environmental factors such as water temperature, population density and food availability are strongly associated with growth rate. For example, maturity rate varies with habitat conditions, changing with different tank settings and natural habitat parameters.
Fourthly, further to the above, data obtained from the natural environment (tag-recapture data) are more challenging to analyse than those obtained from a laboratory environment (tank data). This is because the realisation of moult increments and the intermoult periods can be observed directly in the latter case, whereas the intermoult period is not available for tag-recapture data. In addition, conventional tag-recapture studies focus on ‘single recapture’ data which may lead to misleading results and are prone to biases given that individual heterogeneities exist in the population.
To account for the aforementioned issues including individual heterogeneity and environmental variability, we introduce a special case of L´evyprocess — a subordinator that allows for indefinitely small jumps to be incorporated into growth models. An appealing advantage of a subordinator-based model is that it can ensure a monotonic increase in growth, an important criterion for modelling lengthwise growth. Furthermore, we developed a novel methodology for analysing multiple recapture data, utilising a biologically realistic model that can efficiently describe the correlation between two consecutive moults, including the hidden variables with regard to data derived from multiple recaptures.
To quantify growth parameters of moult increments as well as intermoult periods, a maxi- mum likelihood approach is constructed given that they are conditionally independent of each other. These two probability functions are subsequently integrated through a simulation tech- nique. Our analysis provides a more realistic growth model for crustaceans from which critical information can be deduced, including how often an individual moults and how large can a moult increment be. In addition, the rate at which a crustacean reaches asymptotic size, and the variability of individual asymptotic size can be determined via a mean population growth curve. This thesis has contributed novel methodologies for quantitative modelling of crustaceans growth under both tank and (single/multiple) recapture scenarios, providing much more realis- tic analyses that will undoubtedly be useful for environmental sustainability, marine crustacean industry, and future research. Declaration by author
This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.
I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award.
I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School.
I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis. Publications during candidature
Conference abstracts and presentations
Chuan Hui, Foo and You-Gan, Wang (2013). Modelling growth data from crustaceans. Young Statisticians’ Meeting (YSI). Hong Kong, 23-24 August. (Poster)
Chuan Hui, Foo and You-Gan, Wang (2014). Estimation of molting growth parameters from reared lobsters. Applied Statistics and Public Policy Analysis Conference. Wagga Wagga, 11-12 December. (Oral presentation)
Publications included in this thesis
Chuan Hui, Foo and You-Gan, Wang (2013). Stochastic growth models for analyzing crus- tacean data. In: J. Piantadosi, R. S. Anderssen and J. Boland, Proceedings: MODSIM2013, 20th International Congress on Modelling and Simulation. 20th International Congress on Modelling and Simulation (MODSIM 2013), Adelaide, SA, Australia, (566-572). 1-6 December 2103.
Incorporated as Chapter 5 and Chapter 6.
Contributor Statement of Contribution Author Foo Developed the idea (50%) Wrote the paper (100%) Data analysis (100%) Author Wang Developed the idea (50%) Edited paper (10%) Contributions by others to the thesis
No contributions by others.
Statement of parts of the thesis submitted to qualify for the award of another degree
None. Acknowledgements
Firstly, I would like to express my special appreciation and thanks to my principal advisor Prof. You-Gan Wang who skillfully guided me through the stormy seas of methodological and statistical challenges with his remarkable forbearance, demonstrating high level of scientific precision and intellectual rigor, and ensuring that I did not sink along the way. I could not have imagined having a better advisor and mentor for my Ph.D study. Thank you also to my associate advisor Prof. Geoffrey McLachlan for his generous support that he has provided over the years.
I thank the Ministry of Higher Education (Malaysia) and the Sultan Idris Education University (UPSI) for their financial support during my candidature.
I acknowledge the dedication and assistance of all fellow postgraduates in the School of Mathe- matics and Physics who engaged with me either formal or informal discussion of issues relating to my research studies. The journey to completion of this thesis was a memorable one. The challenges were daunting but stimulating, with peaks to be scaled and valleys to be crossed. All works could never have been accomplished successfully without the backing of key people in my life, who fuelled my motivation and determination to succeed.
Above all, for their constant support, patience and understanding I acknowledge my family especially my beloved parents; for believing in me, showing me what is decent and supporting me spiritually throughout writing this thesis and my life in general. Keywords
Crustacean, moult, von Bertalanffy growth function, subordinator, growth model, heterogeneity.
Australian and New Zealand Standard Research Classifications (ANZSRC)
ANZSRC code: 010401 Applied Statistics, 60% ANZSRC code: 010405 Statistical Theory, 20% ANZSRC code: 060205: Marine and Estuarine Ecology, 20%
Fields of Research (FoR) Classification
FoR code: 0104, Statistics, 60% FoR code: 0103, Numerical and Computational Mathematics, 20% FoR code: 0501, Ecological Applications, 20% Contents
List of Figures xii
List of Tables xvi
Chapter 1 Introduction1
1.1 Overview...... 1
1.2 Background...... 2
1.3 Motivation...... 6
1.4 Aims and Contribution...... 8
1.5 Thesis Organization...... 11
Chapter 2 Crustaceans and Stepwise Growth 16
2.1 Characteristic of Crustaceans Growth...... 19
2.1.1 The Moult Increments...... 21
2.1.2 The Intermoult Periods...... 22
2.2 Crustacean Growth Data...... 25 CONTENTS vii
2.2.1 Tank Data...... 26
2.2.2 Tank or Laboratory Methods...... 27
2.2.3 Tagging Methods...... 28
2.2.4 Tag and Recapture Data...... 29
2.3 Challenges in Modelling the Growth of Crustaceans...... 32
Chapter 3 Growth Models 35
3.1 Deterministic Growth Models...... 36
3.1.1 The von Bertalanffy Growth Function (VBGF)...... 36
3.1.2 Generalised von Bertalanffy Growth Function (GVBGF)...... 42
3.2 Stochastic Growth Models...... 44
3.2.1 Incorporation of Individual Variability in Growth...... 45
3.2.2 Incorporation of Environmental Covariates in Growth...... 46
3.2.3 Stochastic Version of the VBGF (sVBGF)...... 50
3.2.4 L´evyProcess...... 53
3.2.5 The Characteristic of the L´evyProcess...... 54
3.2.6 Subordinators: A Special Case of L´evyProcess...... 63
3.2.7 Time Transformation Model...... 67
3.2.8 Dol´eanExponential Model...... 68 CONTENTS viii
Chapter 4 Modelling Growth for Crustaceans 70
4.1 Modelling of Moult Increments (MI)...... 72
4.1.1 Linear Mixed-Effects Models...... 72
4.1.2 Time Transformation Model...... 76
4.1.3 Dol´eanExponential Model...... 80
4.2 Modelling of Intermoult Periods (IP)...... 82
4.2.1 Cox Proportional Hazard Model...... 83
4.2.2 Modelling IP via Lognormal Distribution...... 85
4.2.3 Modelling IP via Gamma Distribution...... 85
4.3 Parameter Estimation via Nonparametric Approach...... 86
4.3.1 Simulation Studies...... 88
Chapter 5 Crustacean Growth Models for Multiple MI and IP 90
5.1 Introduction...... 90
5.2 Modelling Moult Increments With Individual Variability...... 91
5.2.1 Gamma-Gamma (GG) Model...... 92
5.2.2 Gamma-Lognormal (GL) Model...... 93
5.2.3 Inverse Gaussian-Gamma (IGG) Model...... 94
5.2.4 Inverse Gaussian-Lognormal (IL) Model...... 95
5.3 Modelling Moult Increments via Beta Distribution...... 96 CONTENTS ix
5.3.1 Likelihood Function With Beta Model...... 98
5.3.2 Beta-Gamma (BG) Model...... 100
5.3.3 Beta-Lognormal (BL) Model...... 101
5.4 Parameter Estimation via Numerical Integration...... 102
5.5 Convolution of MI and IP for Crustacean Growth Models...... 105
5.5.1 Introduction...... 105
5.5.2 Likelihood Approach...... 106
Chapter 6 Growth Estimation from Tag and Recapture Studies 108
6.1 Estimation from Single-Recapture Data...... 110
6.1.1 Estimation via Modelling Intermoult Periods...... 111
6.1.2 A Maximum Likelihood Approach...... 113
6.1.3 Individual Variability in Growth via an Improved Fabens Approach... 115
6.2 Estimation from Multiple-Recapture Data...... 117
6.2.1 Generalised Estimating Equations Approach...... 119
Chapter 7 Data Analysis of Lobster Species from Different Environments 122
7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 122
7.1.1 Modelling Moult Increments With Nonlinear Mixed Effects Models... 129
7.1.2 Modelling Moult Increments With Subordinator-based Models...... 133 CONTENTS x
7.1.3 Modelling Intermoult Periods With Cox Proportional Hazard Model... 143
7.1.4 Modelling Intermoult Periods With Generalised Linear Models (GLM). 147
7.1.5 Population Mean Growth Curve via Monte Carlo Simulation...... 151
7.2 Data Analysis of the Single Recapture Data from Field Experiments...... 154
7.2.1 IP Estimation Using Spiny Lobsters (Panulirus Argus)...... 154
7.2.2 Simulation Study...... 155
7.2.3 Maximum Likelihood Estimation for Nephrops norvegicus ...... 156
7.2.4 Simulation Study...... 158
7.2.5 Unbiased Estimation Using Regression Approach for Panulirus ornatus . 159
7.2.6 Simulation Study...... 160
7.3 Data Analysis of the Multiple Recapture Data...... 161
7.3.1 Slipper Lobsters (Scyllarides Latus) from Field Experiments...... 161
7.3.2 Generalised Estimating Equation Approach for Scyllarides Latus ..... 165
7.3.3 Simulated data...... 167
Chapter 8 Conclusion 172
8.1 Major Findings and Remarks...... 172
8.2 Future Directions...... 175
Appendix A The codes of capture-recapture 193 CONTENTS xi
A.1 The analytical method...... 193
A.1.1 For single-recapture data...... 194
A.1.2 For multiple-recapture data...... 195 List of Figures
1.1 The growth trajectories by length increase of ten individuals over time...... 7
1.2 A range of growth models for crustaceans...... 14
1.3 Ornate rock lobster (Panulirus ornatus) ...... 15
3.1 Special cases of the Richards model...... 40
3.2 Compound Poisson Process trajectories with exponentially distributed jumps.. 59
3.3 Gamma Process trajectories with different shapes, β ...... 61
3.4 Standard Brownian Motion trajectories...... 62
4.1 Length of an individual over time...... 78
5.1 Processes that exceeded the asymptotic length...... 96
5.2 Summary of modelling individuals moult increments...... 103
6.1 Panel A shows that a specimen has moulted...... 109
7.1 Lobster growth data over years...... 124 LIST OF FIGURES xiii
7.2 Distribution of carapace length, CL and growth mode...... 125
7.3 Intermoult periods against carapace length...... 126
7.4 Carapace lengths against time in days...... 126
7.5 Increments against time in days...... 127
7.6 Moult increments versus carapace length...... 127
7.7 Intermoult period distribution of reared lobsters...... 128
7.8 Moult increment distribution of reared lobsters...... 129
7.9 Observed versus fitted values plot for moult increment...... 131
7.10 Scatter plots of standardized residuals...... 132
7.11 Quantile for standard normal...... 132
7.12 Predicted moult increment based on cluster (density)...... 133
7.13 Estimated mean (Gamma increments and fixed L∞)...... 135
7.14 Estimated mean (inverse Gaussian increments and fixed L∞)...... 135
7.15 Estimated mean (gamma increments and gamma L∞)...... 136
7.16 Estimated mean (gamma increments and lognormal L∞)...... 136
7.17 Estimated mean (inverse Gaussian increments and gamma L∞)...... 137
7.18 Estimated mean (inverse Gaussian increments and lognormal L∞)...... 137
7.19 Estimated mean (beta increments and fixed L∞) with error bounds...... 138
7.20 Estimated mean (beta increments and gamma L∞) with error bounds...... 139 LIST OF FIGURES xiv
7.21 Estimated mean (beta increments and lognormal L∞) with error bounds.... 139
7.22 Comparison between individual growth paths and estimated lengths of females.. 140
7.23 Comparison between observed trajectories and estimated lengths of males.... 141
7.24 Comparing the mean of DE (blue) and TT (red), with observed mean (dashed). 142
7.25 Simulated growth trajectories with the subordinator be a gamma process..... 142
7.26 Survival curves for both sexes...... 143
7.27 Plots of scaled Schoenfeld residuals against time (days) for each covariate.... 145
7.28 Dfbeta based on index for each predictor...... 146
7.29 Martingale residuals to assess the linearity assumption...... 147
7.30 Estimating intermoult periods against premoult length...... 148
7.31 Individual growth curves at each temperature 25 − 30oC...... 149
7.32 The expected intermoult periods versus premoult length...... 150
7.33 The expected intermoult periods versus premoult length...... 151
7.34 Growth with fixed L∞ (top), random L∞ (bottom)...... 153
7.35 Log-likelihood for log(σ) of the lognormal intermoult period model...... 154
7.36 Estimated intermoult periods from the lognormal model...... 155
7.37 Carapace length over time for male tagged lobsters...... 163
7.38 Carapace length over time for female tagged lobsters...... 164
7.39 Increments(mm) over time for male tagged lobsters...... 165 LIST OF FIGURES xv
7.40 Increments(mm) over time for female tagged lobsters...... 166
7.41 Carapace lengths for males, 1994 − 1997...... 169
7.42 Carapace lengths for females, 1994 − 1997...... 170
7.43 A randomly selected sample with multiple recaptures over time(years)...... 171 List of Tables
1.1 A summary of different stages of the moulting process...... 3
2.1 Major groups of crustaceans...... 20
3.1 Different growth models in the form of a differential equation...... 41
4.1 Parameter estimates with the true values of l∞ = 120, k = 0.5...... 89
5.1 An individual i with Ni moults in a simulated data...... 97
6.1 The binary moulting status for each of the N individuals with k recaptures.... 118
6.2 Table of six types of recapture records in the form of probability function, Px(L, t).119
7.1 The number of individuals with different moulting times...... 123
7.2 Table of parameter estimates using mixed-effects model...... 130
7.3 Parameter estimates for different processes in male and female lobsters ...... 134
7.4 Process with beta increments and L∞ (gamma distribution)...... 138
7.5 Comparison of AIC measures among different models...... 139 LIST OF TABLES xvii
7.6 Table of the parameter estimates for Dol´eanexponential (DE) model...... 141
7.7 Table of the parameter estimates and p-values...... 144
7.8 Table of the χ2 and p-values for model 4.5...... 144
7.9 Parameter estimates for intermoult periods of lognormal function...... 148
7.10 Parameter estimates for intermoult periods of gamma function...... 149
7.11 GLM with mean estimates (ME) and standard errors (SE)...... 156
7.12 MLE approach with parameter estimates from tagged Nephrops norvegicus.... 157
7.13 Comparison of parameter estimates using MLE and Fabens methods...... 157
7.14 Maximum likelihood approach versus Fabens method...... 158
7.15 Comparison of parameter estimates...... 159
7.16 Unbiased regression approach and Fabens method...... 161
7.17 The number of individuals with different moulting times...... 164
7.18 Comparison of parameter estimates using GEE and improved Fabens methods.. 166
7.19 Parameter estimates using GEE and improved Fabens methods...... 168 List of Abbreviations
AFTA ccelerated Failure Time
AICA kaike Information Criterion
BICB ayesian Information Criterion
CLC arapace Length
COXC ox Proportional Hazard
DED ol´ean Exponential
EME xpectation Maximization
GEEG eneralised Estimating Equation
GLMG eneralised Linear Model
GVBGFG eneralised Von Bertalanffy Growth Function
IBMI ndividual Based Model
IPI ntermoult Period
MIM oult Increment
MLM aximum Likelihood
MLEM aximum Likelihood Estimation
REMLR estricted Maximum Likelihood LIST OF TABLES xix
SDES tochastic Differential Equation
SES tandard Error sVBGFs tochastic Von Bertalanffy Growth Function
VBGFV on Bertalanffy Growth Function List of Notation and Symbols
L∞ : Asymptotic length of an animal
l∞ : Estimated asymptotic length of the animals
W∞ : Asymptotic weight of an animal k : Growth rate to reach the asymptotic length of an animal
L0 : Initial length at time equals zero
Lt : Body length of an animal at age t
Wt : Body weight of an animal at age t
t0 : Gestation age at which length equals zero
Bt : Brownian motion which is described by Wiener process
Zt : Subordinators
Xt : Levy process ν : Pure jump in Levy process
γ : Drift coefficient of a Levy process
σ : Diffusion coefficient of a Levy process
φ(θ) : Characteristic exponent of a Levy process where θ ∈ R α : The frequency of jumps of a gamma process
λ : The scale that controls the size of jumps in a gamma process LIST OF TABLES xxi
∧ : Wedge product in differential geometry
ε : Random errors
Iij : Moult increment of animal i at jth measurement
Tij : Moult time interval of animal i at jth measurement
− Lij : Premoult length of animal i at jth measurement
LG : Individual carapace length at tagging
LR : Individual carapace length at recapture ρ : Time at liberty for a tagged specimen
Y : The time from capture to the next moult Chapter 1
Introduction
1.1 Overview
The contemporary methodology for growth models of organisms is based on continuous trajec- tories, and thus, they hinder us from modelling growth in crustacean populations. Crustaceans must moult in order to grow. The growth of crustaceans is a discontinuous process due to the periodical shedding of the exoskeleton when moulting.
To describe the discontinuity in growth, our growth models need to incorporate a stochastic component. Therefore, a subordinator is introduced, since it can be used naturally for mod- elling crustacean growth. As a non-decreasing L´evyprocess, the subordinator enables us to characterise the stepwise growth trajectories for crustaceans.
Two approaches are commonly used for growth parameter estimation: laboratory experiments for tank data and field experiments for tag-recapture data. For tank data, we have the exact moulting times from each individual over a period of time. These kinds of experiments are generally costly, and it is time consuming for an individual to complete its moult cycle. For tag-recapture studies, we do not observe the exact moulting time. However, data such as time interval between release and recapture (time at liberty) and carapace length at tag and recapture (increment length) are available. The tag and recapture method refers to a single 1.2 Background 2 recapture study. We can consider a generalisation of single recapture data, namely multiple recaptures for longitudinal data.
To obtain more realistic growth parameters, we must account for environmental factors in a growth model. We therefore incorporate covariates via a link function into the growth func- tion. Additionally, we develop semi-parametric approaches for data analysis, which offer al- ternatives for growth estimation besides parametric methods, such as a mixed-effects model. Semi-parametric methods may be preferable since the next moulting times remain unknown (right censoring) at the end of the experiment. This alternative approach is more robust, as no assumptions about the distribution of the variables of interest are required in a population. This thesis aims to address how growth discontinuities in crustaceans can be modelled and this information subsequently used to obtain a population mean growth curve in the form of a continuous trajectory.
1.2 Background
The growth of a crustacean is called ecdysis (Needham, 1946), more commonly known as moulting. A prerequisite for studying moulting crustaceans is the subdivision of a moult cycle into specific phrases. Moult and growth of individuals can be examined accordingly by studying the changes in the moulting physiology of crustaceans, provided there is sufficient knowledge of the moult cycle (Buchholz, 1991).
The moulting process was first defined by Drach (1939) and categorized into five major stages. These stages were revised by later researchers. As noted by Chan et al. (1988), the moulting process of many crustaceans can be categorized in four stages: metecdysis, anecdysis, proecdysis and ecdysis. The aforementioned five stages (A to E) can be classified as follows. During metecdysis, the soft shell (stage A) and thin layer period (stage B) are formed after moulting, and the animal expands its size by consuming great amount of water. This is followed by substantial food intake for tissue growth at anecdysis (stage C). Proecdysis or stage D can be identified from a series of colorations on an individual body (e.g., the fifth pereiopod of a blue crab). The onset of ecdysis is stage E, in which an animal absorbs water and the hard 1.2 Background 3 shell or exoskeleton is shed. A sudden increment after an inactive period of time renders a stepwise pattern of successive moults. Growth remains a continuous process after moulting. The various stages of crustaceans are summarized in Table 1.1.
Stages Description
Proecdysis (Premoult) This stage occurs before exuviation. A new soft exoskeleton is formed beneath the old shell. The epidermis separates from the old cuticle. A new epicuticle and exocuticle develop.
Ecdysis (Moult) This stage lasts only a few minutes. Moulting fluid is secreted when the new epicuticle emerged. The endocuticle is digested and absorbed by the fluid. The old exocuticle and epicuticle are shed.
After shedding, the only layers remaining are the Metecdysis (Postmoult) epicuticle and exocuticle. The body enlarges due to absorbing water and air. Hours later, the endocuticle is secreted. After hours or days, the soft cuticle hardens. (Chang, 1992)
Anecdysis (Intermoult) No growth occurs during this period. Tissue water is replaced with protein. All tissue growth is fully developed.
Table 1.1: A summary of different stages of the moulting process.
Substantial studies had been conducted over the years, and yet the researchers still encounter insufficient information concerning crustaceans’ growth parameters and trajectories in the wild. 1.2 Background 4
This is because a complex life cycle and a rich diversity of reproductive strategies exist in crustacean physiologies. The complexity of life histories is greatly shaped by both intrinsic and extrinsic factors. When examining the growth of a crustacean from larval to the juvenile phase and following maturity, researchers find that each of the stages may vary in growth characteristics. For instance, Kurata (1962) used a Hiatt growth diagram to relate post moult to premoult size in crustaceans, and found that the temperature and length of the animals were the factors of the intermoult periods. Additionally, the intermoult periods of ostracod Cyprinotus decreased as the temperature increased from 9◦ to 31◦C in Kurata (1962). The time scale used within the Hiatt growth diagram is logarithmic, as the intermoult periods of an animal increases logarithmically. Leffler (1972), on the other hand, found that blue crab growth is temperature-dependent under laboratory conditions; the highest growth rate occurs when the temperature falls between 20◦ C and 27◦C.
Marine crustaceans vary greatly in maximum size. The Japanese spider crab can be 4.3 m long while the tiniest copepods are only 0.1mm in size. Various factors affect the growth pattern in crustaceans. Most crustaceans, after some successive moults, will reach a stage with asymptotic length and do not moult thereafter. Knowing the terminal stage or the end of a moult cycle is crucial in determining the size at age of an animal. One of the reasons is to ensure that the age corresponds to the von Bertalanffy growth function (VBGF). An asymptotic length will be produced at a definitive moulting time.
The growth rate of individuals can be determined by the interaction of the two fundamental processes of moult increments (MI) and intermoult periods (IP). Hence, incorporating both processes into a growth model enables us to obtain a better estimation of population growth parameters. Hiatt (1948) estimated that the intermoult period increases with size for the lined shore crab Pachygrapsus crassipes. It is not likely to moult multiple times within a relatively short period for reared animals (Bianchini et al., 1998). Therefore, a juvenile crustacean moults more often than adults from the same species. Generally it can be observed from previous studies that the growth rate of an adult is slower than that of a juvenile. For example, Hartnoll (1982) observed that a crab usually grows slower when it has matured sexually, as energy has been allocated for reproduction purposes. Brewis and Bowler (1982) found that the increments at moult and the moult frequency both decrease over time. Robertson (1938) found that 1.2 Background 5 reared crabs smaller than 5 mm moult three times faster than those 12-25 mm, and the moult frequency slows down to 20-50 days after the individuals reaches a carapace length of 101 mm.
As mentioned by Hirose et al. (2010), an artificial environment (reared animals) can defer growth in terms of MI and IP compared to wild fisheries. Models for a complete moult cycle can be obtained directly from the IP and MI of reared crustaceans. In contrast, the intermoult period for wildlife crustaceans is estimated indirectly by measuring the length increment during time at liberty.
In 1948, Hiatt suggested plotting premoult size against postmoult size (Hiatt diagram) to estimate the average moult increment in size. This growth produced a linear function in a form of
Ln+1 = a + bLn , where Ln, Ln+1 are premoult and postmoult length, while a and b are the constant rates for the increments. The MI for inshore lobsters (Cooper, 1970) were smaller than those of the offshore lobsters of the same size (Cooper & Uzmann, 1971). A comparison of moult frequency showed that the growth rate for offshore individuals is higher than for inshore counterparts. Mauchline (1977) found that the logarithm of the IP and carapace length shows a linear relationship for the growth of shrimps, crabs and lobsters. These findings are in accordance with the studies done by R. Winstanley (personal communication), where an increase in size will decrease the MI, showing a nonlinear relationship in adult J. novaehollandiae from Tasmania.
Most crustaceans reproduce sexually and can be categorized as male or female. However, characteristics of the moult cycle are different for the two sexes. For example, fiddler crabs show a differential growth rate between sexes when males and females reached a carapace width of 7 mm and 9.2 mm, respectively (Hirose & Negreiros-Fransozo, 2007). Male crabs mate only when the exoskeleton is fully hardened, whilst among some species, the females will mate as soon as they have moulted (Costa & Negreiros-Fransozo, 1998). Additionally, blue crabs will reach their maturity after approximately 18 to 20 moults. Once reaching maturity, female blue crabs will only mate once in a lifetime, whereas male blue crabs can mate many times (Van Engel, 1958). In general, the pattern of growth and reproduction of a population may vary among the similar species. 1.3 Motivation 6
In the past, researchers have focused mainly on the continuous growth of animals rather than on discontinuous growth processes (Chang et al., 2012). Hence, we need to account for the discon- tinuities or jumps due to the moulting process in individuals. The main focus of this research is to develop stochastic growth models via subordinators that can accommodate individual step- wise processes to estimate the growth parameters of individuals. Subsequently, we can produce the corresponding population growth curve through a simulation of the subordinated-based model.
The analysis of discontinuous growth is used to mathematically estimate the asymptotic length of crustaceans (Hiatt, 1948; Kurata, 1962; Mauchline, 1976). Essentially, parameter estimation can determine the optimal length for harvest, which is crucial for a sustainable fishery. The most comprehensive description of the growth curve is the von Bertalanffy (1938) growth function, or VBGF. We will employ VBGF to estimate the growth of crustaceans, and further discussion will be provided in a later section. We will illustrate the moulting process of 10 individuals following the VBGF, where the asymptotic length is set at 150 mm, as shown in Figure 1.1.
An individual’s growth trajectory has a monotonic increment during its early moult stage. Subsequently, a decrease in the growth rate of the individual after several moulting times implies that it takes an animal longer time to moult as the carapace length gets larger. We notice that all individuals approach to their asymptotic length eventually.
1.3 Motivation
Crustacean fisheries play a vital role both economically and ecologically in global markets. Adult crustaceans are considered economically important to humankind due to the high demand as a food source. It is necessary to model precisely the growth estimates for sustainability of crustacean aquaculture. The growth in fisheries assumes a continuous change in the size of the animals. Nevertheless, the stepwise growth of crustaceans through the moulting process renders growth estimations more complex. Therefore, a different type of model is required for quantitative description of crustacean growth. In view of this, stochastic approaches can be sought as one of the solutions for modelling discontinuous growth or what are commonly known 1.3 Motivation 7
Figure 1.1: The growth trajectories by length increase of ten individuals over time. as “jumps”. However, in a stochastic growth model, we need to ensure that it results in only positive jumps. Hence, we will introduce a subordinator that is a special case of a Levy process. A subordinator can model a jumpy process with only positive jumps and a positive drift.
Another crucial part for modelling the growth of crustaceans is to account for growth variability, both internal and external. When we ignore the stochasticity, a biased estimation would be misleading in modelling the growth pattern and would provide inaccurate information in stock assessment if a large variability exists in the growth of the population. According to Hartnoll (1982), a significant variability exists within and between crustaceans for the entire growing 1.4 Aims and Contribution 8 process. In a context of an individual (internal) variability, the growth of crustaceans is not always consistent throughout a moulting process. Many literature reviews stated that smaller- sized individuals tended to moult more frequently and increase more in length compared to the older counterparts. For external variability, a range of environmental factors need to be considered. These include water temperature, salinity, food availability and population density.
Since all hard shells are shed frequently, no one has yet found a way to determine the exact age of a lobster. However, we are still able to estimate growth parameters for the aforementioned moult processes based on various types of data, such as tank data and tag-recapture data.
The motivational dataset is from the ornate rock lobster, Panulirus ornatus, commonly known as the painted tropical lobster, which can be found between Australia and Papua New Guinea. Lobster fishery is a major commercial enterprise for the inhabitants of the Torres Strait and was initiated in the late 1960s along the far northeast coast of Queensland. The annual migration of lobsters occurs in early August to early September and induces great changes in the population. The larger lobsters tend to emigrate while the smaller lobsters stay in the Torres Strait. The female lobsters are more likely to migrate than the males, leading to a sex ratio imbalance (Pitcher, 1992). Because of the effects of sex on growth rates (Munday et al., 2004), we therefore estimate the growth parameters separately for each sex.
Figure 1.2 shows how growth parameters for crustaceans can be estimated from two types of studies, tagging and reared laboratory studies, through the stochastic process approach.
1.4 Aims and Contribution
The aim of this thesis is to apply stochastic growth models to characterize the individual stepwise growth trajectories in crustaceans. We introduce a unified approach that explicitly displays the stochastic nature of crustacean growth. A growth curve can be yielded by inte- grating both conditionally independent functions, moult increments and intermoult periods. For estimating the growth parameters, we focus on two types of datasets: reared (tank) data and tag-recapture data. Once we obtained the estimated growth parameters, we present a 1.4 Aims and Contribution 9 realisation of the variables of interest by forming a continuous population mean growth curve.
The discontinuous growth paths occur throughout the moulting process. We may look into a model for which the growth path of each individual in the population incorporates a total of no growth and instantaneous growth through a period of time. By viewing this discontinuity, we will investigate a L´evyprocess, specifically the subordinator, to characterize the jumps mathematically for each individual growth trajectory.
Specific aims include the following:
• To develop stochastic growth models that can accommodate discontinuity in the growth of crustaceans.
• To estimate the growth parameters when the age of the individuals is not available in the data.
• To investigate different types of data for parameter estimation, both tank data and tag- recapture data.
• To model growth via two interrelated stochastic processes, one for the time between two successive moults and one for the increments when moulting. We will derive the joint density functions for statistical inferences.
• To incorporate individual and environmental variabilities into a growth model.
Growth trends in crustaceans are complex. Due to the intrinsic variability in crustaceans, where the moult frequency decreases over time and the time interval between moults increases over time, estimating growth is difficult, especially after specimens reach the maturity stage. Females will have a lower growth rate than males, as more energy will be stored for reproduc- tion. Furthermore, we have discrete growth paths in crustaceans species. Crustaceans attain increments when they moult, implying positive “jumps” in a moult cycle. Despite the wide variety of references in the literature concerning growth modelling, the traditional models are not guaranteed to represent an increasing growth function over time for the crustacean pop- ulation. For this reason, we propose a special case of L´evyprocess (subordinator) into the 1.4 Aims and Contribution 10 candidate model. This approach is advantageous in describing the discontinuity, as a L´evy process contains discontinuous paths. Moreover, a subordinator is an increasing L´evy process which is in good agreement with a positive, segmented growth pattern.
The ages of fish can easily be determined by an assessment on their otoliths, spines and scales (Smith, 1992). Yet, the absolute age of crustaceans is most unlikely to be identified by the removal of the hard shell when they moult. The age of crustaceans remain unknown, but we can look for alternatives such as Faben’s method to find out their relative length at a particular age. Previously, the aforementioned model assumed the maximal length for all individuals as a parameter. Here, we make an improvement in the growth model by assuming the asymptotic length as a variable. This modelling framework accounts for the natural heterogeneity of growth, as each individual posseses a different terminal length in reality. We claim that this approach provides a more reliable and biologically comprehensive representation of growth in the estimation of growth parameters.
Parameter estimation becomes relatively straightforward when the moulting times are observed, although random effects or frailties are necessary to take account of individual variability. Alternatively, we may consider tag-recapture data for which time at liberty and increment length are given. In this context, time at liberty indicates the time between tagging and recapture in which a specimen is released to the wild.
Two core stochastic processes which determine the individual growth are the MI and the IP. MI indicates the change in size at moult, while IP means how often an animal moults. Since both MI and IP occur concurrently during the process, they are interrelated to each other. Therefore, the growth parameter estimation may not be straightforward. First, we deconvolute both MI and IP density functions with an assumption that each process is conditionally independent. Subsequently, we employ a maximum likelihood estimation method to attain parameter esti- mates for each function. Finally, we establish a unified likelihood approach in such a way that the corresponding processes will be integrated through simulation. A population mean growth curve will then be produced to describe the growth trend for the population. This modelling approach is applicable to both types of datasets aforementioned. 1.5 Thesis Organization 11
1.5 Thesis Organization
The organization of this thesis is as follows.
Chapter 2 introduces some general structural characteristics as well as the population dynamics of marine crustaceans. In addition to that, the components of growth in crustaceans are studied, and a series of models proposed by certain researchers will be reviewed in the discussion. Various approaches are used for modelling the growth of crustaceans. Different environmental conditions can result in different growth patterns of a population. In this chapter, two of the most common methods will be considered: (1) laboratory experiments, and (2) tag-recapture experiments. Both types of data collection are discussed for different settings.
Chapter 3 reviews and discusses some of the fundamental concepts related to modelling growth with stochastic processes. In general, there are two types of growth models, the deterministic and stochastic models. In deterministic models, the outcomes are conclusively determined through a known relationship between the parameter values and the initial conditions without random variations. Deterministic models usually describe processes by differential equations, which essentially model the average behaviour of the system. In reality, the actual world is stochastic and subject to various dynamic random components, such as intrinsic stochasticity and extrinsic stochasticity. Intrinsic stochasticity describes the individual variabilities while the extrinsic stochasticity describes the changes due to environmental conditions.
Chapter 4 develops and models the individual growth parameters with the association of stochastic components to describe the discontinuous behaviour of crustaceans. Crustaceans grow discontinuously, but traditional growth models focus on continuous functions, and hence the validity of these models has been questioned for certain types of organisms (Miller & Smith, 2003). The traditional models — such as VBGF, Faben’s method and the Gompertz curve — do not account for stepwise trajectories in individual growth processes, specifically for crus- taceans. Stepwise growth paths occur throughout the life of crustaceans. The sudden changes in length can be modelled by a process that contains jumps at different times. One option is to apply a subordinator-based model to mathematically characterize the jumps for individual growth trajectories. Emphasis will be placed on growth models associated with a stochastic 1.5 Thesis Organization 12 component known as a subordinator. Through our modelling framework, we highlight two most important intercorrelated stochastic processes of this work; one is to model the moult time interval and the next is to model the moult increment. These two parameters can be es- timated through laboratory and tag-recapture studies. In this chapter, we focus on laboratory experiments (tank data) in which the growth of crustaceans is studied in artificial conditions. The growth rate is believed to depend on some commonly measured covariates (Wang, 1998a; Wang, 1999). Therefore, we will investigate the effects of environmental factors, such as wa- ter temperature and density, into each given growth function. We make use of our model’s parameters to describe individual growth trajectories, and hence the population mean growth.
Chapter 5 is the core of the thesis; it adds distinctive and novel findings to the primary research that was undertaken in chapter 4. The contemporary models assume that the growth parame- ters are to be fixed when heterogeneity exists in a population, resulting in a biased estimation. In nature, each individual is buffeted by stochasticity and thus we take into consideration the individual variability in quantifying the growth of crustaceans. By integrating a random vari- able into candidate models, it becomes difficult to obtain an analytical solution explicitly for a complex model. To this end, we employ a numerical integration technique as a solution to the aforementioned problems. In addition, a likelihood approach is developed to deconvolute the crustacean growth components based on the Markov property. Once the parameter values are obtained, each individual growth trajectory can then be characterised by a Monte Carlo convolution method.
Chapter 6 investigates the tag-recapture approach to estimate growth parameters of wild crus- taceans. For tank data, we have the exact moulting times for each individual over a certain period of time. Tank experiments are generally costly. The other common datasets come from field experiments (tag-recapture data). A sample of the population is captured, tagged and released to its natural environment. Once the tagged animals are recaptured, a set of time-at- liberty data and length of carapace will be recorded. The exact time of a moulting event usually is unobserved. This is the most commonly used method to estimate the growth parameters for wildlife crustaceans. For the multiple recaptures data, an unknown set of moulting times and moult increments may exist between the captures. A more pragmatic mechanism to model the longitudinal data in multiple recaptures will be included in this chapter. We explore some of 1.5 Thesis Organization 13 the problems that one can encounter in parameter estimation and how a statistical method can be applied to infer the latent variables, such as the unobserved moulting times and the number of moulting times (since an animal may moult more than once during its time at liberty).
Chapter 7 focuses on the application of both types of data, the tank data and the tagging data. Several case studies will be investigated; for examples, the first tank dataset was obtained from the ornate rock lobster, Panulirus ornatus. It is commonly known as the painted tropical lobster (Figure 1.3), which can be found between Australia and Papua New Guinea. Besides tank data, single-recapture data and multiple-recapture data are included in the case studies. Three modelling approaches — the linear regression method for unbiased estimation (Wang, 1998b), the maximum likelihood estimation (Wang et al., 1995) and the deterministic and log- normal intermoult period models (Millar & Hoenig, 1998) — are studied for single-recapture data. Three different datasets will be used: (1) the ornate rock lobsters, Panulirus Ornatus, provided by Wang (1998b), (2) the Norway lobsters, Nephrops norvegicus, reported by Shelton and Chapman (1995), and (3) the spiny lobsters, Panulirus argus, obtained from Little (1972). Further, the multiple-recapture data will be modelled through the generalised estimating equa- tion approach (Wang, 2004) along with data for the slipper lobster, Scyllarides latus, provided by Bianchini et al. (2001). Simulations will aim to test the variations and bias of the estimated parameters corresponding to the modelling approaches. Further, the application to real data is presented and analysed in accordance with the candidate models.
Chapter 8 summarises the main findings of the thesis, which include a marginal probabil- ity approach to incorporating individual asymptotic length into moult increment models via numerical techniques, the moult increment model via beta distribution that overcomes the problems encountered in a simulation study, and the hidden variables in multiple-recapture data. In addition, some of the recommendations for future research that originate from the proposed methods will be specified. 1.5 Thesis Organization 14 Figure 1.2: A range of growth models for crustaceans 1.5 Thesis Organization 15
Figure 1.3: Ornate rock lobster (Panulirus ornatus) Chapter 2
Crustaceans and Stepwise Growth
Crustaceans are a very large group of arthropods that are known to exist mainly in aquatic habitats. In general, arthropods are invertebrate animals with bodies that are divided into a head, a thorax and an abdomen, a pair of eyes, two pairs of antennae, segmented parts, joined limbs, and external hard shells, called exoskeletons. Since these exoskeletons are not stretchable, they restrict the growth of crustaceans. As a result, the old skeletons need to be replaced by new cuticles, or skins, through periodical moults. Although the majority of crustaceans are aquatic animals, many of them, such as the beachhoppers, can be found living on land.
Classification
Crustaceans vary in size from microscopic plankton of less than 1mm in length to enormous crabs up to 4.3 m long. Diverse forms of crustaceans can be distinguished from other groups of arthropods through the following sub-classes, which all have different characteristics as featured in Table 2.1. 17
Branchiopoda
Many of these are small marine species, such as clam shrimp and water fleas, eat mainly plankton. In most cases, Branchiopoda have a carapace-stalked eye antenna with at least four pairs of trunk limbs.
Clam shrimps are usually found in fresh water, and their carapaces are enclosed within two shells that are hinged together, which allows them to open and close alternately. Water fleas are very small, ranging between 0.5 and 3 mm in size. They swim by leaps to enable them to grasp their food through their second pair of antennae. Those that live in a freshwater habitat are filter feeders in contrast to some Branchiopods that exist in ocean environments and are predators.
Maxillopoda
Maxillopoda is a class of crustacean that includes Barnacles, Branchiura and Copepods. These crustaceans have short bodies and usually do not have appendages, most of their bodies being comprised of five head segments, six thoracic segments and four abdominal segments, followed by a telson, or tail.
Branchiura, or fish lice, are external parasites that exist on fish. They can grow to around 7 cm in diameter while Copepods are usually less than 2 mm in length. In addition, Branchiura possess flat, oval bodies that are almost completely covered by a wide carapace.
Barnacles are usually found living in shallow and tidal waters. When they have grown into adults, they live permanently fixed to hard surfaces including rocks. Due to the harsh conditions in which they exist, they remain in their calcium carbonate shells to protect them from predators and to prevent them from drying out. Adults Barnacles have one eye so they can detect light and dark. The majority feed on plankton, although some species are parasites.
Copepods have been found living on the sea floor, and almost all of them have freshwater habitats. Most of them are between 1 to 2 mm in length. Since they are small in size, 18
Copepods do not need any cardio-system or gills; they absorb oxygen from the water directly into their bodies through diffusion. There are two larval stages and many moults that occur before Copepods reach maturity. Depending on which type of this species they are, it can take from approximately a week to a full year from the time they are hatched until they reach maturity.
Ostracoda
The habitat of Ostracoda is mainly in the sediments at the bottom of oceans or lakes, and they grow from 0.1 to 32 mm in length. Ostracoda can be carnivores, herbivores, or filter feeders. Their bodies are protected by a shell that is replaced with each moult. They rely on two pairs of antennae to swim. Most species have no gills or heart, and their major sense is provided by their appendages. Some Ostracoda have light-producing chemicals in their body system. This light is treated as a defence against predators, and some use it for mating purposes.
Malacostraca
Malacostraca is the largest among the sub-groups of crustaceans. These are segmented to include Decapods (crabs, lobsters, and shrimp), Stomatopods (mantis shrimp), Euphausiids (krill), Amphipods (sandhoppers) and Isopods (woodlice).
The main features of Malacostraca include a five-segmented head, an eight-segmented thorax and a six-segmented abdomen. They have eight pairs of thoracic legs, and the first few pairs are often treated as feeding appendages. The abdominal appendages, called pleopods, are generally used for swimming. Malacostraca usually have two-chambered stomachs and internal gills and a circulatory system with a large brain and sensory organs that include compound stalked eyes. 2.1 Characteristic of Crustaceans Growth 19
2.1 Characteristic of Crustaceans Growth
Crustaceans do not grow in a smooth and continuous fashion like other marine species. Because their exoskeletons are hard and not expandable, they need to be shed in order for the crustaceans to grow. The process of getting rid of the shells from their bodies is called moulting.
Growth in crustaceans is limited to the moulting, resulting in stepwise trajectories rather than a linear form of growth path. The moulting process includes a series of phases. Once the hormone called ecdysone has been secreted and released into the body, moulting will automatically come into play. Initially, a new exoskeleton overlays the existing one, and the animal will subsequently absorb large amount of water rapidly, enlarging its body to withdraw the old exoskeleton from the new cuticle. After moulting, the crustacean appears to be vulnerable to predators, and this has been attributed to the new exoskeleton. These processes require extra energy to produce many layers of cuticle before a new exoskeleton can be formed during moulting.
The period between one moult and the next is called the intermoult, and during this time no growth occurs. However, at the intermoult phase, changes in the body are continually in progress. Initially, because the animal is full of water, it is soft and takes time to harden. This water will be replaced with mineral and protein decomposition in the body. A chemical reaction with the oxygen or water causes the cuticles to expand while the exoskeleton is gradually becoming harder through a bio-chemical process. The carapace will be the first to harden, followed by the legs and cuticles.
Because there is no growth during intermoult periods, the growth trajectories of crustaceans are discontinuous, with a spontaneous increment in each successive moult. Consequently, we describe the growth of crustaceans through two core processes called intermoult periods and moult increments. 2.1 Characteristic of Crustaceans Growth 20
Class Examples Features
Brine shrimp Clam shrimp Branchiopoda Notostraca Water fleas Fairy shrimp Clam shrimp
Copepod Maxillopoda Barnacle
Copepod
Ostracoda Seed shrimp
Seed shrimp
Prawn Krill Crab Malacostraca Lobster Crayfish Fish lice Mantis shrimp
Spiny rock lobster
Table 2.1: Major groups of crustaceans 2.1 Characteristic of Crustaceans Growth 21
2.1.1 The Moult Increments
Moult increments in crustaceans are generally described through a Hiatt diagram (Hiatt, 1948).
This is the first model that describes the carapace length after the moult (post-moult, Lj+1) based on the premoult carapace length, Lj of the crustacean. The Hiatt equation is defined as
Lj+1 = β0 + β1Lj + ε ,
where β0, β1 are parameters, Lj is the carapace length at jth time and is an error term that is presumed to be normally distributed. This model has been employed by many authors, including Kurata (1962), Donaldson et al. (1981), and Somerton (1980). Mauchline (1977) as well as Easton and Misra (1988) have objected to the use of such linear models as the carapace length is believed not following a linear regression in reality.
Mauchline (1976, 1977) has criticized the supposition that both the increment and the growth factor, Ij/Lj, are not constants but decrease logarithmically with the premoult length in the form of
log(Ij) = β0 + β1Lj + ε , and
log(Ij/Lj) = β0 + β1Lj + ε ,
where Ij = Lj+1 − Lj.
According to Easton and Misra (1988), the linear relationship of postmoult and premoult length is not allowed to describe population mean growth as it does in VBGF. As a result, a modified version of the Hiatt model is addressed in the following form:
β0+β1Lj β4 Lj+1 = β3Lj e + ε .
In addition, Wainwright and Armstrong (1993) considered the Misra equation for modelling the
β0+β1Lj β3Lj moult increments of the Dungeness crab in the form Ij = β2Lj e − Lj . The resulting 2.1 Characteristic of Crustaceans Growth 22 estimates indicate that the randomness between juveniles and adults was significant in the studies.
Another proof by Luppi et al. (2004) has suggested a quadratic model Ij/Lj = β0 + β1Lj + 2 β3Lj + ε to describe the growth pattern, since the moult increments decreased slowly with the lengths. As Botsford (1985) also noted, a more sensible approach to model growth is to use moult increment as a response variable, because the postmoult length or growth factor may not represent a true correlation between variables. Furthermore, Castro et al. (2003) has claimed that since the increments become smaller as the lengths grow larger, it is not a linear form of regression. The Lj+1 was therefore reparameterised to Ij as a function of the premoult length in all candidate models. For example, one of the candidate models from Wilder (1953), the
β1 β1 log(Lj+1) = β0Lj + ε, has been converted into Ij = β0Lj − Lj + ε.
Corgos et al. (2007) quantify the growth of tag-recapture spider crabs of Maja brachydactyla and then compare these to two other data from tank and culture studies done by Gonz´alez- Gurriar´anet al. (1995). They have constructed the following linear models that can accom- modate three different environmental conditions for this data:
Ij = β0 + β1Lj + ε , and (2.1)
log(Ij/Lj) = β0 + β1Lj + ε . (2.2)
The resulting estimates show that the log growth rate in Eqn (2.2) is the preferred model because of the lower value of the coefficient of variation compared to Eqn (2.1).
2.1.2 The Intermoult Periods
A straightforward way to estimate the intermoult periods is through a measurement from the tank data in aquaria. Growth studies from tank animals are not biologically comparable to those taken from wild populations, because they have different growth rates. However, the estimations for tagging data are not as direct, since the exact moulting time is generally unknown. The intermoult period is much more difficult to determine when it is studied in 2.1 Characteristic of Crustaceans Growth 23 nature. According to Hancock and Edwards (1967), tagged animals will be marked, measured and released into the wild. After a period of time, the information that is of interest can be gathered by recapturing the animals from the wild.
Kurata (1962) has developed a growth model that describes a linear correlation between the intermoult period for a crustacean species and the cube of its body length. We have the inter- moult period at jth time as a function of the length of a form as follows:
3 Tj = β0 + β1Lj + ε .
An exponential growth curve can be observed based on the variables of the intermoult periods and the length of the data. A transformation to a linear regression can be done by taking log intermoult periods based on body length.
In addition to this, Castro (1992) modified and replaced a parameter to the cube of the length, as in Kurata’s model, for modelling the intermoult period:
β3 Tj = β0 + β1Lj + ε .
Additional information about estimating the intermoult period were collected from different authors such as Thomas (1965), Hillis (1971), Figueiredo (1975) and Sard´a(1983, 1985) to study the growth trend of crustaceans. Overall, the estimated intermoult period of Norway lobster Nephrops norvegicus revealed an exponential increment in terms of the length.
Mauchline (1977) has suggested two explanatory variables, the body length at jth time, Lj, and the number of moults, N, for estimating the intermoult periods of shrimps, crabs and lobsters. The following represents a linear correlation between the intermoult period and the independent variable:
Tj = β0 + β1Lj + ε ,
log(Tj) = β0 + β1Lj + ε, or
log(Tj) = β0 + β1N + ε . 2.1 Characteristic of Crustaceans Growth 24
Mauchline plotted the log intermoult period against the body length as well as the number of moults and produced a significant relationship in the large-sized crustaceans. However, the log intermoult period is only applicable for juvenile specimens, because adults may need a longer period (once annually) to moult in this context (Wainwright & Armstrong, 1993). As a result, Wainwright and Armstrong (1993) confined data to include juvenile crabs from aquaria and described growth rate using the log intermoult period according to Mauchline (1977).
Some crustaceans may moult at any time throughout the year, and it is therefore assumed that the accumulated time since the previous moult be uniform for randomly selected crustaceans (Hoenig & Restrepo, 1989). The intermoult period is defined as
Tj = β0 exp(β1Lj) + .
However, when the exact intermoult periods have been observed and quantified from a group that includes those of same size, the durations of the moults differ from one to the other. There- fore, a lognormal distribution is likely to be a more accurate way to describe the intermoult period, rather than assuming it as a uniform function (Restrepo, 1989). We have the following generalized model:
2 log(Tj) = exp(µj + σj /2) + ε ,
2 2 where µj = log(β0) − σj /2 + β1Lj, while µj and σj are parameters of the lognormal model.
Gonz´alez-Gurriar´anet al. (1998) have examined the tank Norway lobsters and described the intermoult period in relationship to the premoult length using Tj = β0 +β1Lj +ε and log(Tj) =
β0 + β1Lj + ε from Mauchline (1977). An average intermoult time for females and males reared in a tank ranged within (100,260) days. The estimates of Tj based on premoult length has a better fit than the log (Tj) estimates for Norway lobsters. 2.2 Crustacean Growth Data 25
2.2 Crustacean Growth Data
This thesis focused mainly on marine crustacean species of decapods, specifically lobsters. Lobsters are ten-legged crustaceans that live on rocky, sandy, or muddy bottoms away from the shoreline. They use claws to defend themselves, attack, and catch their prey. When a claw is lost, a new one grows in the next moult. A lobster will moult up to a maximum of 25 times in the first five years of its life; after becoming an adult, it will moult about once a year. Most lobsters are 25 to 50 cm in length. The colour of their exoskeletons very depending on the surrounding environment to prevent them from being attacked by predators: lobsters are blue in deep European water, while American lobsters are brown or green. Female lobsters carry their eggs under their abdomens for about a year before releasing the larvae. Lobsters are very sensitive to changes in temperature; they will migrate on a yearly basis to find the best hatchery for their fragile babies.
Kurata (1962) has noted that a crustacean has three phases: the larval, juvenile and adult phase. He also stated that growth in arthropods is not a continuous process.
There are two main groups of lobsters that are marine arthropods: Rock lobsters and clawed lobsters. Both types of lobsters typically exist in saltwater. Rock lobsters, also known as spiny lobsters, are known to have extremely long antennae but do not possess a distinctive claw. In order for rock lobsters to protect themselves, they have sharp thorns all over their bodies and two horns on their heads. Rock lobsters live in warmer waters in the Pacific, Atlantic and Indian oceans, and also in the Red and Mediterranean seas.
Unlike rock lobsters, clawed lobsters or maine lobsters have large claws but relatively smaller antennae. Clawed lobsters live in cold, rocky crevices and shallow water. They can be found in Newfoundland and North Carolina.
Crayfish, a freshwater crustacean which are sometimes mistaken for lobsters due to their some- what similar outer appearances, are found in vastly different habitats including rivers, lakes, dams, streams and ponds. 2.2 Crustacean Growth Data 26
2.2.1 Tank Data
The characterisation and quantification of crustacean growth can be achieved by both direct and indirect approaches. The growth per moult of animals that are reared in tanks are observed and measured directly. In addition to obtaining direct measurements, there are other advantages such as having factorial experiments that can be used to determine the internal and external effects in individual growth (Van Olst et al., 1976). In addition, growth studies from tank data provide an opportunity to identify the key factors that influence the transition from intermoult to premoult stages in a moult cycle (Aiken, 1980).
The larval development is essentially temperature sensitive, and the larval size can therefore vary with temperature. For example, growth among American larval lobsters is greater at optimum temperatures, from 15◦C to 18◦C, than it is among lobsters reared at lower or higher temperature regimes (MacKenzie, 1988). Although the cool water larvae, Jasus and Palinurus, grow well at 20◦C, and the warm water larvae, Panulirus, can be reared successfully at 25◦C, the larval spiny lobster can hardly be reared for tank growth studies at all (Kittaka, 2000). The moulting frequency of the Bermuda spiny lobsters appears to be highest during the summer months, when the water temperature has reached 28◦C to 30◦C (Travis, 1954). In contrast, the number of those that are moulting has to stop as water temperatures decline to below 19◦C during the winter.
Wang and McGaw (2016) realized that a low moulting process can be attributed to insufficient food intake and low temperatures. High quality complementary feeding is needed to yield a commercial catch of lobsters.
Numerous factors such as temperature, food availability and salinity influence the growth rates, and social conditions control the growth rates as well. For example, when lobsters are reared in a tank (living in a community), the growth rates are aggressive on average (Van Olst et al., 1980). This phenomenon occurs as the existence of dominant lobsters prompts additional intermoult times among the subordinates as a result of competing for food and oxygen in the tank, implying a lower growth rate based on social interactions (Cobb & Tamm, 1975).
Habitat can also be correlated to the growth rate in a population. An environmental condition 2.2 Crustacean Growth Data 27 that is more vulnerable to predation correlates to a higher mortality rate than in a secluded place. As noted by Wahle and Steneck (1992), tank lobsters that are sheltered with cobble manage to survive twice as long as those in “open” or shelter-less tanks. In addition, the moult increments of slipper lobsters in laboratory settings (tank data) can deteriorate, most likely as a result of captivity conditions (Bianchini et al., 2003).
The level of the food supply also has a significant effect on the frequency of moulting, espe- cially when limitations in food render a low growing rate in juvenile rock and spiny lobsters (Chittleborough, 1975; Vijayakumaran & Radhakrishnan, 1986). In addition, when the food supply is limited, moult increments will be reduced while intermoult periods will be increased in the larvae of spanner crab as well as in the European green crab (Hartnoll, 2001).
The growth of animals that live in laboratory settings varies from their counterparts in the wild. For example, juvenile lobsters in aquaria that imitate the ocean bottoms grow better than those in natural conditions (Jørstad et al., 2001).
2.2.2 Tank or Laboratory Methods
Animals were collected from the wild, brought into the laboratory and marked with an individ- ually numbered tag that included a description of the individual details of each of them. The initial weights, and lengths (in mm) as well as the sex of the animals were measured, identified and noted.
Feeding was practised according to different diets ranging from a natural mixed diet (shrimp, fish, squid, scallop or crab) to a diet of frozen mussel. Tank animals were normally fed two to three times per week. After one or two weeks, the tanks were cleaned, and any remaining feed and dead animals were removed.
In laboratory experiments, various selective factors such as water temperature, food nutrients and the number of feeding times can be used to determine how likely these are to affect the survival and moulting of the animals. 2.2 Crustacean Growth Data 28
Tank animals are usually checked on a daily basis. Any animals that have moulted or not survived are noted and recorded. In experimental tanks, a constant flow of ocean water is generally supplied to the animals, representing temperature experienced by those living in sea conditions.
The tank animals were maintained at certain temperatures, about 10◦C, 15◦C, and 20◦C, with different compartments equipped for different temperature settings. The water temperature regime in each tank was monitored closely everyday, while water oxygen remained at a consistent level through the control of an air system inside the tank.
The growth of crustaceans can be measured through their weight or the length of their carapace. To measure the body mass of animals such as lobsters, they were taken out of the tank and their bodies are dried before they are weighed, to minimize measurement errors. For the carapace length measurement, an accurate ruler is placed between the antennal horns in such a way that it fits the notch well.
2.2.3 Tagging Methods
Tagging and marking are useful techniques for studying the life histories of marine fishes. Animal tagging enables biologist to learn how rapidly an animal grows, the length of its life span and the actual sizes of populations in certain regions. Essentially, fish tagging experiments are conducted with a sample of captured fish being marked by a tag and then released. Once the tagged animals have been recaptured, a wide variety of information can be tracked, providing biologist with a better understanding of some phase of the animals’ life pattern.
The key issue during capture is the survival, reproduction and growth of the animal that is being tagged. For example, commercial fishing is aimed at harvesting rather than at keeping animals alive and considering their well-being. Fish may have been under stress for certain periods of time and, therefore, not be appropriate for tagging purposes. An excess of lactic acid can be produced continuously in the blood after the fish have undergone stressful conditions (Wendt & Saunders, 1973). In addition, trawling itself can render damage to individual animals, such as when rocks or sharp garbage causes injury to the captured species (Jones, 1979). 2.2 Crustacean Growth Data 29
Over the years, various tagging or marking approaches have been used. There are two common categories of modern tagging techniques. The tag-recovery approach involves a sample of ani- mals that are tagged, then released, and have died as soon as they were recaptured. Normally, the process of tagging involves the efforts of scientists and a large number of commercial fisher- men as well as volunteer fishermen. Conversely, capture-recapture experiments are essentially done directly by scientists, with the tagged animals being recaptured when they are alive. This method is more labour intensive than the tag-recovery studies.
A variety of tag types have been used to mark the animals, based on the fact that although an identical tag applies for many different species, other tags can be documented for particular types of animals. Regardless of the types of tags used, all tags have a common characteristic, with the following information printed on them: a serial number and the name of the agency that is conducting the experiment. Some examples of different types of tags include external tags, internal tags and chemical and biological marks. Each of these types of tags will link the tagging effect to individual growth.
For crustaceans, the main problem with tagging results from the moulting process itself. In order to solve this problem, new tagging approaches are aimed at inserting the tags in muscles by using tags such as an anchor tag, the “streamer” tag, “spaghetti” tags and a coded-wire tag. In cases where the tagging method needs surgical implantation into the crustaceans and will lead to extensive blood loss, it may not be applicable for crustacean tagging. To this end, the type of tag used should be chosen for optimal handling processes, based on the animal species.
2.2.4 Tag and Recapture Data
Several studies have been conducted on the use of tag-recapture approaches to quantifying growth in mud crabs (Hill, 1975; Hill et al., 1982; Barnes et al., 2002). Slow-growing or larger species are preferable for these tag-recapture experiments (Chang et al., 2012.) Although tagging experiments have been widely implemented in fishery studies, there is still insufficient information about crustacean growth trends in wild conditions. The following statement, that “ It could hardly be expected, moreover, that lobsters kept under artificial conditions would 2.2 Crustacean Growth Data 30 grow as rapidly as when free in the ocean” (Herrick, 1911), indicates that the growth of animals is believed to vary and they would have a higher growth rate in nature.
This tag-capture method is different from the tank experiments, where the real intermoult periods and moult increments can be directly observed in a moult cycle. Nevertheless, for laboratory studies, “this is a time-consuming procedure (one has to wait two intermoult cycles to obtain the results) and the results may not be indicative of growth in the wild” (Hoenig & Restrepo, 1989). In addition, tank studies are rarely representative, as their grow rates may underestimate those of wild lobster in the ocean (Cobb & Phillips, 1980). These studies also concluded that it was impossible to determine which of the animals had moulted twice and which had moulted even more times. However, Shelton and Chapman (1987) discovered an accurate way of determining the number of times an animal has moulted, by implanting a cuticle with an attached epidermis into the abdomen of a crustacean. Once the animal moulted, another layer of cuticle and epidermis would be formed and the same phenomenon repeats for the subsequent moults.
For tag-recapture data, we first measure the carapace length of the tagged animals, L0, and record the initial time, t0, once they are released. The recaptured animals will be measured again for their lengths, L1, and times, t1. Therefore, we are able to obtain information about the increment, L1 − L0, and the time-at-liberty, t1 − t0. An animal will be considered to have moulted when the increment or soft shell has been discovered during recapture time (Chang et al., 2012). In the next section, we will examine another approach developed by Millar and Hoenig (1997).
There are numerous statistical methodologies that can be used to estimate growth parameters from tag-recapture data. A well-known example, the Fabens’ method (1965), uses a least squares method that incorporates VBGF in the model. Suppose that a tagged animal was captured, the length, L1, and initial time, T1, were recorded for the first time, and then the animal was released into the wild. When the animal was recaptured, the length, L2, and time at recapture, T2, would be measured once again. We would then have binary data, of which the length increment denotes I = L2 − L1 while time at liberty denotes δT = T2 − T1, respectively. 2.2 Crustacean Growth Data 31
We have shown the re-parameterisation as
−k(υ+δT ) −kυ I = L2 − L1 = L∞(1 − e ) − L∞(1 − e )
−k(υ+δT ) −kυ = L∞ − L∞e − L∞ + L∞e
−kυ −kδT = L∞e (1 − e )
−kδT = (L∞ − L1)(1 − e ).
Through this formula, we can model individual growth parameters without considering the age information. The outcome produces a biased estimator as it does not consider randomness among samples of the same age, where the parameters L∞ and k are assumed to be constants.
To address this, James (1991) has modified the least squares method for tag-recapture data by developing an estimator that eliminates the assumption about the distribution. This time, the parameters in VBGF, L∞, vary for all individuals, while k remains constant. Methods were then developed to model the age at tagging (Wang et al., 1995), which proved to be a better approach where both length at tagging, L1, and length at recaptured, L2, were included in the study. There are only small biases in the estimates because an efficient growth model accounts for the correlation between both lengths.
Wang (1998b) has proposed a non-linear regression model as a way to improve Faben’s method by considering individual variability where L∞ is assumed to be a random variable. Com- pared to the other two models (Faben’s and James’ models), Wang’s modified model, which is specifically for tagging data, has successfully reduced biases and shows its robustness when the constant growth rate is being violated substantially.
The oscillations in growth among many species appear to be of a seasonal nature, and it has been noted that, in particular, wild animals that live in temperate habitats have distinct annual moult cycles. Churchill (1921) reported that juvenile blue crabs are inactive and cease to grow in the winter; their normal growth will resume when the water temperature rises during the spring or summer. Based on this, a modified VBGF was constructed to account for season- ally oscillating growth (Pauly & Gasch¨utz,1979; Hoenig & Choudary Hanumara, 1982). The Pauly-Gasch¨utzmodel can be written as 2.3 Challenges in Modelling the Growth of Crustaceans 32
ck L = L 1 − exp −k(t − t ) − sin 2π(t − t ) , (2.3) t ∞ 0 2π s
where c is the growth oscillation amplitude and ts is the time when the first sine wave (maximum growth) occurred. However, direct age is not available with tag-recapture data and, therefore, a modified version of the model in Eqn (2.3) has been proposed by Appeldoorn (1987) in the form of
Lt ck ∗ ∗ Lh = L∞ 1 − 1 − exp −kh + sin(−πh) cos π(2t + h − 2ts) . L∞ π
∗ ∗ We denote h = ∆t as the increment time while t and ts are the adjusted times of the first measurement and the maximum growth occurrence from the experiments.
2.3 Challenges in Modelling the Growth of Crustaceans
From the wide variety of previous and ongoing research works that has been done, most growth studies currently assume a continuous function for animals such as fish. However, the valid- ity of such models is questionable where the growth trajectories of the crustaceans are of a discontinuous nature (Miller & Smith, 2003).
There have been many attempts to describe the growth of crustaceans in mathematical terms using different approaches. For example, we use the rate of change in length, ∆L/∆t, to quantify the growth of animals, with ∆L as the increment length while ∆t is the interval time of the increment. This method can be used to describe the mean population growth. In addition, a deterministic function VBGF as in Eqn (3.3) is known to be the most widely used model for estimating size at specific age. This model assumes that each animal is similar in asymptotic length and that they possess a constant growth rate. In spite of the fact that it is not biologically realistic, the VBGF can be applied to approximate the mean population growth in continuous curve. 2.3 Challenges in Modelling the Growth of Crustaceans 33
We investigate two general problems that exist in determining the growth estimations of a moult process. First, due to the fact that growth in crustaceans occurs in a stepwise fashion, and this has hindered us from understanding the growth clearly. Second, we face difficulty in determining their age. Traditionally, the age of fish can be deduced in a comparatively simple and straightforward manner using information from their scales, spines, vertebrea and otoliths is available (Smith, 1992). However, due to the loss of hard parts from the body due to moulting, it is not possible to obtain such direct measurements.
As a solution to these problems, we have taken the two stochastic components into considera- tion. One is the time between two consecutive moults, the intermoult period. The other is the increment obtained during moulting, the moult increment (Botsford, 1985). A stepwise growth function is made up of these two processes, which occur simultaneously during a moult.
A unified likelihood approach will be proposed to estimate growth parameters for both inter- moult periods and moult increments without the need for age information. Not only can we describe the average population growth, but we are also able to characterize the trajectories of each of the animals mathematically to obtain the data about the crustaceans. We consider two environmental conditions for the moult increments and intermoult periods: the tag-recapture experiments (tagging data) and the rearing experiments (tank data).
For the tank data, we have observed moulting times that enable us to quantify the growth parameters directly for the intermoult periods and moult increments (Kurata, 1962). We derive one joint density to form a function of the moult increments and another one as a function of the intermoult periods. We claim that both of these are conditionally independent, given the information for the premoult lengths and the intermoult periods. Therefore, the parameters in each function can be estimated separately.
We then integrate both of the functions through a Monte Carlo simulation and are able to obtain a population mean to describe the overall growth trend for crustaceans. In reality, however, laboratory experiments are generally costly, and the other common datasets come from tag-recapture studies (Wilder, 1953). In these studies, the lengths at mark and recapture have been collected and multiple recaptures are sometimes available as well. 2.3 Challenges in Modelling the Growth of Crustaceans 34
For tagging data, although the exact moulting time is unknown, but we have time at liberty recorded, between the time when the animals were tagged and when they were subsequently recaptured. Based on the changes in length, we may assume that we know whether an animal has moulted or not during the time at liberty. If it has moulted, we then claim that the time from recapture until the next moult (time to moult) is less than the time at liberty; otherwise, the time to moult is longer than the time at liberty. This censored data can be used to approximate the intermoult periods based on, for example, a likelihood approach (Hoenig & Restrepo, 1989). Note that moulting could have occurred more than once during the time at liberty. Estimation becomes more interesting when we have multiple recaptures, and we will model the moulting status for each time interval through a correlated binary variable. The moulting times will be modelled by a latent variable. In addition, with this approach, we can estimate the probability distribution of moulting times within a specified period of increment through the expectation-maximization (EM) algorithm. This idea will be essential in terms of the validity of the parameter estimation. Chapter 3
Growth Models
Stochastic growth models have attracted great attention in many areas including economics, finance, medicine and biology. In medical research studies, either the description of infant growth or the growth of tumours as well as the effect of the relevant treatments are of interest. In farming and agriculture, the problems are normally focused on knowing how fast a crop grows, how much it could possibly be yielded, and how the growth rate relates to the environmental conditions. In fisheries research, modelling the growth of animals is fundamental in stock assessment to assure sustainable fisheries and optimal management (Wang & Die, 1996; Somers & Wang, 1997).
To construct a basic framework of a growth model, we state our assumptions as true and these assumptions will be validated in future analysis. For instance, we assume that in the absence of environmental variability, population growth rate is proportional to a population’s size in population studies. In general, a differential equation can be applied to determine the rate of change in the growth of an animal in continuous time.
It is crucial to comprehend the relationship between an individual’s age and size for the optimal harvesting of aquatic animals. Mathematically, size can be measured in magnitude or different dimensions, such as length, width, weight, volume, diameter or mass. All these measurements can be applied to the full range of species, including aquatic creatures, elephants, bears and plants as well as tumours in the human body. A wide range of studies have used deterministic 3.1 Deterministic Growth Models 36 models to describe the growth of fish and crustaceans (Rafail, 1973; Maller & DeBoer, 1988).
3.1 Deterministic Growth Models
Essentially, a general growth function can be expressed as
dL t = g(L , θ), (3.1) dt t
where Lt refers to the size at time t and g(·) is a deterministic function of parameters θ, which also depends on the size. Once we change the form of function g(·), another model can be produced, such as the Richards curve, Gompertz, the logistic model or the von Bertalanffy curve. All these processes exhibit an increment that approaches a maximum point before levelling off when time goes to infinity. The inflection point leads to a sigmoidal fashion (S- shaped) in the growth curve.
3.1.1 The von Bertalanffy Growth Function (VBGF)
The most well-known growth model commonly used to describe the relationship between size and age in fisheries is the VBGF (1938):
g(Lt, θ) = k(L∞ − Lt), (3.2)
where L∞ is the asymptotic length, and k is the growth rate. There have been many attempts to mathematically describe the change in size over time using a VBGF across a wide range of animals, especially for fishery species (Russo et al., 2009a; Russo et al., 2009b; Chang et al.,
2012; Pardo et al., 2013). Once the g(Lt) in Eqn (3.2) is integrated, we have
−k(t−t0) Lt = L∞{1 − e }, (3.3) 3.1 Deterministic Growth Models 37
where t0 is an age at the size of 0 (Ricker, 1975). For parameter estimation, both L∞ and k were said to be strongly correlated (Piling et al., 2002; Helser & Lai, 2004; Helser et al., 2007). The growth rate will decrease when an animal approaches asymptotic size. The growth parameters can be estimated using the likelihood approach when an absolute size-at-age is given (Kimura, 1980).
Apparently, not all populations grow exponentially following a VBGF. For example, the growth of the red abalone Haliotis rufescens in California (Rogers-Bennett et al., 2007) and juvenile rock lobsters in New Zealand (Starr et al., 2009) was most appropriately described by a lin- ear model. The difference in estimation of the growth curves is mainly due to physiological assumptions (Brody, 1945). Richards (1969) concluded that biologically, many of the growth models fulfill a form of deterministic differential equation
1 dLt = g(Lt, θ), Lt dt
−1 where Lt dLt/dt is a relative growth rate. For example, Richards (1959) proposed
( 1/δ) Lt g(Lt, θ) = k 1 − , k > 0 (3.4) L∞
with the unknown parameters of k, L∞ and δ. The underlying function is initially employed to quantify the relative growth rate of an individual or plant size at time t.
While the VBGF tends to have a linear decline in growth from the juvenile to adult stages; the Gompertz growth model and the logistic model tend to predict growth rate where there is an initial increase for juveniles and followed by a decline when they grow larger.
Verhulst (1838) empirically derived the logistic equation to quantify the size of an organ and the growth of a population. In logistic growth, an individual growth rate becomes lower as the length reaches a terminal point due to environmental factors. 3.1 Deterministic Growth Models 38
The Gompertz model was empirically formulated by Wright (1926) and modified by Medawar (1940) into an animal growth model to study the heart of a chicken. In the later years, the Gompertz curve was applied in studies on the growth of tumours (Laird, 1964). Additionally, Gompertz curve can take different kinds of measures; for instance, weight, length, area and surface all have the same growth pattern. The form of these sigmoidal curves was:
k g(Lt, θ) = (L∞ − Lt)Lt, L∞ for the logistic model and
L∞ g(Lt, θ) = k log Lt, Lt for the Gompertz curve (1925).
The Gompertz and logistic models can be employed to quantify the growth rate of heifers when they are at the puberty stage (Budimulyati et al., 2012). They assume that the growth rate reduces over time exponentially, and its rate is typically well described when growth is relatively slow during the juvenile phase. However, neither model characterises a change in the growth trend (increase or decrease) in juveniles (Day & Fleming, 1992). Furthermore, Haddon et al. (2008) concluded that both the VBGF and Gompertz models are, biologically, inappropriate to describe the growth of blacklip abalone juveniles. A more plausible growth model, the inverse logistic, was introduced and was claimed to be statistically optimal (minimum relative Akaike’s Information Criterion, AIC) compared to the VBGF and Gompertz models. The inverse logistic model predicts constant increments at an early stage and follows a decline when the juveniles become mature at a later phase.
From Eqn (3.4), we can have different shapes of curves with differences in the inflection param- eter of δ. For example, the Richards equation of g(Lt, θ) can be integrated to form a length at 1/(1−a) time t function Lt = L∞(1 − b exp(−k(t − t0)) ) where k, a, b > 0 are constants. Hence, the Richards function yields the VBGF (in length), the VBGF (in weight), the Gompertz and 3.1 Deterministic Growth Models 39 logistic models when a = 0, a = 2/3, a → 1 and a = 2 as summarised in Figure 3.1 (Berry et al., 1988; Birch, 1999; Tholon & Queiroz, 2007).
Another example of a sigmoid function would be the Weibull function. For Weibull, the growth rate is given by
k t − µk g(L , θ) = (1 − L ) , t υ t υ where µ, υ are constants and k is the shape parameter. The shape parameter can be interpreted as follows:
• If k < 1, indicating that the growth rate reduces over time, for example, the child mortality rate over time;
• If k = 1, indicating that the growth rate that is stable over time. For example, the human population will grow constantly if the number of births and deaths remains at the current rate. The Weibull is equivalent to a simple exponential growth curve;
• If k > 1, indicating that the growth rate increases over time. For example, the aging process is growing over time.
Exponential growth is common in a population if there are no limitations on resources or if predators do not exist. The exponential growth curve can be expressed in
g(Lt, θ) = kLt .
An example of exponential growth would be bacteria grown in a flask, showing an increasing growth rate over time.
From Eqn (3.1), a different functional form of g(Lt, θ) can be displayed, as in Table 3.1.
These different models make a great difference in estimating growth size, due to the error assumption differences for each model. For example, the exponential model assumes that error 3.1 Deterministic Growth Models 40 Figure 3.1: Special cases of the Richards model. 3.1 Deterministic Growth Models 41
dLt Differential equation, dt = g(Lt, θ) Growth Model
g(Lt, θ)=k(L∞ − Lt) VBGF −1 g(Lt, θ)=kL∞ (L∞ − Lt)Lt Logistic equation
g(Lt, θ) = k log(L∞/Lt)Lt Gompertz curve 1/δ g(Lt, θ) = k{1 − (Lt/L∞)} Richards curve −1 −1 k g(Lt, θ) = kυ (1 − Lt){υ (t − µ)} Weibull function
g(Lt, θ) = kLt Exponential curve
Table 3.1: Different growth models in the form of a differential equation follows a log normal, whereas the VBGF model assumes a normal error with a mean equivalent to zero and a constant variance.
A deterministic model may not be realistic because a growth process consists of a complicated stochastic process, such as environmental perturbation. Therefore, a stochastic differential equation (SDE) is more advantageous in accommodating heteroscedasticity in a growth pro- cess. Sandland and McGilchrist (1979) modified the Richards’ model by applying the Ito or Stratonovich calculus which appears to be of stochastic version of Eqn (3.5).
To fit a model, it is important to identify if there is any deviation throughout a growing process. Thus, Brown et al. (1975) defined a set of recursive residuals to test, based on the general linear model, if the changes in errors are independent. Then McGilchrist and Sandland (1979) extended the recursive algorithm proposed by Brown et al. (1975), allowing the errors to posses a correlation structure and a set of independent recursive residuals. When the residuals were not significantly independent, the assumption of an autocorrelation structure in the residuals needed to be considered. Subsequently, the method of scoring was developed to estimate the correlation parameters.
However, one aim when developing a growth function is to minimize the number of parameters, whilst obtaining the best fit for the given data. Suppose a set of models for the data exists; we will account for AIC to measure the quality of the statistical models, relative to each model. In selecting a model, we consider the one with the minimum AIC value as the preferred model. 3.1 Deterministic Growth Models 42
3.1.2 Generalised von Bertalanffy Growth Function (GVBGF)
Due to a relatively simple mathematical function for estimation, the VBGF has been extensively applied in fishery studies. However, the equation does not allow the flexibility of quantifying growth due to the fixed parameters in the asymptotic length L∞ and the growth rate k. Hence, a GVBGF was reconstructed to yield a more sensible growth equation, such as the one by Gulland and Holt(1959), Fabens (1965), Pella and Tomlinson (1969) as well as Pauly (1979). For example, when the VBGF was applied to the data of the Atlantic bluefin tuna Thunnus thynnus; in certain cases, the parameter estimates L∞ yielded a value greater than the observed maximum length. As a result, Pauly (1979) modified and generalised the VBGF to form:
−kD(t−t0) 1/D Lt = L∞{1 − e } for length,
−kD(t−t0) 3/D Wt = W∞{1 − e } for weight,
where D > 0 is an additional parameter to the VBGF, defined as the ratio of the gill surface area to body weight. When D = 1, the resulting Lt and Wt are versions of the VBGF. In contrast, the GVBGF is a form of a curve from Richard (1959) when D 6= 1. Richards (1959) generalised the VBGF by adding another parameter to the model to be a biological considera- tion in individual growth. The three-parameter Richards curve is
−k(t−t0) 1/(1−a) Lt = L∞{1 ± be } ,
where b, k and a are the parameters. A different growth (sigmoid) shape can be noticed where initially an individual increases gradually and speeds up the process at an exponential growth rate and levels off at the final phase.
Another GVBGF proposed by Pauly (1979) is equivalent to four parameter sigmoidal curves. It has additional parameter that provides a more informative growth process in the form of:
L∞ Lt = , {1 + (1/θ)e−k(t−t0)}θ 3.1 Deterministic Growth Models 43
where θ is a dimensionless parameter.
Among the VBGF, GVBGF, Gompertz, logistic and Richards models, the last one best fit (min- imum AIC) for the length of the longtail tuna (Thunnus tonggol) by Griffiths et al. (2009). Due to its flexibility, not only can the model detect the high growth rate of longtail tuna and long-lived temperate reef fish (Ewing et al., 2007) during an early phase, but it can also discover the low growth rate of large temperate pelagic fish (Stewart et al., 2004). The five-parameter Schnute-Richards model (Schnute & Richards, 1990) can be expressed as
−ktν 1/γ Lt = L∞(1 + δe ) .
Despite the flexibility in fitting complex growth trajectories for individuals, the model is com- plicated because it contains many parameters and thus may cause over-parametrisation. Unless sufficient longitudinal data are provided, the model is computationally intractable for the un- derlying models in practice.
A more recent work by Ohnishi et al. (2012) extended the VBGF in which the energy allocation for reproduction was incorporated because it was of fundamental biological importance. The general form of the resulting growth function was given as
−kTt Lt = L∞(1 − e ) ,
R t where Tt is a function of time t defined by Tt = {1 − p(u)}du in which 0 ≤ p(u) ≤ 1 can be t0 the arbitrary density function. The authors proposed two types of models: (1) discontinuous p(u) that occurs at the time of maturity tm and (2) continuous p(u) that is performed as a logistic curve.
For the case of a discontinuous function at maturity, when t < tm , p(u) = 0; otherwise, p(u) = δ(t ≥ tm) for 0 ≤ δ ≤ 1. In this context, Tt is 3.2 Stochastic Growth Models 44
t − t0 t < tm Tt = . t − t0 − δ(t − tm) t ≥ tm
As for continuous function, p(u) is assumed to follow a logistic function such that p(u) =
δ/[1 + exp{−r(t − tm)}]. Hence, Tt is defined as
δ T = (1 − δ)(t − t ) − log{1 + e−r(t−tm)} − log{1 + e−r(t0−tm)} , t 0 r where δ is the upper bound of the rate of energy allocation whereas r is the rate to maturity.
Through the aforementioned model, a difference in the growth rate among individuals can be justified.
3.2 Stochastic Growth Models
Despite the most commonly used deterministic growth model of the VBGF, which assumes that no individual and environmental variability occur in the population from a biological point of view, the VBGF is unrealistic when the variations are caused by environmental perturbations that result in variability in growth rates. Therefore, a biased estimation could happen if the effects of the fluctuating environment had been neglected.
Incorporating stochasticity can have a great impact on quantifying the population mean growth rate (Lv & Pitchford, 2007). Each individual experiences different environmental conditions, such as food availability (Rilling & Houde, 1999), which will introduce environmental variability in growth rates. To this end, numerous studies developed individual-based models (IBMs) that take into account variations between individuals and environmental factors in a growth process (Letcher et al., 1996; Wang, 1999; Wang & Thomas, 1995; Gudmundsson, 2005; Lv & Pitchford, 2007). 3.2 Stochastic Growth Models 45
Normally, financial or economic studies will account for an exchange rate or stock price as the variable over time based on a stochastic differential equation in the form of
dLt = g(t, Lt)dt + h(t, Lt)dWt , (3.5)
where Wt is a standard Brownian motion whereas g(·), h(·) are the functions of time and current state. The function g, the deterministic part of the model, is known as a drift. The function h is known as the diffusion coefficient and dWt defines the noise term.
3.2.1 Incorporation of Individual Variability in Growth
Much less is known about the growth variation in crustaceans. Under physiological conditions, all animals differ from one another due to individual heterogeneity. Variation may be in physical features (different asymptotic lengths), different rate of reproduction, metabolism and rate to maturity. In this case, differences between individuals yield individual variability.
In reality, each specimen possesses its own growth rate to reach the maximal length. As noted by many authors, L∞ and k in Eqn (3.2) have a significant converse correlation (Pilling et al., 2002; Eveson et al., 2007; Shelton & Mangel, 2012). A different growth rate among individuals implies different asymptotic lengths for different specimens in a population. To be biologically realistic, L∞ and k should be random variables.
For somatic growth, ignorance of variation among individuals can have serious implications for the evolutionary and ecological dynamic. Each individual owns a specific genetic factor, resulting in genotypic variation. For example, King et al. (2016) found a significant difference in terminal sizes of Lake Erie watersnakes, Nerodia sipedon insularum. Kirkwood and Sommers
(1984) proposed an IBM by setting L∞ to be random.
From Eqn (3.3), the non-linear deterministic function may not be realistic if high variability exists in growth among individuals. Large variability in a model can cause bias estimation and the results of stock assessment could be misleading in fishery. Thus, Sainsbury (1980) proposed 3.2 Stochastic Growth Models 46
an inter-individual variability model, where both k and L∞ are independent variables such that k, L∞ followed a gamma and normal distribution, respectively. We express the length at age as
−ki(t−t0,i) Lt = L∞, i {1 − e } .
Despite including individual variability, the equation above does not consider environmental factors, which is another key aspect to be concerned about in growth models. Wang and Thomas (1995) pointed out that the approaches of Kirkwood and Sommers as well as Sainsbury do not account for individual variability appropriately, where the mean increment size for a population
−kt −kt Ep(I| l, t) = l∞ − l + lE(e ) − E(L∞e ) (3.6)
may be unrealistic for parameter estimation. Here, p indicates the population, l, l∞ are denoted as the mean of length and asymptotic length of an individual. In contrast, Wang and Thomas introduced an improved method of Eqn (3.6) where the mean increment is for an individual
−kt −kt Ei(I| l, t) = Ei(L∞| l, t) − l + lEi(e | l, t) − Ei(L∞e | l, t). (3.7)
When both findings from Eqn (3.6) and Eqn (3.7) are compared, the mean increment lengths from a population exhibit a linear regression, whereas the latter’s size is nonlinear. In this regard, the pronounced differences indicate inconsistency and bias in estimates based on method of Sainsbury as well as Kirkwood and Somers.
3.2.2 Incorporation of Environmental Covariates in Growth
Growth plays a vital rule in productive and sustainable aquaculture. Shrimp is one of the most discussed topics in fishery studies. However, shrimp culture is still poorly understood by many researchers (Jackson & Wang, 1998; Bureau et al., 2000). In particular, the term “growth” with respect to the shrimp species refers to an increase in weight or length. Each individual has its 3.2 Stochastic Growth Models 47 own unique genes that affect its metabolism and result in individual variability in growth. The individual variability leads to variation in the sizes of the animals.
In addition to gene variation, animals often experience different environmental conditions, such as food availability (Rilling & Houde, 1999), for rearing, which may affect their actual growth rate. In fishery studies, numerous evaluations have proved that environmental factors, such as water temperature, show significant results with respect to the growth rate (Sumpter, 1992; Dennis et al., 1997; Moser et al., 2012). Meanwhile, Swain et al. (2003) concluded that large variations in a variety of populations with the same species are caused by environmental variability, such as temperature, density and food availability.
Mia and Shah (2010) concluded that a significant result of a different level of salinity affected the growth of mud crabs. Many other external influences of the individual growth rate exist in the ecology system, yet their parameters remain unknown. It is known that the environmen- tal factors may vary depending on the region and ecological behaviour of the particular size (juvenile or adult) category.
Failure to incorporate individual or environmental variables into the growth model may create biases in estimation. For instance, Wang and Thomas (1995) demonstrated how individual variability may produce biases in growth parameter estimates. Consequently, including influ- ential covariates and a comprehensive understanding of the environmental impact on animal growth is crucial for long-term sustainability and productivity in fishery research.
As noted in Eqn (3.1), the parameter θ consists of growth variability that controls the rate of how an individual reaches its maximal length. It can be due to the environmental impact on growth, such as water temperature. For instance, pronounced seasonal oscillations can be detected from aquatic animals in temperate waters because of the fluctuations in temperature (Shul’man, 1974). Therefore, seasonal changes can be interpreted as the key environmental factor. A seasonal version of the VBGF by Hoenig and Choudary Hanumara (1990) is used to describe the seasonal effects in the form of:
Lt = L∞{1 − exp(−k[t − t0] − W sin2π[t − tw] + W sin2π[t0 − tw])} . 3.2 Stochastic Growth Models 48
We denote W = ck/2π, where c is the amplitude of seasonal oscillation, while tw is the interval time between the initial time of t0 and the time when the maximum growth rate happens.
Subsequently, Wang (1998a) proposed a seasonal growth model that generalised from the VBGF in the form of
g(Lt, θ) = (L∞ − Lt)st,
where st = k + u cos(2πt + v) is an annual seasonal oscillations such that st = 0 when cos(2πt + v) < s0, else st = k. The parameters v and s0 determine the switch-off period in growth, annually. Wang considered the growth rate k as a function of t since tempera- ture affects the individual growth rate to reach the L∞. Apart from environmental factors, the physical interference from tagging may affect the growth of marked animals (Sandland & McGilchrist, 1979). Tagging may cause changes in the growth rate of an individual, temporar- ily. Wang suggested the use of the following form
−(t−t0)ω 0 st = k0 + η{1 − e } ; t ≥ t ,
where k0 is the growth rate after tagging with positive η that implies a prolonged tagging effect, whereas a negative measurement shows the rapid growth of the tagged individual. And ω−1 could be referred as an average recovery time. Data of barramundi were used to estimate the growth parameters associated with the tagging effect in which the specimens had approximately 169 days of recovery once they were tagged.
Maguire and Allan (1992) found that the optimum growth of Penaeus monodon Fabricius hap- pened within 27◦C to 33◦C. Jackson and Wang (1998) extended the Gompertz model (Seber & Wild, 1989) to predict the rates of shrimp growth for environmental parameters, such as water temperature, mortality and pond age, with the model in the form of:
R t exp[− 0 (a+bx)dy] L0 Lt = L∞ . L∞ 3.2 Stochastic Growth Models 49
The terminal weight is denoted as w∞, b is a vector of the coefficients of environmental variables x, a is the basic growth rate under general conditions and w0 is the initial weight. Once we take a logarithm to the equation above, it will be a form of the VBGF. The water temperature is said to reach an optimum for growth around 30◦C generally for Fabricius shrimp in ponds. Water temperature influences the individual growth rate, but there are limited studies on both aspects.
In fact, the growth rate for shrimp in ponds may vary from that of shrimp in the wild. For instance, Jong, Kou and Chen (1993) found that Penaeus monodon reared in ponds gained a smaller maximal length than wild shrimp. Both in farms and natural conditions, the growth rate of shrimp relied on their population density in the system (Caillouet, Norris, Heald & Tabb, 1976; Sedgwick, 1979; Maguire & Leedow, 1983). According to Edwards (1977), the growth rate of Penaeus vannamei Boone was depressed when the population density of shrimp was above 2.5 m−2 in coastal lagoons. Nevertheless, the shrimp population density hiked up to 200 m−2 without a depression in the rate under experimental conditions (Sandifer, Hopkins, Stokes & Browdy, 1993).
The difference in both conditions may be affected by the water quality or feeding strategy. Cultured species have been appropriately fed and provided with quality water, such as water with dissolved oxygen and the optimal pH compared to the wild species’ food and water con- sumption. The growth rate may be slowed if the food provided is lower than the optimum level (Hartnoll, 1982; Hartnoll, 2001). Shrimp is density dependent due to the variable mortality during growth. Population density and mortality are inversely correlated in that the popula- tion density of shrimp decreases when mortality increases. Therefore, it is essential to consider aquaculture conditions for the realisation of growth potential.
The number of crops from the pond (pond age) have a big impact on growth rate (Jackson & Wang, 1998). The optimum growth of shrimp occurs when approximately six to seven crops have been harvested but then start to deteriorate thereafter. As mentioned in Briggs and Funge-Smith (1994), a decline in productivity was attributed to the degradation conditions at the bottom of the pond. 3.2 Stochastic Growth Models 50
Although many of the growth models have focussed on deterministic functions to describe size-at-time, stochastic models have been constructed to take into account the variability in individual growth processes. Subsequently, we first review the continuous stochastic process and then review the discontinuous models for crustaceans as a description of the full variability in growth.
3.2.3 Stochastic Version of the VBGF (sVBGF)
Conventional models consider stochastic growth by adding random error to the deterministic model, allowing substantial deviations from a realistic description of the data. However, it should be noted that the proposed model avoids the assumption about the short-term individ- ual variability of the conventional models. Gudmundsson (2005) employed a joint estimation of genetic and environmental effects to quantify the growth of cod fish studies. Primarily, the deterministic function g(Lt, θ) is assumed to follow the VBGF in the form of
g(Lt, θ) = k(L∞ − Lt). (3.8)
By adding stochasticity to the underlying model, we have g(Lt, θ) = k(L∞ − Lt) + εt, where the derivative of Wiener process is equivalent to εt. Thus, the solution to the stochastic growth model is
Z t −kt −kt −k(t−s) Lt = L∞(1 − e ) + L0 e + e εs ds, (3.9) 0
where L0 is the initial length at time t = 0 with a constant growth rate of k. Consider the case that when both individual variability and environmental variability are taken into account in the proposed model, we can express the growth using partial differential equations 3.2 Stochastic Growth Models 51
∂L t = g(L , K, θ ), (3.10) ∂t t 1
∂K = h(K, θ ) + ε , (3.11) ∂t 2 t
where K is the random environmental perturbation with its function of h, and θ1 and θ2 are parameters.
The author modified the model in Eqn (3.8) by incorporating the environmental variable K into the function as presented in Eqn (3.10). The L∞ is genetic-specific rather than being affected by the changes in environmental conditions. Under different environmental conditions, the maturity rate k in the VBGF will vary. Hence, we denote K as the variable in a form of differential equation
∂K = φ(k − K) + ε , ∂t t where φ and k are constants.
Similar to the solution of ∂L/∂t in Eqn (3.9), using the same technique, the solution of the
Eqn (3.11) with k = k0 at time t = 0 is
Z t −φt −φt −φ(t−s) K = k(1 − e ) + k0 e + e εs ds. (3.12) 0
Subsequently, we look for the solution of Eqn (3.10) in the form of
Z t Lt = L∞ − (L∞ − L0) exp(− Kds). (3.13) 0
Knowing K from Eqn (3.12), therefore 3.2 Stochastic Growth Models 52
Z t Z t Z s −φs −φs −φ(s−u) K ds = k(1 − e ) + k0 e + e εu du ds 0 0 0 Z t Z s 1 −φt −φ(s−u) = kt + 1 − e (k − k0) + e du ds. φ 0 0
Ultimately, both Eqns (3.12) and (3.13) were unified to describe individual growth in the form of
Z t Z s 1 −φt −φ(s−u) Lt = L∞ − (L∞ − L0) exp − kt + 1 − e (k − k0) + e du ds . φ 0 0
Wang (1999) incorporated stochastic components into animal growth models to produce un- biased estimating functions for parameter estimation in generalised VBGFs. By including the environmental variability and a link function f incorporating the covariates xt, a sVBGF is
dL t = g(L )f(x , θ) + h(t, L )dB , (3.14) dt t t t t
where the case g(Lt) = (L∞ − Lt) is the VBGF, f(xt, θ) determines the growth rate of ap- proach to maturity, a function h is known as an error process σt and dBt as the environmental perturbations.
Lv and Pitchford (2007) also proposed three forms of diffusion coefficients that h(t, Lt) is set to be decreasing, constant and increasing stochasticity, whereas the deterministic part g(t, Lt) follows a VBGF corresponding to Eqn (3.5). The deterministic function g is not merely lim- ited to the VBGF, it can also be any specific functional form of growth models, such as the
Richards, the Gompertz or the logistic model. Considering that the h(t, Lt) is a constant σ, the differential equation is written as
dLt = k(L∞ − Lt)dt + σdBt, (3.15)
representing the Vasicek model (Prakasa Rao, 1999). The solution to the Eqn (3.15) is called 3.2 Stochastic Growth Models 53 the Ornstein-Uhlenbeck process in the form of
Z t −kt −kt −k(t−u) Lt = e L0 + L∞(1 − e ) + σ e dB(u). 0
The underlying model may be suitable for financial or economic considerations that tend to fluctuate around equilibrium values, such as interest rates or stock prices. However, it is not appropriate for modelling the non-negative growth of crustacean species.
Russo et al. (2009a) introduced the use of a subordinator that shows the non-decreasing growth paths of individuals. Hence, the resulting model can be used to track the increasing growth paths in the form of:
dLt = (L∞ − Lt−1)dZt .
A subordinator Zt is strictly increasing in a stochastic process, rendering the Lt that follows an increment also.
3.2.4 L´evyProcess
A L´evyprocess, X = (Xt)t≥0, is a stochastic process that satisfies the four properties,
(i) P(X0 = 0) = 1. (ii) X paths are right-continuous with left-limits, and discontinuities happen at random times.
(iii) For s ≤ t, Xt − Xs has the same distribution as Xt−s, stationary increments.
(iv) For s ≤ t, Xt − Xs is independent of {Xh : h ≤ s}, independent increments.
A L´evyprocess is a stochastic one that has right-continuous paths with left-limits, allowing a number of jumps that can be counted on a finite interval. Hence, an individual’s growth curve can be modeled by a L´evyprocess. Some well-known examples of L´evyprocesses are Brown- ian motion, Poisson process, stable process and subordinators. The L´evyprocess generalises 3.2 Stochastic Growth Models 54 random walks to continuous time. We can use it to characterise individual growth paths (a discrete process) as well as the population mean growth (a continuous process).
3.2.5 The Characteristic of the L´evy Process
Let (Xt)t≥0 be a L´evyprocess, thus the distribution of Xt for t > 0 is infinitely divisible. For 2 Pn−1 example, say X ∼ N(µ, σ ) or we can write as X = r=0 Yr where Yr is independent and identically distributed with N(µ/n, σ2/n).
We claim that Xt is a class of infinitely divisible distributions if for all t > 0, n ∈ N consist of independent and identically distributed random variables Xt/n, ..., (Xt − X(n−1)t/n).
Xt = Xt/n + (X2t/n − Xt/n) + ... + (Xt − X(n−1)t/n).
A general L´evyprocess, Xt is a mixture of (γ, σ, ν) with a truncated function. Apparently, X is infinitely divisible whereby the characteristic γ is the drift coefficient, σ2 ≥ 0 the diffusion coefficient and the L´evymeasure also known as the pure jump process ν satisfies ν(0) = 0 and R (1 ∧ |x|2)ν(dx) < ∞. R
The L´evy-Itˆo decomposition states that one can represent a L´evyprocess as a summation of drift, a Brownian motion, an independent compound Poisson process with jumps of the in- tensity of at least one (a finite size of jumps) and independent compound Poisson processes with jumps of a magnitude of less than one (infinite small jumps). Hence, the characteristic exponent φ(θ) of any L´evyprocess is
1 Z Z φ(θ) = iγθ + σ2θ2 + (1 − eiθx)I ν(dx) + (1 − eiθx + iθx)I ν(dx), 2 (|x|≥1) (|x|<1) ZR R 1 2 2 iθx = iγθ + σ θ + 1 − e + iθxI(|x|<1) ν(dx), (3.16) 2 R where I denotes the indicator function, A = σ2 is a symmetric positive n × n matrix, γ ∈ Rd, t ≥ 0 and θ ∈ R (Cont & Tankov, 2004). 3.2 Stochastic Growth Models 55
To describe L´evy processes, criteria for characteristic functions are essential. Suppose there is an infinitely divisible distribution F on Rd; its characteristic function Φ can be expressed in the form of
Z iθx d e ν(dx) = ΦX (θ), θ ∈ R . (3.17) R
The function Ψx is said to be the cumulant generating function or log-characteristic function of X such that
ΨX (θ) ΦX (θ) = e , = E(eiθx) ,
where ΦX (0) = 0.
If φ is the cumulant generating function of X1 : φ = ΨX1 , then ΨXt = tΨX1 = tφ. A function Φ is called the characteristic function of an infinitely divisible distribution if and only if it satisfies
Φ(θ) = e−φ(θ).
It is known that the cumulant generating function is the logarithm of the moment generating function, M(·). Therefore,
ΨX (θ) Mt(θ) = e t , θ ∈ R. = e−tφ(θ) .
Given the expression of the characteristic function as shown in Eqn (3.17), hence 3.2 Stochastic Growth Models 56
Z eiθxν(dx) = E(eiθXt ) R
= ΦXt (θ) = e−tφ(θ).
Considering the triplet (γ, σ, ν) above, thus there is a probability space (Ω, F, P) with the existence of three independent L´evyprocesses, X(1),X(2) and X(3) where X(1) is a Brownian motion, X(2) is a compound Poisson process and X(3) is a pure jump martingale whereby the sum of jumps on each time interval of length is less than 1. Having X = X(1) + X(2) + X(3), the probability space on the L´evyprocess X = (Xt)t≥0 with the characteristic exponent is defined as in Eqn (3.16). To be specific, this process is known as L´evy-Itˆo decomposition.
Each L´evyprocess can be written as
Xt = γt + σBt + Dt + Ct,
where D is an independent compound Poisson process with intensity measure ν, at t ≥ 0
Dt = Σu≤t∆XuI|∆u|>1 ,
where ∆t = Xt−Xt− and C is the limit of compensated compound Poisson processes with jumps
∆Ct = ∆XtI|∆Xu|≤1 .
A random variable X ≥ 0 is said to be infinitely divisible if and only if
−θX −γ0θ−R ∞(1−e−θx)ν(dx) E(e ) = e 0 . 3.2 Stochastic Growth Models 57
For instance, there is a gamma process Xt ∼ Γ (t; α, β). For the positive size of jumps, it is an increasing L´evyprocess with an intensity measure ν(x) = f(x)dx such that
α f(x) = exp(βx) , x > 0, x where γ0 = 0. The parameter α controls the frequency of jumps and the scale β controls the size of jumps.
Examples of L´evyProcesses
We proceed to some following examples for a further study of the L´evyprocesses, such as the compound Poisson process, Brownian motion and the gamma process.
Compound Poisson Processes
A compound Poisson process has random jumps following a Poisson process and the jump size is random according to a certain distribution F (Figure 3.2). Let N be a Poisson process with
d intensity λ and a series of independent variables Di which have the same F ∈ R . Suppose a Poisson random variable D is parameterised by λ > 0, with a compound Poisson process
{Xt, t ≥ 0} defined by
N Xt Xt = Di, i=1
where {Nt : t ≥ 0} is a Poisson process, and {Di : i ≥ 1} are independent and identical distributed random variables with distribution function F that is concentrated on (0, ∞) and independent of Nt. For θ ∈ R, the moment generating function of X1 can be given as 3.2 Stochastic Growth Models 58
iθXt X iθ E(e ) = e Pr(Xt = i) i X iθ X = e Pr(Xt = i|Nt = n) Pr(Nt = n) i n X iθ X = e Pr(Xt = i|Nt = n) Pr(Nt = n) i n X λn = E(eiθx) e−λ n! n n X Z λn = eiθx dF (x) e−λ n! n R Z = exp −λ (1 − eiθx) dF (x) . (3.18) R
From Eqn (3.18), the characteristic component is
Z φ(θ) = (1 − eiθx), R where the compound Poisson process has zero drift and diffusion coefficient, and its L´evy measure is ν(dx) = λ dF (x).
From the calculation above, we replace Di with the L´evy-Khintchine formula for the compound Poisson process φ(θ) = λ R (eiθx − 1) dF (x). Suppose a drift rate d ∈ exists in a compound R R Poisson process such that
N Xt Xt = Di + dt, t ≥ 0 i=1 and again, it is a L´evyprocess. The L´evy-Khintchine exponent is given by
Z φ(θ) = λ (1 − eiθx − iθx) dF (x) − i dθ. R
If we shift the compound Poisson distribution as the central point, then d = λ R x dF (x) and R 3.2 Stochastic Growth Models 59
Z φ(θ) = (1 − eiθx + iθx)λ dF (x). R
Figure 3.2: Compound Poisson Process trajectories with exponentially distributed jumps.
Gamma Processes
Suppose that X = {Xt : t ≥ 0} is a gamma process. For α, β > 0, consider that a gamma distribution takes the probability law
αβ dF (x) = xβ−1e−αx, Γ (β) supported on (0, ∞). Hence, 3.2 Stochastic Growth Models 60
Z ∞ iθx 1 e dF (x) = β 0 (1 − iθ/α) 1 n = , (1 − iθ/α)β/n which is an infinitely divisible distribution (Figure 3.3). For the L´evy-Khintchine decomposi- R 1 tion, we have the triplet (γ, σ, ν) such that the drift, γ = − 0 xν(dx), the diffusion coefficient, σ = 0 and the L´evymeasure, ν(dx) = β/xe−αx dx.
The L´evy-Khintchine decomposition above can be obtained through Frullani integral
Z ∞ 1 iθx −1 −αx β = exp − (1 − e )βx e dx (1 − iθ/α) 0 for α, β > 0, thus for θ ∈ R
Z ∞ 1 φ(θ) = β (1 − eiθx) e−αxdx 0 x = βln(1 − iθ/α).
Further, the gamma process can be expressed using a moment generating function in the form of
E(eiθXt ) = e−tφ(θ) θ −βt = 1 + α α + θ−βt = α α βt = . α + θ
The characteristic exponent, φ(θ), belonging to the infinitely divisible distribution, is related
−tφ(θ) to the moment-generating function through MX (t) = e . Therefore, we can calculate the 1 2 mean and variance of Xt via first moment, MX (0), and second moment, MX (0). 3.2 Stochastic Growth Models 61
Figure 3.3: Gamma Process trajectories with different shapes, β
Brownian Motion
A well-known Gaussian distribution with mean µ ∈ R and variance s2 > 0, a Brownian motion can be written as
Xt = µt + sBt, t ≥ 0,
where (Bt)t≥0 is a Brownian motion. Take the probability law
1 2 2 dF (x) = √ e−(x−µ) /2s (dx). 2πs2
Thus, 3.2 Stochastic Growth Models 62
Z 1 eiθx dF (x) = exp − s2θ2 + iθµ R 2 ( ) 1 s 2 µ n = exp − √ θ2 + iθ 2 n n is an infinitely divisible distribution (Figure 3.4) with a triplet (γ, σ, ν) equivalent to (−µ, s, 0). The characteristic exponent is φ(θ) = s2θ2/2 − iθµ.
Standard Brownian Motion B(t) 0.00 0.10 0.20 0.30 0.000 0.005 0.010 0.015 0.020 0.025 0.030 t
Figure 3.4: Standard Brownian Motion trajectories
Brownian motion does not have monotone paths and its trajectories are unbounded variation over time. It may hit any real value, but it will be back to zero after some time eventually or we call it a fluctuation. By adding stochasticity, we can model various environmental randomness that involve the standard Brownian motion process. Next, we will look into one of the special cases of the L´evyprocess, which is generally known as a subordinator. 3.2 Stochastic Growth Models 63
3.2.6 Subordinators: A Special Case of L´evyProcess
A subordinator is a special case of the L´evyprocess that has a L´evydensity. It is a form of distribution that describes the state of no growth, alternating with spontaneous growth of many small jumps (Sato, 1999). Not only does it account for both intrinsic and extrinsic sources of variation to be associated with a growth model, but a subordinator also enables individual growth trajectories to be in monotonic increments. Hence, we consider how to apply the subordinators to characterise the discontinuities in individual growth trajectories.
Every semi-martingale Xt can be expressed as a time-transformation Brownian motion in which
Tt is a positive semi-martingale (Monroe, 1978). Therefore, we have a Brownian motion Bs, s ≥
0 and an increasing random time Tt such that Xt = B(Tt). Since every L´evyprocess is a semi- martingale, by such result, a time-transformation L´evyprocess can be written as Xt = Z(Tt), where the subordinator Zt is an increasing L´evyprocess.
Let {Zt, t ≥ 0} be a subordinator starting from the origin such that Z0 = 0, the process is right continuous and takes the value on [0, ∞). It has independent and stationary increments on [0, ∞]. For t ≥ 0,Zt ≥ 0 and Zt1 ≤ Zt2 at t1 < t2. Furthermore, it does not have a diffusion component and a negative drift, which means the individual never moves backward and has no negative jumps either (see Cont & Tankov, 2004).
A subordinator incorporates a deterministic function and a stochastic process made up of infinite small jumps (Russo et al., 2009a). A L´evyprocess is considered to be a subordinator if and only if its L´evytriplet possesses in the form of (γ, 0, ν). It has a drift and a L´evymeasure; however, the diffusion coefficient does not exist throughout the process.
It is more convenient to look at the Laplace transform of Zt
−θZt Mt(θ) = E(e ), θ ≥ 0 (3.19) = etφ(θ),
where φ(θ) is the Laplace exponent in the form of 3.2 Stochastic Growth Models 64
Z ∞ φ(θ) = iγθ + eiθx − 1 ν(dx), 0 where γ ≥ 0 and L´evymeasure ν satisfies the requirements of
ν(−∞, 0) = 0,
Z ∞ x ∧ 1 ν(dx). 0
The Laplace transform in Eqn (3.19) will be able to estimate the mean and the variance of the
2 subordinators easily. If Zt has a finite mean µ and finite variance σ , then the first moment (mean) of the subordinator can be written as
E(Zt) = µt, (3.20)
and
2 Var(Zt) = σ t, (3.21)
2 where µ and σ are the mean and variance of Zt at t = 1.
A subordinator can also take into consideration the individual and environmental variations.
Examples of Subordinators
The gamma process and inverse Gaussian process are examples of subordinators and, as such, they do not have Brownian or martingale components. Both processes can be seen as an infinite 3.2 Stochastic Growth Models 65 superposition of compound Poisson processes that lead to infinite jump activity. We can use the subordinators to model the growth of animals based on their statistical framework.
Gamma Processes
A gamma process is a subordinator following a gamma distribution at time t with density in the form of
λαt f(x) = xαt−1 e−λx ∼ Γ (αt, λ), x > 0, Γ (αt) for α, λ > 0 and f(x) = 0 elsewhere. The parameter α is the average rate of jumps and the scaling parameter λ controls the distribution of jump sizes.
The Laplace transform of the gamma distribution is
−θZt Mt(θ) = Ee λ αt = . (3.22) λ + θ
Hence, the mean of subordinator following a gamma distribution is
−dMt(θ) E(Zt) = dθ θ=0 αt = , λ and the variance is
2 d Mt(θ) 2 Var(Zt) = 2 − {E(Zt)} dθ θ=0 αt = . λ2 3.2 Stochastic Growth Models 66
A natural extension of the traditional von Bertalanffy growth model is,
dLt = g(Lt−, θ)dZt, (3.23)
where Zt is a subordinator (such as a gamma process). Russo et al. (2009a) provided an interesting “jumpy” paths with g(Lt) = L∞ − Lt− (see their Figure 4, pp.526). We will fur- ther investigate how to apply such models to both tank and tagging studies. Incorporating the effects of covariates on growth in a similar way to Eqn (3.14) is also an interesting step forward.
Inverse Gaussian Processes
While a Gaussian process describes a Brownian motion’s level at a fixed time, an inverse Gaus- sian process describes how long a Brownian motion with positive drift takes to reach a fixed √ positive level. Here, the subordinator given by Zt = aBt + bt, t ≥ 0 where (Bt) is a standard
Brownian motion with drift b > 0 and a > 0. Then the first passage time of Zt for a fixed level α > 0 is
Tα = inf{t ≥ 0|Z0 = 0,Zt ≥ α}.
It is well known that Tα follows an inverse Gaussian distribution, where IG(µ, λ), with the mean and scale parameters α α2 µ = , and λ = . b a
The first passage time Tα for fixed-level α has probability density function
αebα/a b2 α2 fα(x) = √ exp − x − . (3.24) 2πax3 2a 2ax
Its Laplace transform of fα(x) is 3.2 Stochastic Growth Models 67
( r !) b2 2θ b M (θ) = exp −α + − . α a2 a a
3 3 The corresponding subordinator has a mean E(Zt) = µ = α/b, and variance = µ /λ = aα/b , where a, b > 0.
3.2.7 Time Transformation Model
Let {Zt, t ≥ 0} be a subordinator; the time transformation of the L´evyprocess is Lt = ZVt with the size-at-age of an animal denoted by Lt at time t, and the time transformation
−k(t−t0) Vt = 1 − e is followed a von Bertalanffy (VB), where k is the growth speed and t0 is the age of fish at zero length (Russo et al., 2009a). As discussed earlier, the size-at-age can be esti- mated as long as Zt has a finite mean and a variance based on Eqns (3.20) and (3.21), such that
E(Lt) = µVt, and
2 Var(Lt) = σ Vt,
2 where µ and σ are real numbers. Here, µ plays the role of l∞; thus, the estimates follow the VBGF and possess monotonic increment behaviour. However, the variance by time change is an increasing function, although biologically, the variances ought to be non-monotonic in time. As a result, the models could still be improved.
In particular, for a gamma process, Zt follows a distribution of Γ (αVt , λ), where µt = αVt /λ, 2 2 √ √ and σt = αVt /λ . For an inverse Gaussian process, Zt ∼ IG(Vt/ a, b/ a), where µt = Vt/b, 2 3 and σt = aVt/b .
In the later part, we will show how to estimate the growth parameters of these processes based on data in §4.1.2. 3.2 Stochastic Growth Models 68
3.2.8 Dol´eanExponential Model
The Dol´eanexponential (Russo et. al, 2009a) is a more interesting growth model, which is believed to be a better model compared to the aforementioned time transformation by Faben’s method.
Suppose that a stochastic equation is written in the form of
dLt = (L∞ − Lt−)dZt, (3.25)
where Zt is a subordinator and length at time of conception, Lt0 = 0. If we define Yt = L∞ −Lt, knowing that Lt0 = 0, then Yt0 = L∞. Hence, the differential equation of Yt can be written as
dYt = −dLt
= −(L∞ − Lt−)dZt
= −Yt− dZt .
Given that Yt0 = L∞, the solution of dYt = −Yt−dZt is a Dol´eanexponential in the form of
−Z˜t+t Yt = L∞e 0 ,
˜ where Zt is restricted to jumps not greater than 1. Here Z is another subordinator that contains
L´evymeasureν ˜ by l → log(1 + l) and driftγ ˜ = γ. Since Yt = L∞ − Lt, we simplify
−Z˜t+t Lt = L∞(1 − e 0 ), (3.26)
˜ where Z = k = γ + α log((λ + 1)/λ). It is known that the Laplace transform of Zt is
−θZ˜t −Z˜t Mt(θ) = E(e ). When θ = 1, we have Mt(1) = E(e ) leading to the mean size at age t as 3.2 Stochastic Growth Models 69
−(t+t0)φ˜(1) E(Lt) = L∞(1 − e ) ,
where φ˜ is the Laplace exponent. Chapter 4
Modelling Growth for Crustaceans
In this thesis, we mainly focus on the growth rate of crustacean species. The growth of crus- taceans is significantly different from that of fish. Basically, crustaceans must moult so that they can grow. Moulting is a complicated process that involves changes fuelled by hormones and enzymes. Before moulting, an individual will undergo an inactive period for expanding and swelling its body to force the old cuticle to be loosened and removed from the epidermis. The process involves muscular contraction and fluid secretion from the body to ensure that there is sufficient room for a new, larger cuticle to grow. Consequently, the growth of crustaceans is a discontinuous process due to the periodical shedding of the exoskeleton and a rapid increase in body size.
Growth is a complex process involving intrinsic and extrinsic variability in the different temporal lead to the discontinuous process. The stepwise growth of crustaceans through the moulting process makes estimating growth more complex. Some species cease to moult at maturity stage, whereas others continue to grow indefinitely when they sexually mature. Most crustacean species have a finite size as growth proceeds, until death. The physiological mechanism will cause variation in asymptotic length for each one in a population. Therefore, a discontinuous component must be incorporated in the growth models of crustaceans.
In reality, most of the studies related to the moulting and growth of crustaceans, until now, have poorly understood these unique differences. Thus, intense research is required to better 71 quantify the growth of crustaceans. In a discontinuous stepwise model, the growth pattern of the crustacean species can be described by two fundamental growth processes, namely, IP and MI. In addition to using a subordinator to estimate the growth parameters for a MI, we can also fit a nonlinear mixed-effect model that incorporates environmental factors to form a stochastic process. We are considering two examples from a class of subordinator that fit the MI from tank data: the gamma process and the inverse Gaussian process. By estimating the population mean and variance of the size increment, we can then average the discontinuous growth paths to yield a population growth trend.
For modelling the IP, various models are available, such as the Cox proportional hazard model (Cox, 1972) and the generalised linear model (GLM) approach.
Parameter estimations can be done in many ways, including three approaches we have accounted for: (1) The parametric approach, where we assume that IP or MI follows a distribution (e.g., gamma or lognormal); (2) The moment approach, where the estimation of IP or MI is a function of its condition variables of premoult interval periods Tij−1 and premoult length Lij−1; and (3) The nonparametric approach, when the density functions of IP and MI are unknown, thus an estimating equation will be considered for parameter estimation.
Comparatively, the data-driven or non-parametric approach is much more sensible and realistic than the model-driven or parametric approach. However, the available dataset we have may not be possible since the spline method requires a big data set to yield a meaningful result. For tank spiny lobsters, the data need to be collected in a long-term process besides they can be costly in a practical manner. Furthermore, the model-driven is a set of parameters without visual deformations (ease of computing speed), which is in favor of our aim to estimate growth parameters of MI and IP.
Throughout the entire work in the thesis, our dataset were analyzed using the statistical soft- ware R version 2.10.
First, we model the MI by: 4.1 Modelling of Moult Increments (MI) 72
• The linear mixed-effects approach.
• The subordinator-based model.
• The nonparametric approach.
Second, we model the IP of individuals using:
• The Cox model.
• The GLMs approach.
In addition to glm function, there are R packages that can fit a large variety of parametric distributions, e.g., gamlss and VGAM.
4.1 Modelling of Moult Increments (MI)
4.1.1 Linear Mixed-Effects Models
Several models have been proposed to describe the intercorrelation of both the intermoult period and the MI on the premoult length for crustacean species. One of the candidate models for the MI analysis is the mixed-effects model. There are two assumptions about a fitted mixed-effects model:
1. The within-group errors are independent and identically normally distributed (iid), and they are independent of the random effects.
2 2. The random effects, u ∼ N(0, ψ = σu), are independent of the group and other different groups.
A model with p fixed covariates and the random intercept could be written as (see Pinheiro & Bates, 2000), 4.1 Modelling of Moult Increments (MI) 73
Iij = xijβ + ui + εij, i = 1, 2, ..., n, j = 1, 2, ..., ni,
where xij = (xij1, xij2, ..., xijp). We can express the random intercept in matrix notation as
2 2 Ii = Xiβ + Ziui + εi, ui ∼ N(0, σu), εi ∼ N(0, σ I), where Ii1 β1 . . Ii = . , β = . , Iini βp ni×1 p×1 xi11 ········· xi1p 1 εi1 x x x ··· x i21 i22 i23 i2p . . Xi = , Zi = 1 = . , εi = . . ...... . . . . . 1 εini ni×1 ni×1 xini1 xini2 xini3 ··· xinip ni×p
It is assumed that both random effects and the error term have a zero mean so that E(I)=Xβ. We denote the variance-covariance matrix for the residual errors, ε as Q, whereas variance- covariance matrix for the random effects, u is denoted as P. Therefore, P and Q form a set of diagonal matrices due to the independence of ε and u,
2 P = σuI
2 Q = σε I , where I is an identity matrix. Hence, the variance-covariance matrix V in the mixed model for increments is
V = ZPZT + Q, 4.1 Modelling of Moult Increments (MI) 74
of which ZT be the transpose of matrix Z.
In this section, we assume a continuous growth study for the animals since the population growth, overall, is a continuous function. This approach is to model the individual variabilities in the analysis of repeated measures. It does not need an equal number of repeated measures among individuals.
Nonlinear mixed-effects models describe a response variable as a function of predictors (Pin- heiro & Bates, 2000), allowing the parameters to vary for each individual (between-individual variability) and considering the correlation among measurements in each individual (within- individual variability). We can use a mixed-effects model that contains both fixed effects and random effects in the analysis of MI.
We express the model for MI as
Iij = g(Xij, βi) + εij, i = 1, ..., n, j = 1, ..., ni,
where Iij are the ni moult increments for animal i, the nonlinear function g models the rela- tionship between response variable Iij and predictor Xij, the p × 1 parameter vector βi. The random errors εij are independent and identically distributed, the estimated MI for animal i at time j, given the parameter βi are
E(Iij|βi) = g(Xij, βi).
Example 1: Linear Models With Random Intercepts
We proposed a simple regression using the previous length with quadratic terms. One could also try a basic regression spline, however, it requires a large number of samples to be meaningful. But as a trade-off in complexity, we consider more straightforward models which are easier to interpret biologically and also easier to derive a final population growth model rather than considering a more complex spline model. 4.1 Modelling of Moult Increments (MI) 75
Suppose the MI of ith individual are
2 2 I = δi + β1Li + β2Li + β3SEXi + β4Ti + β5Di + β6(SEX ∗ L )i + β7(SEX ∗ T )i + εi,
where we denote the random effects as δi ∼ N(0, ψ), L as the premoult length, T is the water temperature, D is the density, L2 is the second order of premoult length, SEX ∗ L2 is the interaction between the sex and premoult length, and SEX ∗ T is the interaction between the sex and temperature, and ε is the random errors.
We define β to be a vector of fixed effects, X is a matrix of fixed effects, u is a vector of random effects, Z is a matrix of random effects, and ε is the iid random error term.
To account for individual variability, pre-moult length L is assumed to be random. Here, we consider that when the length of the rock lobster is very small, an unusually small increment (close to 0) is expected. As a result, the intercept is constrained to be 0. Figure 7.12 displays predictions for increment of the females and males according to different densities. The tendency is to get a greater increment when the population has a lower density.
For parameter estimation in this model, the variance components have to be estimated initially. When we obtained the P and Q estimates, the parameters β and u could then be estimated. To fit the random-effects model, the lme function can be used either by maximum likelihood (ML) or restricted maximum likelihood (REML). As it has been already pointed out, the intercept is not available in both fixed and random effects. The samples formed as a group variable in our study.
Example 2
Let g to be a function of L; the model can be written as
α+βL Iij = e + εij, i = 1, ..., n, j = 1, ..., ni, 4.1 Modelling of Moult Increments (MI) 76
where α, β are the parameters, and L is the premoult length of the lobsters.
4.1.2 Time Transformation Model
Following from §3.2.7, Lt = ZVt where Lt is the size-at-age of the animal t and V follows a VB
−k(t−t0) function Vt = 1 − e ; then the estimated size-at-age t is
E(Lt) = E(ZVt ) = µVt, (4.1)
is the VBGF when µ plays the role of l∞, and variance of the size-at-age t is
2 Var(Lt) = σ Vt.
Example 1 : Increments of the Gamma Process
The likelihood function will also be seen in §5.5.2, the MI of an individual consists of {I|T,L−},
− where L is the premoult length of the individuals and T is the moult time interval. Then ZVt
−k(t−t0) follows a gamma distribution Γ (αVt , λ), where V follows a VBGF such that Vt = 1−e .A VBGF is constructed based on the ages known initially. However, we do not have information on age t0 for crustacean species. The Fabens’ (1965) method for estimating the size of animals will be used under the assumption that the ages are all unknown.
For the purpose of illustration, Figure 4.1 exhibits the different time points t1, t2, t3... for the growth of an organism (measured in carapace length, L). Let the initial age t0 = 0 for all individuals; hence, the lengths at time t2, t3 are
−kt2 L2 = L∞(1 − e ), (4.2)
−kt3 L3 = L∞(1 − e ). 4.1 Modelling of Moult Increments (MI) 77
We denote a MI at time t as It = Lt − Lt−1, t ≥ 1; then the increment at time t3 is
−kt2 −kt3 I3 = L3 − L2 = L∞(e − e )
−kt2 −k(t3−t2) = L∞e {1 − e }.
−kt2 From Eqn (4.2), L∞e = L∞ − L2. Therefore, the increment at time t3 can be reparame- terised as
−k(t3−t2) I3 = (L∞ − L2){1 − e },
or it can be simplified in a more general expression as
−k(tij −ti,j−1) Iij = (L∞ − Li,j−1){1 − e },
−kTij = (L∞ − Li,j−1)(1 − e ),
where i = 1, 2, ..., N, j = 1, 2, ..., ni and Tij = tij − ti,j−1.
−kT 2 −kT 2 Now, a gamma process has a mean µT = α(1 − e )/λ, and a variance σT = α(1 − e )/λ . − Thus, α = λ(l∞ − L ), the estimated increment length after time interval T , is
− − −kT E(I|T,L ) = (l∞ − L )(1 − e ),
and the variance of the increment length after time interval T is
(l − L−)(1 − e−kT ) Var(I|T,L−) = ∞ , λ where L− is the premoult length.
Subsequently, the likelihood of the increment lengths from the N animals is 4.1 Modelling of Moult Increments (MI) 78
Figure 4.1: Length of an individual over time.
−kT N ni − ij ! λ(l∞−Lij )(1−e ) − −kTij λ (λ(l∞−L )(1−e ))−1 − Y Y ij −λIij L(I|T,L ; l∞, k, λ) = − Iij e , Γ (λ(l − L )(1 − e−kTij )) i=1 j=1 ∞ ij where i is the individual, and j is the number of measurements in each individual.
The log-likelihood after time interval T is
N ni X X −kTij −kTij l(l∞, k, λ) = λ(l∞ − Li,j−1)(1 − e ) log λ − log(Γ {λ(l∞ − Li,j−1)(1 − e )} i=1 j=1
−kTij + [{λ(l∞ − Li,j−1)(1 − e )} − 1) log(Iij) − λIij].
Example 2 : Process With Inverse Gaussian Increments
If we assume an inverse Gaussian distribution, then following from the Eqn (3.24), the mean
−kT and variance of a subordinator following an inverse Gaussian process are µT = (1 − e )/b, 2 2 − and σT = aµT b , where 1/b plays the role of (l∞ − L ). Assuming the increment length to − − −kT √ be {I|T,L }, ZVT follows an inverse Gaussian process IG{(l∞ − L )(1 − e ), b/ a)}, where L− is the premoult length of the individuals, and T is the moult time interval. The estimated 4.1 Modelling of Moult Increments (MI) 79 increment length after time interval T can be written as in Eqn (7.1.4),
− − −kT E(I|T,L ) = (l∞ − L )(1 − e ),
while the variance of the increment size after time interval T is
− − 3 −kT Var(I|T,L ) = a(l∞ − L ) (1 − e ).
We could write the joint likelihood function on increment length at the time of N individuals as follows
N n i −kTij A −kTij 2 Y Y (1 − e )e Iij (1 − e ) L(I|T,L−; a, l , k) = exp − − , ∞ q 2 3 2a(l∞ − Li,j−1) 2aIij i=1 j=1 2πaIij
−kTij where we denote A = (1 − e )/a(l∞ − Li,j−1) while the estimated parameters are a, l∞ and k. The log-likelihood of the increment length after time interval T forms
N n X Xi (1 − e−kTij ) 3 l(a, l , k) = (log(1 − e−kTij ) + − log(I ) ∞ a(l − L ) 2 ij i=1 j=1 ∞ i,j−1
−kTij 2 Iij (1 − e ) 1 − 2 − − log(2πa)). 2a(l∞ − Li,j−1) 2aIij 2
The parameters will be estimated using the maximum likelihood (MLE) approach for both examples above. The method of MLE is carried out in two steps: first, we develop a model that the likelihood function contains a set of parameters given the observed data. Subsequently, we select the parameter estimates that maximise the likelihood function.
Maximising L(θ) with respect to all the parameters (θ) will result in their estimates. The underlying process I(t) is not a L´evyprocess anymore because of its non-stationarity and dependence between increments induced from the conditional mean dependence. 4.1 Modelling of Moult Increments (MI) 80
4.1.3 Dol´eanExponential Model
The mean length from the Dol´eanexponential model follows a von Bertalanffy function, pro- vided that its L´evymeasure 0 ≤ ν ≤ 1 holds. However, the deficiency of Eqn (3.26) shows that when t is at infinity, it approaches L∞ for the entire population. This fact indicates that all organisms have the same limiting size at time t, which seems to be unrealistic due to the individual variability. Therefore, we will assume L∞ to be a random variable in the following discussion.
Also, the aforementioned model §3.2.8 is constructed for length at a given age t + t0, where t0 is the time of conception. From Eqn (3.26), the Dol´eanexponential model is Lt = L∞(1 − ˜ exp(−Zt+t0 ). However, we face the difficulty to age directly, especially for crustaceans. There- fore, we consider that all individuals have an initial time t0 = 0 such that Lt = L∞(1 − ˜ ˜ exp(−ZTt ), where T is the relative age at at time t while assuming ZTt ∼ Γ (αTt, λ). By using the MLE approach, we can estimate the parameters α, λ, γ and E(L∞) in which L∞ follows ˜ a density function (e.g., gamma distribution). In this case, we presume L∞ and ZTt are both independent.
Making the assumption that Z˜ follows a gamma process with jumps constraint in (0,1], the L´evymeasure of Z˜ has a density of f(u) such that
Z 1 uf(u)du < +∞. 0
Let
−λu αe ; 0 < u ≤ 1 f(u) = u (4.3) 0 ; u > 1.
Both α and λ are positive parameters. Having the L´evymeasure f(u) and drift γ, the solution for Eqn (3.25) can be yielded through the Dol´eanexponential to form
˜ Lt = L∞{1 − exp(−γt − ZTt )}. 4.1 Modelling of Moult Increments (MI) 81
Assuming that the lengths of an individual are independent, the likelihood can be expressed as
n αT αTtk −1 −λL Z λ tk L e tk Y tk δ δ−1 −ηL∞ L(α, λ; Lt1 ,Lt2 , ..., Ltn ,Tt1 ,Tt2 , ..., Ttn ,L∞) = η L∞ e dL∞, Γ (αTt ) k=1 k
2 in which L∞ ∼ Γ (δ, η) such that E(L∞) = δ/η and var(L∞) = δ/η . Apparently, both Lt and
L∞ follow a gamma density function.
As L∞ is a random variable, it is mathematically intractable and a numerical integration is utilised as a solution of the likelihood function above. Therefore, the negative log-likelihood for i = 1, 2, ..., N individuals of the observed data {Ttik ,Ltik }, k = 1, 2, ..., ni is l(α, λ|L ,L , ..., L ,T ,L ) ti1 ti2 tini tik ∞,i
N n X Z Xi = log exp αTtik log(λ) + (αTtik − 1) log(Ltik ) − λLtik − log{Γ (αTtik )} i=1 k=1 +δ log(η) + (δ − 1) log(L∞,i) − ηL∞,i − log{Γ (δ)} dL∞.
Therefore, the mean length is
−γTt−φ˜(1)Tt E(Lt) = E(L∞)(1 − e ) ( ) λ αTt = E(L ) 1 − e−γTt , (4.4) ∞ λ + 1 whereas the variance of size is 4.2 Modelling of Intermoult Periods (IP) 82
˜ ˜ 2 −Zt−t0 2 −Zt−t0 2 Var(Lt) = E(L∞(1 − e ) ) − {E(L∞(1 − e ))} h ˜ ˜ ˜ i −Zt−t0 2 −Zt−t0 2 −Zt−t0 2 2 = E(1 − e ) − {E(1 − e )} + {E(1 − e )} E(L∞)
−Z˜t−t 2 −{E[L∞(1 − e 0 )]} ˜ ˜ ˜ −Zt−t0 2 −Zt−t0 2 2 2 −Zt−t0 2 = Var(1 − e )E(L∞) + [E(1 − e )] E(L∞) − (EL∞) (E[1 − e ]) ˜ ˜ −Zt−t0 2 −Zt−t0 2 = Var(1 − e )E(L∞) + (E[1 − e ]) Var(L∞) n o −(t−t0)˜l(2) −2(t−t0)φ˜(1) 2 −(t−t0)φ˜(1) 2 = e − e {Var(L∞) + (EL∞) } + Var(L∞)[1 − e ] .
Subsequently, we simplify equation above with the help of Eqn (3.19) and Eqn (3.22) in a form of and the variance of length
" # λ αTt λ 2αTt Var(L ) = E(L2 )e−2γTt − t ∞ λ + 2 λ + 1 " #2 λ αTt +Var(L ) 1 − e−γTt . ∞ λ + 1
As noted from Eqn (4.4), the growth rate k can be expressed in the form of γ +α log{(λ+1)/λ}.
4.2 Modelling of Intermoult Periods (IP)
This chapter introduces a stochastic growth component into a growth model that account for the stepwise growth of crustaceans. One important stochastic process is the well-known Markov process, where future events are determined by the present. Here, stochastic growth models that we study are closely related to Markov property, where future and the past moults are independent, provided the current moult is known. Also, we aim to develop a new stochastic growth model that provides a few valuable insights by improving the existing stochastic growth models.
Despite substantial literature reformulating growth models as stochastic differential equations, some models may predict negative size increments and, thus, may not be able to guarantee a 4.2 Modelling of Intermoult Periods (IP) 83 monotonically increasing function in growth (Garcia, 1983). To this end, we need to ensure that the stochastic growth model considers only positive jumps. Therefore, we introduce a subor- dinator, which allows nondecreasing measurement in growth, e.g., the length of crustaceans. For example, Russo et al. (2009b) introduced a subordinator to characterise the growth of Atlantic herring populations with a mean growth curve following a VBGF. A subordinator is said to have two parameters that control frequency and size of jumps, both of which are inversely correlated. As the size of jumps getting closer to the origin, the number of jumps will get larger. The aforementioned L´evyprocess describes a subordinator in terms of a stochastic process. It is an important process since the L´evyprocess generalises the random walks to continuous time, which enables us to quantify the growth rate in a smooth curve.
Besides using these two stochastic processes MI and IP, it is imperative to account for envi- ronmental factors in growth studies, as highlighted above. We therefore incorporate covariates into the growth function. Also, we develop a semi-parametric approach to data analysis, which offers an alternative method for growth estimation. More often, a misspecification of distri- butional assumption results in biased parameter estimates and could be misleading. Hence, a semiparametric approach is more robust because they do not make assumptions about the distribution of the variables of interest in a population. Furthermore, nonparametric methods may be preferable because the subsequent moulting times remained unknown at the end of the experiment (right-censoring).
4.2.1 Cox Proportional Hazard Model
Estimating the moulting time is crucial for the purpose of stock assessment, such as forecasting recruitment to ensure the sustainability in the population. There is a variety of statistical approaches to analysing such “time to event” data. For example, Hougaard (2000) provided an excellent review in the context of survival data analysis. The time intervals of a moult can be treated as a survival data while moulting itself is assumed to be an event. Thus, we can account for a semiparametric method, such as the Cox model. The Cox model and a generalised lognormal model will be introduced for quantifying IP of lobsters. 4.2 Modelling of Intermoult Periods (IP) 84
The Cox model is the most interesting survival modelling that assesses the relationship between survival and one or more covariates in the survival-analysis literature. In addition, there is no assumption of a baseline hazard and so the parameter estimate for a baseline hazard is not required. In comparison, this semiparametric model is therefore more robust without assump- tions about the parametric distribution of the survival times (Kleinbaum & Klein, 2005). The hazard function is
0 g(t|x) = g0(t) exp(x β),
where g0(t) is an unspecified baseline hazard function with independent covariates x; the ex- ponential part is attractive because it ensures that the ratio of the hazards g(t|x) is always non-negative and time-independent. Parameter estimates in the Cox model are obtained by maximising the log-partial likelihood. The partial likelihood function is given by
Y exp(xiβ) L(β) = P , j≥i exp(xjβ) i,δi=1
where δi is an indicator function implying δi = 1 is uncensored, δi = 0 for censored data and the partial likelihood has no ties to the data set. The log partial likelihood is given by
" ( )# X X l(β) = log L(β) = xiβ − log exp(xjβ) .
i,δi=1 j≥i
First, we plot the survival curves using the Cox model to study the correlation between inter- moult period, T and the covariates, x of the lobsters. The hazard function of T has the form
− − 2 g(t|x) = g0(t) exp{β1L + β2(L ) + β3T + β4D} , (4.5)
− − 2 where g0(t) is the baseline hazard function, L is the premoult carapace length, (L ) is the second order of premoult length, T is the tank water temperature, and D is the density of lobsters in the tank. 4.2 Modelling of Intermoult Periods (IP) 85
4.2.2 Modelling IP via Lognormal Distribution
Here, we know the exact moulting time T of the animals. We will utilise the model proposed by (Restrepo, 1989), where the intermoult period T is a function of the premoult length L−.
An estimator of the IP following a lognormal distribution will be considered in the growth
2 model. Suppose log(T ) is distributed N(µL, σ ); the mean of intermoult period T is based on premoult length L− ,
2 E(T ) = exp(µL + σ /2) = α exp(βL−), (4.6)
therefore, we have
2 − µL = log(α) − σ /2 + βL ,
where α > 0 and β ≥ 0 are constants. There are three parameters to be estimated: α, β and σ. The likelihood for IP is
N ni " 2 − 2 # Y Y 1 − log(Tij) − (log(α) − σ /2 + βLij) L(T |L−; α, β, σ) = √ exp . 2σ2 i=1 j=1 σTij 2π
For simplicity, we assume that all the observations in j are independent for each sample i. Parameter estimation can be carried out by maximising the log{L(T |L−; α, β, σ)}.
4.2.3 Modelling IP via Gamma Distribution
Suppose the moulting interval Tij is a variable with a conditional density function, g(t|Lij−1), where Lij−1 is the premoult length. Let Tij ∼ Γ (γµij, γ) or it could be other sensible functions. 4.3 Parameter Estimation via Nonparametric Approach 86
Consider that the estimation of IP uses a GLM framework; the mean is thus given by µij = exp(β0 + β1Lij−1) and variance µij/γ.
Parameter estimates are obtained by maximising the log-likelihood given by
PN Pni l = i j=1 log{g(Tij|Lij−1)}
PN Pni β0+β1Lij−1 β0+β1Lij−1 = i j=1 [(γe ) log(γ) − log[Γ (γe )]
β0+β1Lij−1 +(γe − 1) log(Tij) − γTij],
where γ, β0 and β1 are parameters.
4.3 Parameter Estimation via Nonparametric Approach
If likelihood functions are difficult to specify or justify, as in §4.1, one can also consider the following generalised estimating equations (GEE) for (l∞, k) (James, 1991; Wang, 2004).
Assuming that all lobsters are reared in a tank, for animal i, denote the premoult length as
− L , and the carapace length of crustacean i at time interval Tij, j = 1, 2, ..., ni can be written as
− Λij = (L∞ − Lij)[1 − exp(−kTij)] + εij,
where the asymptotic length is L∞ and εij is the random error.
Suppose we presume that all animals were observed at the same initial time point, Ti1 = 0. Therefore, we can express the MI in a matrix form as
L− Λi = Xi + εi, L∞ 4.3 Parameter Estimation via Nonparametric Approach 87
whereby
0 0 exp(−kTi2) 1 − exp(−kTi2) Xi = . . . exp(−kTini ) 1 − exp(−kTini )
− Since the L∞ is an individual variability, we have E(L∞) = l∞. Let Ii = (l∞ − Li ){1 − exp(−kTi)} be the mean of Λi and all the residuals are independent and identically distributed, 2 iid such that ε ∼ N (0, σe ).
Under the working assumption that all the repeated measurements of an individual are inde- pendent, thus we can present our estimating functions for the parameters, α = (l∞, k) as (see Wang, 2004)
X T −1 U(α) = Si Vi (Λi − Ii) = 0, i in which
0 0 1 − exp(−kTi2) Ti2exp(−kTi2) Si = . . . 1 − exp(−kTini ) Tini exp(−kTini )
Consistency does not require correct specification of the covariance function Vi. Since Vi can be arbitrary, it is also known as an unstructured error matrix in this case. The covariance of T ∆i can be expressed as Vi = XiVcXi + Ψni in which Ψni is the covariance matrix of εi. The − covariance between two random variables Cov(L∞,L ) can be written as 4.3 Parameter Estimation via Nonparametric Approach 88
− 2 − (σ ) ρσ σ∞ Vc = . − 2 ρσ σ∞ σ∞
− 2 2 2 T Let the parameters in Vi be represented as θ = {(σ ) , σ∞, ρ, σe } . To obtain estimates of θ, we consider the estimating function of
X T G(θ) = Wi Ri(Zi − ei), i
th where Ri is the j diagonal elements of Vi, 1 ≤ j ≤ ni, Zi = (∆ij − Iij)(∆ik − Iik) whereas
Wi = (∂ei/∂θ).
4.3.1 Simulation Studies
Parameter estimates are obtained from a set of simulated data with repeated measurements.
Here, the growth rate of individuals k was set to be fixed at 0.5 while asymptotic length L∞ followed a lognormal distribution with a mean 120 and a standard deviation of σ∞.
2 2 2 2 Denote log(L∞) ∼ N {ln(l∞) − 0.5σ , σ = ln(1 + σ∞/l∞)}. The growth of each lobster was assumed to follow a VB function. Three values of σ∞ (5, 10, 20) will be used to compare the different sizes of individual variability. Sample sizes from small to large scale (N=100, 200,
500) were considered, and we evaluated through the standard error of the estimates (l∞, k), respectively.
The number of captures was assumed to be (2 ≤ ni ≤ 5) for each individual. The time interval between two consecutive moults was sampled from a lognormal distribution that Tij ∼ logN 2 (meanlog=log(α) − σT /2 + βLij−1, sdlog=σT ) of which α = 0.1, β = 0.01, and σT = 0.01. A 2 measurement error was normally distributed with σe = 5.
Let the initial length of animals range between 10mm and 30mm, and the carapace length of a lobster can be generated if the information on increment lengths I is known to follow a VBGF.
Suppose L∞ varies from one to another while k remains constant. 4.3 Parameter Estimation via Nonparametric Approach 89
(σ∞,N) l∞ k
(5,100) Mean 119.17 0.50 SE 0.78 0.01 (5,200) Mean 119.39 0.50 SE 0.55 0.004 (5,500) Mean 119.36 0.49 SE 0.35 0.002 (10,100) Mean 119.07 0.50 SE 0.78 0.01 (10,200) Mean 119.28 0.50 SE 0.55 0.004 (10,500) Mean 118.51 0.49 SE 0.48 0.003 (20,100) Mean 117.97 0.48 SE 0.92 0.01 (20,200) Mean 118.56 0.49 SE 0.70 0.01 (20,500) Mean 118.22 0.48 SE 0.38 0.003
Table 4.1: Parameter estimates with the true values of l∞ = 120, k = 0.5.
From 100 simulations, we had different sets of the means and standard errors, as presented in the following table. We assume the true values of (l∞, k) = (120, 0.5). As the sample size increases, the standard error of (l∞, k) decreases.
As the individual variability increases, the standard error will increase, too (Table 4.1). The estimated standard error decreases when the sample size increases. Although the standard error increases for larger σ∞, it does not have much effect on the estimates of (l∞, k). Chapter 5
Crustacean Growth Models for Multiple MI and IP
For tank data, we observed directly the changes in growth, such as length increment and intermoult periods throughout the moulting process of a population. Therefore, it was relatively straightforward to estimate the size-at-age of individuals. Animals more suitable for laboratory experiments (reared in tanks) are small crustaceans or organisms that grow rapidly. For large adult crustaceans, the time it takes to moult can take a year or even longer (Sard`a,1985). In this regard, juvenile crustaceans are generally used for tank experiments, which appears to be cost-effective.
5.1 Introduction
First, it is useful to discuss the moulting processes of crustaceans in general. A crustacean (e.g., lobster) has an exoskeleton that must be withdrawn so that the animal can grow longer. Primarily, the work was concerned with how often lobsters moult and how much they grow. The time required for shedding relies on the size of the animal; the larger the lobster, the longer the time needed to pull out the claws from its joints. The shell breaks between the tail and the body is called the carapace length. We will refer to carapace length when we address 5.2 Modelling Moult Increments With Individual Variability 91 the size or length of a lobster in the following discussion. As the lobster grows bigger, the number of moulting events decrease over time. The frequency of moults can be observed only when the lobsters are at the juvenile stage, because an adult may take more than a year to moult once. Since crustaceans shed periodically, such individuals cannot age directly. However, some techniques can determine the nature of the growth of the animals, such as tag-recapture studies, time intervals between two consecutive moults and the length increment of captive animals. Two types of data sets will be considered: tank data and tagging data; each type of data set, both observed and simulated will be examined in the numerical studies.
We propose a modification of the time transformation model as a solution to the problems that arose from the previous findings in §4.1.2. Note that it is possible to have a length Lij− larger than the mean L∞, which will cause a negative increment Iij. This will no longer be an issue when L∞ is assumed to be random because each L∞ will be guaranteed to be larger than any of the observed lengths. We now incorporate intrinsic variability to allow L∞ to vary among individuals (cf., Russo et al., 2009a; Wang et al., 1995) in a conditional joint probability function.
5.2 Modelling Moult Increments With Individual Vari- ability
Suppose p(l∞) is the probability density for L∞. In this context, the values of L∞ for each individual is not known and thus we typically integrate the joint probability distribution f(·) over L∞. Assuming L∞ conditional independence for the increments, we have the following log-likelihood function in which the vector θ collects all the parameters,
N ( n ) X Z Yi lI (θ) = log f(Iij|L∞) p(L∞) dL∞ . (5.1) i=1 j=1
This approach is biologically more realistic since each individual owns a different terminal length. In the application to our data set, we will assume that p(l∞) is a lognormal or a gamma function. The density function of increments f(·) can either be a gamma process or an inverse 5.2 Modelling Moult Increments With Individual Variability 92
Gaussian process.
5.2.1 Gamma-Gamma (GG) Model
− − −kTij Suppose {Iij|L∞,Lij,Tij} ∼ Γ (λ(L∞ − Lij)(1 − e ), λ) and L∞ ∼ Γ (α, β). If I and
E(L∞) = l∞ are independent, the log-likelihood given by Eqn (5.1) can then be written as
−kT N Ni − ij Z λ(L∞−Lij )(1−e ) − −kTij X Y λ λ(L∞−Lij )(1−e )−1 lI (θ) = log − × Iij × exp(−λIij) Γ {λ(L − L )(1 − e−kTij )} i=1 j=1 ∞ ij βα × × Lα−1 × exp(−βL )dL , Γ (α) ∞ ∞ ∞ where λ, α, β, k are non-negative parameters.
Implementation:
For simplicity, it is often easier to take a logarithm equation and rewrite it into an exponential form as follows:
N Z h Ni X X − −kTij lI (θ) = log exp λ(L∞ − Lij)(1 − e ) log(λ) i=1 j=1
− −kTij − −kTij −logΓ {λ(L∞ − Lij)(1 − e )} + {λ(L∞ − Lij)(1 − e ) − 1} log(Iij) i −λIij + α log(β) − log{Γ (α)} + (α − 1) log(L∞) − βL∞ dL∞.
One of the numerous ways to improve a numerical analysis is by using an iterative algorithm, such as Newton’s method. This method applies only when a differentiable function with the derivative exists. In general, two types of equation are to be determined, either linear or non- linear. These iterative methods are generally needed for large problems that cannot be resolved in a few steps. Starting from an initial value, the iteration ends when successive approximation converges onto the true solution.
For numerical quadrature, Simpson’s rule is one of the popular numerical solutions to the defi- 5.2 Modelling Moult Increments With Individual Variability 93 nite integrals. Among various means of deriving Simpson’s rule, the midpoint and trapezoidal rules provide a general framework for numerical approximations. However, in higher dimen- sions, it may be computationally expensive to use the aforementioned methods. Therefore, one can account for Monte-Carlo integration or sparse grids in terms of a problem of large dimensions.
We considered the MLE approach to compute a numerical solution iteratively on an interval
[min(L∞), max(L∞)]. When the numerical approximation converged, the appointed growth parameter estimates can be obtained. Since L∞ ∼ Γ (α, β), therefore E(L∞) = α/β. The resulting parameter estimates are used to compute the first and second moment of the moult increments I in the form of
− −kT E(I) = {E(L∞) − L }(1 − e ),
and
− −kT Var(I) = {E(L∞) − L }(1 − e )/λ.
Similarly, for models 4, 5 and 6, the algorithms will be carried out as discussed above.
5.2.2 Gamma-Lognormal (GL) Model
− − −kTij 2 Assume that {Iij|Lij,Tij,L∞} ∼ Γ {λ(L∞ − Lij)(1 − e ), λ} and L∞ ∼ ln N (µ, σ ). Hence, the LI (θ) can be written as
L(Iij|Tij,Lij−1,L∞)
−kT N Ni − ij Z λ(L∞−Lij )(1−e ) − −kTij Y Y λ λ(L∞−Lij )(1−e )−1 = − × Iij × exp(−λIij) Γ {λ(L − L )(1 − e−kTij )} i=1 j=1 ∞ ij
1 −A × √ e dL∞, L∞ 2πσ 5.2 Modelling Moult Increments With Individual Variability 94
2 2 where A = {log(L∞) − µ} /2σ , µ > 0, σ, k and λ are parameters to be estimated. For L∞ 2 follows a lognormal distribution, and its first moment is E(L∞) = exp(µ + σ /2). Hence, we have
− −kT E(I) = {E(L∞) − L }(1 − e ),
and
− −kT Var(I) = {E(L∞) − L }(1 − e )/λ.
5.2.3 Inverse Gaussian-Gamma (IGG) Model
− − −kTij − 2 Suppose {Iij|Lij,Tij,L∞} ∼ IG[(L∞ − Lij)(1 − e ), 1/{a(L∞ − L ) }] and L∞ ∼ Γ (α, β).
Hence, the LI (θ) can be written as
L(θ|Tij,Lij−1,L∞)
−kT N Ni ij − ( ) Z −kTij (1−e )/a(L∞−Lij ) −kTij 2 Y Y (1 − e )e Iij (1 − e ) = exp − − q − 2 3 2a(L∞ − Lij) 2aIij i=1 j=1 2πaIij βα × × Lα−1 × exp−βL∞ dL , Γ (α) ∞ ∞ where a, k, α and β are positive parameters.
Given that L∞ ∼ Γ (α, β), we obtain E(L∞) = α/β. Thus, the mean of increment lengths is
− −kT E(I) = {E(L∞) − L }(1 − e ),
and
− 3 −kT Var(I) = a{E(L∞ − L )} (1 − e ). 5.2 Modelling Moult Increments With Individual Variability 95
5.2.4 Inverse Gaussian-Lognormal (IL) Model
− − −kTij − 2 2 Suppose {Iij|Lij,Tij,L∞} ∼ IG[(L∞ −Lij)(1−e ), 1/{a(L∞ −L ) }] and L∞ ∼ ln N (µ, σ ). Thus,
N N ( ) Z i −kTij B −kTij 2 Y Y (1 − e )e Iij (1 − e ) L(θ|T ,L ,L ) = exp − − ij ij−1 ∞ q − 2 3 2a(L∞ − Lij) 2aIij i=1 j=1 2πaIij
1 −A × √ e dL∞, L∞ 2πσ
2 2 −kTij − where A = {log(L∞) − µ} /2σ , B = (1 − e )/a(L∞ − Lij), a, k, µ and σ are parameters. If 2 2 L∞ ∼ ln N (µ, σ ), then E(L∞) = exp(µ + σ /2). The first and second moments of the incre- ment lengths can be expressed as
− −kT E(I) = {E(L∞) − L }(1 − e ),
and
− 3 −kT Var(I) = a{E(L∞) − L } (1 − e ).
After formulating the models, the following stage is to fit those models with data accordingly. Subsequently, it is crucial to implement model diagnostics. If there are any violations, an improved model may be taken into consideration.
Diagnostics play an important role in determining the validity of model assumptions. The tests allow us to analyse whether the findings are plausible and can be trusted. The set of candidate models can be tested for the quality of the fit and the best can be selected, such as Akaike’s Information Criterion (AIC) or Bayesian Information Criterion (BIC). 5.3 Modelling Moult Increments via Beta Distribution 96
5.3 Modelling Moult Increments via Beta Distribution
We have modelled the growth of the stepwise function using different distributions in §4.1.2. However, a problem arises when we predict the future growth over years. For example, let the
MI at time interval T is IT ∼ Γ (αVT , λ) in which VT = 1 − exp(−kT ) with k as a constant. For − a random sampling, it is possible that the current length L +IT may exceed asymptotic length
L∞ when T → ∞. Figure 5.1 illustrates how the problem can happen when the gamma or inverse Gaussian distribution comes into play. The problem is that the gamma distribution has a lower bound of zero and is unlimited on the right. Likewise, the inverse Gaussian distribution may have similar consequences because the boundary lies within (0, ∞). For carapace length greater than L∞, the length is therefore an unreasonable estimate.
Figure 5.1: Processes that exceeded the asymptotic length.
To illustrate the aforementioned problems, we considered a simulation of the stepwise growth model for N individuals. First, assuming that the moult increments follow a gamma density
− −kT function, I ∼ Γ (shape = λ(L∞ − L )(1 − e ), scale = 1/λ), where the estimated parameters are λ = 0.6225 and k = 0.328. Subsequently, let the intermoult periods follow a gamma density
− β0+β1L function, T ∼ Γ (shape = γe , scale = 1/γ) with β0 = 2.087, β1 = 0.011, γ = 12.535. 5.3 Modelling Moult Increments via Beta Distribution 97
The initial length and initial intermoult period of N individuals were randomly sampled within (10, 50) mm and (10, 150) days, respectively. Each individual i = 1, 2, ..., N had different moulting times, j = 1, 2, ..., Ni ranging from 4 to 15 moults for all individuals.
For example, we selected an individual from a population; it had the initial length, L0= 30mm and the initial intermoult period, T0 =79 days.
Table 5.1: An individual i with Ni moults in a simulated data.
L∞,i = 158.37
i Moulting times, Ni = 14
j1 j2 ... j8 j9 j10 ... j14
Lij−1 30.00 38.11 . . . 132.47 146.80 159.60 > L∞,i ... Lij−1 > L∞,i
Tij−1 0.22 0.47 . . . 2.16 2.62 3.43 ......
Referring to Table 5.1, we notice that gamma processes produce invalid values during simula- tion. As L− increases indefinitely over time, these iterations result in negative measurements
− for increment length of I = (L∞ − L ){1 − exp(−kT)} and subsequently render a problem in generating T , since I and T happen concurrently in a moulting process.
− In view of these circumstances, we suggest a bounded region by scaling I/(L∞ − L ) to a constraint within (0,1). In this case, we can ensure that the growing process is within the L∞. − Let I/(L∞ − L ) ∼ Beta(α, β), leading to
α−1 β−1 − −1 I I b{I/(L∞ − L )} = {G(α, β)} − 1 − − , L∞ − L L∞ − L in which
Γ (α)Γ (β) G(α, β) = , Γ (α + β) where Γ (·) is a gamma function. 5.3 Modelling Moult Increments via Beta Distribution 98
− If we assume the mean of increment length E(I|T,L ,L∞) follows a VBGF, the estimate of − I/(L∞ − L ) is therefore
1 − e−kT = α/(α + β),
and the variance
αβ Var{I/(L − L−)} = . ∞ (α + β)2(α + β + 1)
Let ζ = α + β + 1, and ζ−1 is an overdispersion parameter to be estimated. Now, we reparam- eterise the α, β as
α = (1 − e−kT )(ζ − 1), (5.2)
β = e−kT (ζ − 1). (5.3)
5.3.1 Likelihood Function With Beta Model
Given f(·) in Eqn (5.1) is a density function of moult increment I, we can derive a cumulative function F as
F (y) = Pr(I ≤ y) y = Pr ∆ ≤ − L∞ − L y = B − , L∞ − L where B(·) is the cumulative function of ∆.
The relationship between functions f and b can be expressed as 5.3 Modelling Moult Increments via Beta Distribution 99
I f(I) = b − kJk L∞ − L = b(∆) kJk,
where J is the corresponding Jacobian matrix of the transformation with respect to ∆.
∂∆1 ∂∆1 ∂∆1 ··· ∂I1 ∂I2 ∂In ∂∆ ∂∆ ∂∆ 2 2 ··· 2 ∂I ∂I ∂In J = 1 2 , .. . ∂∆n ∂∆n ··· ∂∆n ∂I1 ∂I2 ∂In
1 0 ··· 0 L∞−L1 1 0 ··· 0 L∞−L = 2 . .. . 0 0 0 1 L∞−Ln
Therefore, the Jacobian determinant
1 1 1 1 kJk = ... L∞,i − L1 L∞,i − L2 L∞,i − L3 L∞,i − Ln n Y 1 = . L − L j=1 ∞,i ij
Note that (−1)i+j has positive measurements in all entries because J is in the form of a diagonal matrix. 5.3 Modelling Moult Increments via Beta Distribution 100
Therefore, the likelihood of MI can be written as
Z ( ) Y Y 1 Iij L(θ|I ,L ,L ,T ) = b g(L ) dL , ij ∞,i ij−1 ij L − L L − L ∞,i ∞ i j ∞,i ij−1 ∞,i ij−1
where L∞,i is an individual asymptote with density function g(L∞,i).
− Let l∞ be a population parameter and I/(L∞ − L ) follows a beta distribution with pa- rameters α, β as in Eqns (5.2) and (5.3). Subsequently, we have b{Iij/(l∞,i − Lij−1)} ∼
Beta{(1 − e−kTij )(ζ − 1), e−kTij (ζ − 1)}, the likelihood function can be written as
N Ni Y Y 1 Γ [{1 − exp(−kTij)}(ζ − 1) + exp(−kTij)(ζ − 1)] L(θ|I ,L ,T ) = ij ij−1 ij l − L Γ [{1 − exp(−kT )}(ζ − 1)]Γ {exp(−kT )(ζ − 1)} i=1 j=1 ∞ n ij ij I [{1−exp(−kTij )}(ζ−1)]−1 I {exp(−kTij )(ζ−1)}−1 − 1 − − , l∞ − L l∞ − L
where l∞, k, ζ are the parameters.
5.3.2 Beta-Gamma (BG) Model
Considering the individual variability, we denote L∞ as a random asymptotic length for N individuals. Since L∞ is not known in this case, we need to integrate out the variable in the form of a likelihood function
N ( N ) Y Z Yi 1 L(θ|L ,T ,L ) = b{I /(L − L )}g(L ) dL , ij−1 ij ∞,i L − L ij ∞,i ij−1 ∞,i ∞ i=1 j=1 ∞,i ij−1
−kTij −kTij in which b(·) ∼ Beta{(1 − e )(ζ − 1), e (ζ − 1)} and g(L∞) ∼ Γ (γ, λ), whereas ζ, γ, k and λ are the parameters to be estimated. The growth parameters are estimated by employing the MLE approach. 5.3 Modelling Moult Increments via Beta Distribution 101
With the above methods applied to our case, the likelihood function can be rewritten as
N Z ( Ni Y Y Γ [{1 − exp(−kTij)}(ζ − 1) + exp(−kTij)(ζ − 1)] L(θ|I ,T ,L ) = ij ij ∞,i Γ [{1 − exp(−kT )}(ζ − 1)]Γ {exp(−kT )(ζ − 1)} i=1 j=1 ij ij I [{1−exp(−kTij )}(ζ−1)]−1 I {exp(−kTij )(ζ−1)}−1 − 1 − − l∞ − L l∞ − L γ γ−1 ) λ exp(−λL∞,i)L∞,i 1 × dL∞, Γ (γ) L∞,i − Lij−1
in which E(L∞) = γ/λ is the mean of random L∞.
The negative log-likelihood for N individuals is
l(k, ζ, γ|Li,j−1,Tij,L∞,i) N N X Z Xi n = log exp log[Γ {(1 − e−kTij )(ζ − 1) + e−kTij (ζ − 1)}] i=1 j=1 Iij + {(1 − e−kTij )(ζ − 1) − 1} log L∞,i − Li,j−1 Iij + {e−kTij (ζ − 1) − 1} log 1 − L∞,i − Li,j−1 − log[Γ {(1 − e−kTij )(ζ − 1)}] − log[Γ {e−kTij (ζ − 1)}] o + γ log(λ) − λL∞,i + (γ − 1) log(L∞,i) − log{Γ (γ)} − log(L∞,i − Li,j−1) dL∞.
where k, ζ, γ and λ are parameters to be estimated.
5.3.3 Beta-Lognormal (BL) Model
−kTij −kTij Assuming that b{Iij/(L∞,i − Lij−1)} ∼ Beta{(1 − e )(ζ − 1), e (ζ − 1)} and g(L∞,i) ∼ ln N (µ, σ2), we have 5.4 Parameter Estimation via Numerical Integration 102
N Z Ni Y Y Γ [{1 − exp(−kTij)}(ζ − 1) + exp(−kTij)(ζ − 1)] L(∆ |L ,T ,L ) = ij i,j−1 ij ∞,i Γ [{1 − exp(−kT )}(ζ − 1)]Γ {exp(−kT )(ζ − 1)} i=1 j=1 ij ij I [{1−exp(−kTij )}(ζ−1)]−1 I {exp(−kTij )(ζ−1)}−1 − 1 − − l∞ − L l∞ − L 1 −A 1 × √ e dL∞, L∞,i 2πσ L∞,i − Lij−1
2 2 2 where A = {log(L∞,i) − µ} /2σ , k, ζ, µ and σ are parameters. If L∞ ∼ ln N (µ, σ ), then 2 E(L∞) = exp(µ + σ /2).
Hence, the mean of moult increments I can be expressed as
(L − L−)(1 − e−kT )(ζ − 1) E(I) = ∞ , (5.4) (1 − e−kT )(ζ − 1) + e−kT (ζ − 1) and the variance is
(L − L−)(1 − e−kT )(ζ − 1)e−kT (ζ − 1) Var(I) = ∞ , (5.5) {(1 − e−kT )(ζ − 1) + e−kT (ζ − 1)}2{(1 − e−kT )(ζ − 1) + e−kT (ζ − 1) + 1}
To wrap up the aforementioned problems and solution, a diagram in Figure 5.2 summarises the modification of modelling the growth parameters in crustaceans.
5.4 Parameter Estimation via Numerical Integration
The log-likelihood function in Eqn (5.1), however, cannot be solved analytically. The work pre- sented herein addresses the problem of evaluating an intractable function analytically. One of the parameter estimation methods is the application of the aforementioned estimating function without considering any likelihood function. Alternatively, we can use the numerical integra- tion approach to determine the numerical value through approximation to a definite integral 5.4 Parameter Estimation via Numerical Integration 103
Figure 5.2: Summary of modelling individuals moult increments. over an interval [a,b] such that
Z b f(I|l∞) p(l∞) dl∞, a
where L∞ follows a lognormal or a gamma distribution. Given that L∞ is a random variable, the integrand f(I|l∞), mathematically, is not available to compute the antiderivative.
Using the R package, the corresponding log-likelihood function is
ff<-function(Linf,lambda,k,alpha,beta,Ii,Ti,Li) { gLinf<-0 ni<-length(Ii) for(j in 1:ni) { increment <- lambda*(Linf-Li[j])*(1-exp(-k*Ti[j])) log.likelihood <- aa*log(lambda)+(aa-1)*log(Ii[j])-lambda*Xi[j]- 5.4 Parameter Estimation via Numerical Integration 104
lgamma(increment)+ alpha*log(beta)+(alpha-1)*log(Linf)- (beta*Linf)-lgamma(alpha) gLinf<-gLinf+ log.likelihood } return(gLinf) }
Rewriting a logarithm in an exponential form can induce zeros and infinite numbers due to extreme values in the underlying function. The findings may not be valid for estimating the growth parameters; therefore, the extreme values will be scaled to [0, 1] through a division by the largest value of the resulting function, M. A modified model is formed and written as
N Z " ni # X X − lI (θ) = log exp log g Lij,Tij,Iij,L∞,i; θ dL∞, i=1 j=1 N Z Pni − X nexp[ j=1 log{g(Lij,Tij,Iij,L∞,i; θ)}] o = log × exp(M ) dL , exp(M ) i ∞ i=1 i N Z ni ! N X h X − i X = log exp log{g(Lij,Tij,Iij,L∞,i; θ)} − Mi dL∞ + Mi, i=1 j=1 i=1
max n − o where Mi = L∞ ∈ Ω log g(Lij,Tij,Iij,L∞,i; θ) .
Showing in R code:
gg<- function(Linf,lambda,k,alpha,beta,Ii,Ti,Li,MinLinf,MaxLinf) { res <- ff(Linf,lambda,k,alpha,beta,Ii,Ti,Li) Mi <- optimize(ff,c(MinLinf,MaxLinf),lambda=lambda,k=k,alpha=alpha, beta=beta,Ii=Ii,Ti=Ti,Li=Li,maximum = T)$objective return(exp(res-Mi)) } LL <- function(theta,dat,MinLinf,MaxLinf) { lambda<-theta[1] 5.5 Convolution of MI and IP for Crustacean Growth Models 105
k<-theta[2] alpha<-theta[3] beta<-theta[4] all.LL <- 0 for(iid in unique(dat$LOBSTER)) { dati<-dat[dat$LOBSTER==iid,] Ii<- dati$INC Li<-dati$PL Ti<-dati$INT/365.25 Mi<- optimize(ff,c(MinLinf,MaxLinf),lambda=lambda,k=k,alpha=alpha, beta=beta,Ii=Ii,Ti=Ti,Li=Li,maximum = T)$objective res.int <- integrate(gg, lambda=lambda,k=k,alpha=alpha,beta=beta, Ii=Ii,Ti=Ti,Li=Li,MinLinf = MinLinf, MaxLinf =MaxLinf, lower = MinLinf, upper = MaxLinf) res.int<- log(res.int$value) all.LL <- all.LL + res.int + Mi } print(c(lambda,k,alpha,beta)) return(-all.LL) }
5.5 Convolution of MI and IP for Crustacean Growth Models
5.5.1 Introduction
Despite many studies using the VBGF to quantify crustacean growth, however, they assume continual growth for all individuals, which is not true for individual growth in crustaceans. 5.5 Convolution of MI and IP for Crustacean Growth Models 106
In reality, crustacean growth exhibits a discontinuity in the moulting process. Although the VBGF can be used to describe the population growth of crustaceans, it is unlikely to define individual growth dynamics (Chang et al., 2012). Hence, it is more appropriate to take into account an association of MI as well as IP in a stepwise growth function.
The following discussion is a convolution of both stochastic processes based on the Markov property and Monte Carlo technique in modelling the discrete process in crustacean growth paths.
5.5.2 Likelihood Approach
Once we obtained the information from §4.2 and §4.1, the resulting parameter estimates could then be used to quantify the mean of the discontinuous trajectories from all individuals. To do so, we developed a joint density function that enables both processes to convolute using an ap- propriate approach, such as the Monte-Carlo simulation. Through this, a stepwise function can be characterised individually in such a way that it can be known how frequently an individual will moult and how large the size of increment would be once the individual moulted.
ni Suppose an individual i consists of ni repeated measures {Li,j−1,Tij,Iij}j=1, where Li,j−1 is the jth premoult length, Tij is the jth intermoulting time, and Iij is the jth increment length. Since each individual is measured at different times, a general density function is required to describe a sample of different lobsters. We define f(Li0) as an initial length function for an animal. We assume g(·) is a function of IP and f(·) is a MI function, and both are conditioned on premoult length and IP. For example, if lobster i has moulted three times, we can derive the joint density function of IP and MI as
f{(Li2,Ti3,Ii3), (Li1,Ti2,Ii2), (Li0,Ti1,Ii1)}
= f(Li2,Ti3,Ii3,Ti2,Ii2|Li0,Ti1,Ii1)f(Li0,Ti1,Ii1)
= f(Li2,Ti3,Ii3,Ii2|Li0,Ti1,Ii1,Ti2)g(Ti2|Li0,Ti1,Ii1)f(Ii1|Li0,Ti1)g(Ti1|Li0)f(Li0)
= f(Li2,Ti3,Ii3|Li0,Ti1,Ii1,Ti2,Ii2)f(Ii2|Li0,Ti1,Ii1,Ti2)g(Ti2|Li0,Ti1,Ii1)
× f(Ii1|Li0,Ti1)g(Ti1|Li0)f(Li0). (5.6) 5.5 Convolution of MI and IP for Crustacean Growth Models 107
Because Li1 = Li0 + Ii1, and Li2 = Li1 + Ii2, the function (5.6) can be written as f{(Li2,Ti3,Ii3), (Li1,Ti2,Ii2), (Li0,Ti1,Ii1)}
= f(Li2,Ti3,Ii3|Li1,Ti2,Ii2)g(Ti2|Li1,Ti1)f(Ii2|Li1,Ti2)f(Ii1|Li0,Ti1)g(Ti1|Li0)f(Li0)
= g(Ti2|Li1,Ti1)f(Ii2|Li1,Ti2)g(Ti3,Ii3|Li2,Ti2)f(Ii1|Li0,Ti1)g(Ti1|Li0)f(Li0)
= f(Ii3|Li2,Ti2,Ti3)g(Ti3|Li2,Ti2)g(Ti2|Li1,Ti1)f(Ii2|Li1,Ti2)f(Ii1|Li0,Ti1)g(Ti1|Li0)f(Li0)
= g(Ti3|Li2,Ti2)g(Ti2|Li1,Ti1)g(Ti1|Li0,Ti0)f(Ii3|Li2,Ti3)f(Ii2|Li1,Ti2)f(Ii1|Li0,Ti1)f(Li0).
In general, the joint density function for the ith individual can be written as
n n Yi Yi {g(Tij|Lij−1,Tij−1)} {f(Iij|Lij−1,Tij)}f(Li0). j=1 j=1
Suppose there are N individuals in our study. The likelihood function for all individuals is
N ( n n ) Y Yi Yi L(θ) = g(Tij|Lij−1,Tij−1) f(Iij|Lij−1,Tij)f(Li0) i=1 j=1 j=1 N n N n N Y Yi Y Yi Y = g(Tij|Lij−1,Tij−1) f(Iij|Lij−1,Tij) f(Li0), i=1 j=1 i=1 j=1 i=1 where the first term is for the IP function, and the second term is for the MI. Parameters in g(·) and f(·) can be estimated by maximising the two parts separately. Therefore, the first and second terms are asymptotically independent, given the premoult and IP. Chapter 6
Growth Estimation from Tag and Recapture Studies
Crustaceans are commercially important, with captured species of crustaceans, such as mud crabs, providing lucrative incomes for fishing communities (Walton et al., 2006). In many major consumer countries in Southeast Asia, crustacean species, especially mud crabs, are in high demand in China, Taiwan, Singapore and Malaysia as one of their food sources. For this reason, the supply has soared and eventually caused overfishing in certain regions (Kosuge, 2001). The balance between supply and demand is crucial for controlling the density in aquaculture areas. To this end, studying the growth of aquaculture is important for understanding growth sustainability and productivity in stock assessment. However, the estimation of yield per recruit is noticeably difficult due to the unknown ages of crustaceans. In this chapter, we considered estimating the growth of individual crustaceans on the basis of data collected through tag- recapture experiments.
Researchers over the past few decades assumed that the time remains between the last moult to the next one was assumed to be uniformly distributed. However, crustaceans with seasonal modulation are prone to having different intermoult periods, especially during winter. In view of this, crustaceans were assumed to moult independently and not at the same time. With this being the case, randomly selected individual crustaceans can be found in any stage of the 109 moult cycle. Hence, the time from capture to the next moult, Y (Hoenig & Restrepo, 1989), is
Y = U, T where the intermoult period is T , and the fraction of the intermoult period is uniformly dis- tributed as Y ∼ U(0,T ). Suppose there is neither mortality nor migration from the population, while U is distributed independently of T and of other lobsters in the moulting system.
To determine if an individual crustacean is moulted or not, we illustrate the moulting process as shown in Figure 6.1. In denoting ρ as the time at liberty, we claim that a recovered specimen has moulted if an increment in length is found during the recapture time; otherwise, it has not moulted. In other words, say that a specimen is considered to have moulted when Y is shorter than ρ, and otherwise, it is not moulted. In fact, intermoult period T is not known, but Y and ρ are available for moult status determination.
Figure 6.1: Panel A shows that a specimen has moulted, and otherwise is shown for panel B during recapture.
However, biological evidence suggests that organisms with similar lengths have a wide range of intermoult periods that do not follow a uniform random variable. A more realistic lognormal 6.1 Estimation from Single-Recapture Data 110 model is preferable for estimating the distribution of the intermoult periods of asynchronously moulting lobsters (Millar & Hoenig, 1997). The mean of the intermoult period is a function of premoult length L (Restrepo, 1989).
6.1 Estimation from Single-Recapture Data
Information on intermoult periods can be estimated using approaches such as observation of growth in captivity and tag-recapture experiments (Hillis, 1979; Millar & Hoenig, 1997). Crus- taceans raised in captivity can be observed directly through the growth of the moult increments and the intermoult periods of each individual (Kurata, 1962). However, keeping animals in captivity may produce considerable problems if inappropriate conditions exist for individual crustaceans growth in laboratories. Conversely, we do not observe directly the moulting time in tag-recapture experiments. The growth estimates do not describe the growth of the wild population because juveniles in captivity tend to grow more rapidly than do juveniles in the wild (Van Dykhuizen & Mollet, 1992). Furthermore, knowledge on growth between laboratory and field studies is yet to yield conclusive outcomes (Buchholz et al., 1989).
This chapter aims to provide different modelling approaches that have been used to quantify growth and how to incorporate variability in modelling tag-recapture data. Despite the fact that the crustacean growth paths are discontinuous, in general, the growth trend can be characterised by a population growth curve based on candidate continuous growth models.
Data Description for Single-Recapture Data
To facilitate the analysis, we used the tagging data of Little (Florida Department of Natural Resources Marine Research Laboratory, Special Scientific Report No. 31, 1972) to quantify the growth of spiny lobsters (Panulirus argus) from South Florida.
The details of those 68 recaptured spiny lobsters, such as the carapace length at tagging L0, length at recapture L1 and time at liberty (in days), were reported by Little (1972). Little 6.1 Estimation from Single-Recapture Data 111
assumed any of the individuals with length increments of L1 − L0 > 2mm to be moulted; otherwise, they had not moulted.
6.1.1 Estimation via Modelling Intermoult Periods
For the tag-recapture study, we determined the individual moult status through the observa- tion of time at liberty ρi and the carapace length. We observed merely L1 (capture) and L2 (recapture) for each selected individual crustacean. Suppose an animal i = 1, 2, ..., n. If the time from tagging to moulting Y < ρi, the animal has moulted; otherwise, it has not moulted. Thus, this technique produces binary data for each individual. Here, a lognormal distribution
2 is considered in the intermoult period estimation. Suppose log(T ) is distributed as N(µL, σ ): The mean of intermoult period T is referred to by Restrepo (1989) as,
2 E(T ) = exp(µL + σ /2)
= α exp(βL1), (6.1)
and therefore, we have
2 µL = log(α) − σ /2 + βL1,
where α > 0 and β ≥ 0 are constants. There are three parameters to be estimated: α, β and σ.
With a random sample of n observations, let individual i consist of the binary data of time at liberty ρi and remaining time to the next moult yi, whereas mi is an indicator of I(yi < ρi).
When mi = 1, an animal is moulted; otherwise, we denote mi = 0 when yi > ρi. If T is unknown and only mi and ρi are observed, the likelihood can be written as
n Y mi 1−mi f(mi, ρi|L1; θ) = F (ρi; θ) {1 − F (ρi; θ)} , (6.2) i=1 6.1 Estimation from Single-Recapture Data 112
where F is the cumulative distribution since the first capture with length L1, and θ is a set of parameters in cumulative function F .
Assuming Y = TU, where Y is a random fraction of T , Millar and Hoenig (1997) suggested
2 2 that intermoult period T follows a lognormal distribution log(T ) ∼ N(log α − σ /2 + βL1, σ ), and U ∼ U(0, 1). Hence,
H(x; θ) = Pr(− log(Y ) ≤ x) = Pr(− log(T ) − log(U) ≤ x), having the parameters of α, β and σ. Underlying function F will be derived and produce the log-likelihood as below.
2 Let − log(T ) − log(U) = Z + M where Z ∼ N (−µL, σ ) and M ∼ exp(1). Therefore,
H(x) = Pr(Z + M ≤ x) = E{Pr(Z ≤ x − M|M)} 1 Z ∞ Z x−m −(z + µ)2 √ = exp 2 dz exp(−m)dm σ 2π 0 −∞ 2σ 1 Z x −(z + µ)2 Z x−z √ = exp 2 exp(−m)dmdz σ 2π −∞ 2σ 0 1 Z x −(z + µ)2 √ = exp 2 [1 − exp{−(x − z)}]dz σ 2π −∞ 2σ x + µ x + µ = Φ − exp(−x − µ + σ2/2)Φ − σ , σ σ
where Φ(·) is the standard normal distribution function. Based on Eqn (6.2), F (ρi) can be expressed as
F(ρ) = Pr(Y ≤ ρ)
= Pr{− log(Y ) ≥ − log(ρ)} = 1 − H{− log(ρ)} .
Hence, the log-likelihood for (mi, ρi) can be rewritten as 6.1 Estimation from Single-Recapture Data 113
N 2 X di σ di log{f(m , ρ |L ; α, β, σ)} = m log 1 − Φ + exp −d + Φ − σ i i 1 i σ i 2 σ i N 2 X di σ di + (1 − m ) log Φ − exp −d + Φ − σ , i σ i 2 σ i
2 where di = − log(ρi) + log(α) − σ /2 + βL1.
6.1.2 A Maximum Likelihood Approach
In the analysis of tagging data, almost all growth models built in fisheries presume that a fish grows continuously throughout its life. However, the stepwise growth of crustaceans has received less at- tention. A greater research emphasis is needed on how to mathematically describe the individual growth trajectories of crustaceans, specifically for tag-recapture research. Some candidate models will be discussed in the following sections for quantifying growth parameters from tagged animals. The continuous curves produced from the underlying models can be used as the population mean growth of all crustaceans, while each individual crustacean possesses its own discontinuous growth paths.
There have been many attempts to mathematically describe the growth of crustaceans using the VBGF. Nonetheless, age information as in the model is not available for tag-recapture studies. We observed only length at tagging, LG, length at recapture, LR and time at liberty, T . As a result, the Fabens method was constructed to overcome this problem. The growth can be written as
−kT LG − LR = (L∞ − LG)(1 − e ).
However, this method is not appropriate to use when individual variability exists in growth. The Fabens method does not take into consideration individual variability, thus resulting in a biased growth estimation.
Wang et al. (1995) presented an asymptotically unbiased method for estimating both the random variables of maximal length and age at tagging by employing a maximum likelihood approach. In this case, the general form of VBGF is
−k(t−t0) Lt = L∞{1 − e }, 6.1 Estimation from Single-Recapture Data 114
where the parameters of L∞, t and t0 are generalised to random variables, while k remains a constant.
We assume that the length at tagging is
−kA LG = L∞(1 − e ),
whilst the length at recapture is
−k(A+T ) LR = L∞{1 − e }.
−kA As both L∞ and A are unknown, we can express L∞, 1 − e with respect to the observed data of
LG, LR. Subsequently, the likelihood function is
1 f(LG,LR) = −kT g(x)h(LG/x), LR − e LG
where x = {LR − exp(−kT )LG}/{1 − exp(−kT )}, g(·) as well as h(·) are the density of L∞ and 1 − e−kA, respectively. Furthermore, we can express the h(·) in terms of p(·) as
1 −1 h(u) = p log(1 − u) , k(1 − u) k
where p(·) refers to the density of A.
In addition to the individual variability, it is more realistic to incorporate the measurement error into a growth model. For the observed length of
LˆI = LI + εI ,
for I = G, R is an indicator of tagging or recapture, while LI is the expected length, which is condi- 2 tional on L∞; A, εI is assumed to be normally distributed with mean 0 and variance σ . If we denote
εG = s and εR = v, the likelihood function can be expressed as 6.1 Estimation from Single-Recapture Data 115
ZZ exp(−(s2 + v2)/2σ2) f(Lˆ , Lˆ ) = f(Lˆ − s, Lˆ − v)dsdv. G R 2πσ2 G R
6.1.3 Individual Variability in Growth via an Improved Fabens Ap- proach
The Fabens method yields inconsistent estimates because it assumes that all individual crustaceans have the same population growth parameters. In fact, different individuals have different asymptotic lengths due to their individual variability. However, the Fabens method does not account for individual variability; therefore, a biased estimation will be obtained once any perturbation exists among the individual crustaceans growth.
To this end, Wang (1998b) introduced a modified Fabens method in which terminal length L∞ is considered an individual variability in growth. As an illustration, we assume length at tagging, −kA −k(A+T ) LG = L∞(1 − e ), and length at recapture, LR = L∞{1 − e }. A moult increment of the Fabens method can be expressed as
−kT LR − LG = I = (L∞ − LG)(1 − e ), (6.3)
where T is time at liberty. Age information can be omitted in this context. As noted by the author, the Fabens method applied the least squares method to estimate (k, L∞) by minimising
2 −kT 2 (I − Iˆ) = {I − (l∞ − LG)(1 − e )} . (6.4)
The squared residual above can be defined as the difference between the true value of the depen- dent variable, I, and the value of the dependent variable predicted by the corresponding model, Iˆ, is a minimum. The l∞ refers to the mean of L∞. In general form, we can express the residual as y−f(x, β) = I−Iˆ. For a linear model, f(x, β) = β0+β1x, β0 and β1 are the intercept and slope, respec- tively. Following the regression theory, if LG is an independent variable, the Iˆ is E(I|LG) = f(LG, β), −kT equivalent to E[{(L∞|LG) − LG}(1 − e )], where E(L∞|LG) = L∞. Therefore, the least squares 6.1 Estimation from Single-Recapture Data 116 method is expressed as
−kT 2 {I − (L∞ − LG)(1 − e )} . (6.5)
Comparing to Eqn (6.4), l∞ in the model differs from L∞ in Eqn (6.5), resulting in a biased estimation, as the Fabens method does not account for individual variability. As a result, the conditional least squares method is approximated through the first-order approximation of
L∞ − E(L∞) = β{LG − E(LG)}
L∞ = l∞ + β{LG − E(LG)}.
Hence, the regression model can be rewritten as
−kT I = [l∞ + β{LG − E(LG)} − LG](1 − e ) + ε, (6.6)
−kT = {l∞ − β E(LG) + βLG − LG}(1 − e ) + ε,
−kT = [{l∞ − β E(LG)} + (β − 1) LG](1 − e ) + ε,
where ε is a random error in the model, while l∞, k and β are parameters to be estimated.
The expected moult increments based on the model above is
−kT E(I|LG) = (b1 + b2 LG)(1 − e ),
where b1 and b2 are estimated parameters with respect to l∞, k and β.
Referring to Eqn (6.6), β − 1 = b2; thus, β = b2 + 1. We can simplify l∞ in the form of
l∞ − β E(LG) = b1 ,
l∞ = β E(LG) + b1 ,
= (b2 + 1) E(LG) + b1. 6.2 Estimation from Multiple-Recapture Data 117
Therefore, the l∞ = b1 + (b2 + 1) E(LG). A significant difference between Eqns (6.3) and (6.6) can be identified when the tagged animals of a similar age but different L∞ yield a linear regression based −ka −kA on the modified model in Eqn (6.6). For L∞ = β2 LG, β2 = 1/(1 − e ) since LG = L∞(1 − e ).
Hence, moult increment I = b2 LG increases by β2 when LG increases, whereas the moult increment of the Fabens method decreases with length at tagging LG. Apparently, the modified model is equivalent to the Fabens method when there is no significant individual variability β in growth.
6.2 Estimation from Multiple-Recapture Data
The tag-recapture technique was initially used for population censuses in fishery studies before it was developed for other settings. Most of the growth parameter estimation of animals has been done using the tag-recapture approach (Corgos et al., 2007). For tag-recapture data, the estimation of intermoult periods is not straightforward. Normally, intermoult periods can be realised by considering the moulting probability.
The moulting probability is defined as the proportion of an individual crustacean moulted during a given period. In tag recapture, one of the ways of observing if an animal is moulted or not moulted is through the length measurement between capture and recapture. An animal is considered to have moulted when the length at recapture is longer than its length at tagging time, or it is not moulted when the length at recapture is the same as its length at tagging time. Hence, only a single recapture was observed. This assumption is not necessarily true because the number of moults could have happened more than once (e.g., three times) between the first capture and recapture.
Additionally, tag-recapture studies have drawbacks, where the sequential moults, perhaps, are not detected during the experiments (Kurata, 1962). Moreover, there may be measurement errors of length at capture and recapture, often resulting in a misclassification of the moult status. Consequently, the growth parameters may have been underestimated or overestimated.
In view of this, a multiple-recapture study is a more realistic mechanism, as several moulting times might occur within the time at liberty. This idea is imperative for the validity of the parameter estimation. For multiple-recapture data, the application of the EM algorithm can be used to determine the latent moulting times between successive recapture times. 6.2 Estimation from Multiple-Recapture Data 118
We can modify the aforementioned likelihood by taking into account the number of moults that happened since the last moult instead of considering merely the single binary data (moulted or not moulted). We propose a generalisation to the multiple-recapture approach. In most of the cases, the multiple moults between capture and recapture can only be inferred. Referring to Table 6.2, suppose there are k times of recapture for N individuals. For each observation t1, t2,...tk, the moulting times may differ and are unknown. Although moulting may be a complex process, we can still use latent variables and EM algorithms to estimate the intermoult periods.
As previously mentioned, there are variables that cannot be measured directly from experiments. However, given observable variables, such as changes in length and time at liberty, the values of the latent variables (moulting times) can be inferred from measurements of the observable variables. We also can consider the EM algorithm as one of the ways of estimating the probabilities of moulting times from recaptures.
The moulting times may differ from one capture to another. Hence, the probability of moulting times would not be the same for each capture. We will look into addressing this issue by investigating previous literature, such as the multiple-moult model proposed by Hoenig and Restrepo (1989). For multiple recapture data, it can be very complicated and challenging to form a likelihood function for the moult increments, as we do not know the exact number of moults that occurred from the last moulting time to the next moulting time. However, we overcome this problem by considering the aforementioned approaches for growth parameter estimation.
Table 6.1: The binary moulting status for each of the N individuals with k recaptures.
Moult status (0=not moulted, 1=moulted at least once) Lobster Recapture times
t1 t2 t3 ... tk 1 1 1 0 . . . 1 2 0 1 1 . . . 0 3 1 1 1 . . . 1 ...... N 1 0 0 . . . 0 6.2 Estimation from Multiple-Recapture Data 119
In tag recapture, the misclassification of the moult status can happen if there is any improper measure- ment of the size increment of a recaptured crustacean. We could modify the aforementioned likelihood by taking into account the real number of moults instead of binary data (moult or not moult). That is, the likelihood from time at liberty t with size increment ∆s and size of moulting L is
n n n Y0 Y1 Y2 f0(∆s|Li, ti)P0(Li, ti) + f1(∆s|Lj, tj)P1(Lj, tj) + f2(∆s|Lk, tk)P2(Lk, tk) i=1 j=1 k=1 n n n Y3 Y4 Y5 + f3(∆s|Ll, tl)P3(Ll, tl) + f4(∆s|Lm, tm)P4(Lm, tm) + f5(∆s|Ln, tn)P5(Ln, tn) , l=1 m=1 n=1
where the probability function Px(L, t)(x = 0, 1, ..., 5) is the type of record, the initial intermoult pe- riod ∆t0 and the subsequent intermoult period ∆t1 as given in Table 6.2 (Hoenig & Restrepo, 1989).
Type of x Number of moults t vs ∆t Probability, Px
0 0 t < ∆t0 1 − t/∆t0
1 1 t < ∆t0 t/∆t0
2 1 ∆t0 ≤ t < ∆t1 1 t − ∆t1 3 1 ∆t1 ≤ t < ∆t0 + ∆t1 1 − ∆t0
t − ∆t1 4 2 ∆t1 ≤ t < ∆t0 + ∆t1 ∆t0
5 2+ ∆t0 + ∆t1 ≤ t 1
Table 6.2: Table of six types of recapture records in the form of probability function, Px(L, t).
Due to the complexity of determining age directly in crustaceans, growth estimates of crustaceans are normally derived from moult-increment realisation from tag-recapture data under wild or captivity conditions.
6.2.1 Generalised Estimating Equations Approach
In this section, we explore the use of the GEE method for estimating growth parameters from multiple recapture data. GEE is known to be widely used in medical studies; however, it is plausible to use 6.2 Estimation from Multiple-Recapture Data 120 them for growth parameter estimation in fishery research if the specification of the mean and variance components are available in the proposed model.
Suppose the growth model is in the form of VBGF, where observed length LG from crustacean i at time tj is
−ktij Lij = LG + (L∞ − LG)(1 − e ) + εij,
−kt in which initial time ti0 = 0, term (L∞ − LG)(1 − e ij ) is the moult increment and εij is a measure-
2 −ktij ment error such that εe ∼ N (0, σe ). The expected LG = LG + (L∞ − LG)(1 − e ) can be written in the form of a matrix as
Li = Xiβ + εi, 1 0 −kti2 −kti2 e 1 − e LG = + εi . . . L∞ −kt −kt e ini 1 − e ini
The variables written in bold indicate a vector notation.
Furthermore, the mean and covariance matrices of Li can be expressed as
LG E(Li) = Xi , L∞
T Ψi = XiVθXi + Ωi ,
where Vθ is the covariance of β, having a matrix of
2 σG ρσGσ∞ Vθ = 2 ρσGσ∞ σ∞ ,
where ρ denotes the correlation between two variables, and Ωi belongs to the covariance matrix of i. 6.2 Estimation from Multiple-Recapture Data 121
For parameters of interest β = (l∞, k), we consider optimal estimating function Υ (·) for the aforemen- tioned linear regression model when covariance Ψ is known. We have
X T −1 Υ (β) = Λi Ψi (Li − E(Li)), i wherein
0 0 −kti2 −kti2 1 − e ti2e Λi = . . . −ktini 1 − e−ktini tini e
It is important to note that the underlying estimating function is robust even if Ψi is misclassified. In other words, Ψi is an arbitrary matrix in this case. Chapter 7
Data Analysis of Lobster Species from Different Environments
7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments
The ornate rock lobster Panulirus Ornatus (Fabricius) in Torres Strait, northeastern Australia, pro- duces approximately 280 tonnes of lobsters a year, making its population dynamics complicated (Den- nis, Skewes & Pitcher, 1997).
Ornate rock lobsters have been fished by the traditional inhabitants of Torres Strait, probably for several centuries, before commercial fishing began in the late 1960s. The Torres Strait Tropical Rock Lobster commercial fishing season begins from 1 December to 30 September (the following year), each year. Independent Torres Strait Islander involvement in the fishery has also increased. Torres Strait now has 300-500 dinghies, and lobster fishing has become a major source of income (Pitcher & Bishop, 1995). Most commercial fishing for lobster occurs in Torres Strait with some along the far north-east coast of Queensland. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 123
The dataset were collected in Cairns, Australia, from 1995 to 1999, in which the length of lobster carapace, ranged from 7.9 to 158.3 mm. The lobsters were kept in twelve 108-L (60 × 60 × 30 cm deep) aquaria supplied with filtered sea water at 4.5 L /hour. Carapace lengths of the moulted exuviae were measured to minimize stress on the lobsters. Moulted exuviae of individual lobsters were easily identified by distinctive patterns on the rostral horns. The water temperature ranged from 26-30 degrees while the population density varied from 1 to 10. We consider that the first measurements (carapace length, temperature, density and sex) of each individual were taken on January 1, 1995. Lobster growth can be observed in terms of moult interval and moult increment in general.
A total of 75 lobsters, made up of 39 females and 36 males, were reared in tanks over a time-span of 4 years. Table 7.1 exhibits a summary of how many times the 75 male and female tank lobsters molted during the entire experiment. A large number moulted between 1 and 6 times while minority moulted up to a maximum of 15 or 16 times. From year 1995 to 1999, lengths increased monotonically for females and males in step function Figure 7.1. Some lobsters moulted once or twice while a small amount moulted several times throughout the study. All of the lobsters show asynchronous moulting times as their initial points vary from one to another. Certain lobsters grew more rapidly than the others. Overall, the carapace length of spiny rock lobsters can reach to approximately 160mm of maximal length. Subsequently, several plots may describe the growing trend of the data in the following parts to determine the linearity of the growth variables. In the latter part, we will apply stochastic models to study the growth pattern of crustacean species.
Number of Moulting times individuals 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Males 9595450010000001 Females 6528430311100010
Table 7.1: The number of individuals with different moulting times. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 124
Figure 7.1: Lobster growth data over years
Initially, a difference in length up to 15 mm occurred substantially when the carapace length reached approximately 50 mm, but the number of lobsters decreased when the carapace length grew from 100 mm to 150 mm, as shown in Figure 7.2. Additionally, the duration of growth for the carapace length up to 150 mm, mainly occurred by the time the lobsters were two months old, while a few of the lobsters meet this growth around 7-9 months old. Therefore, the resulting plots indicate that a majority of the lobsters grew rapidly in the first year, especially for carapace lengths around 50 mm. Nevertheless, the growth of lobsters increased monotonically between 1995-1999. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 125
Figure 7.2: Distribution of carapace length, CL and growth mode
Many lobsters moulted in high frequency during the first three months. As the carapace length grew larger, the animals were more likely to spend more time moulting as shown in Figure 7.3. For data visualization, a smoothing approach is constructed that aims to capture important patterns in the data. In females, the intermoult period tends to increase with size, and this growth levels out after carapace lengths reach 130 mm. In this case, towards the end of the study, there were insufficient female samples to show growth trends. Males are relatively slower than females in reaching the asymptotic length, as they take more than 200 days to reach 140 mm.
While the carapace length grows larger with time, eventually the lobster will reach an asymptotic length where growth idles (Figure 7.4). The larger a lobster’s carapace length, the longer time it takes to moult. Hence, the moult increment increases linearly with increasing intermoult periods (Figure 7.5). 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 126
Figure 7.3: Intermoult periods against carapace length
Figure 7.4: Carapace lengths against time in days 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 127
Figure 7.5: Increments against time in days
Figure 7.6 shows the greater the carapace length, the slower the moult increment process will be. Apparently, male lobsters have a bigger increment size compared to females.
Figure 7.6: Moult increments versus carapace length 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 128
From Figure 7.7, the highest density intermoult periods took place at day 50. The intermoult period for males was slightly ahead of females, ranging between 50 and 150 days. This circumstances occurs because when female lobsters reach maturity, a compromise between reproduction and growth causes a delay (Nelson, 1991). Subsequently, a longer period is required for males to moult — within 150-300 days. However, the intermoult periods for both females and males were almost equivalent thereafter. Figure 7.8 shows that the density of moult increment for female lobsters was twice as much compared to the males by an increment of 5 mm. Males manage to overtake the females in relation to size increments of 10 mm and 30 mm.
Intermoult period distribution
Female Male Density 0.000 0.002 0.004 0.006 0.008 0.010 0 100 200 300 400 500
Time (days)
Figure 7.7: Intermoult period distribution of reared lobsters 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 129
Moult Increment Distribution
Female Male Density 0.00 0.05 0.10 0.15
0 10 20 30 40
Length Increment (mm)
Figure 7.8: Moult increment distribution of reared lobsters
7.1.1 Modelling Moult Increments With Nonlinear Mixed Effects Models
Referring to the model in §4.1.1, the parameters estimates are shown in Table 7.2 below.
The covariate SEX : T is tested to be non-significant. As shown in Table 7.2, we reduced the model and yielded the mean of the fixed effects (random effects) as follows:
The non-significant value of p for the likelihood ratio with smaller values of AIC and BIC in the simpler model would be preferred. The parameter estimates of the fixed effects for both REML and
ML are almost the same. For REML estimates, the standard deviation of random effectσ ˆL is 0.07761, while standard deviation of sampling errorσ ˆ is 5.413734. ML estimates show thatσ ˆL is 0.07368 and σˆ is 5.416548. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 130
Value approx. S.E. p-value L 0.442 0.047 < 1e−5 L2 -0.004 0.001 < 1e−5 D -0.100 0.184 < 1e−5 T -0.075 0.077 0.336 SEX 11.012 5.080 0.035 SEX : L2 0.001 3.38×10−4 0.030 SEX : T -0.364 0.184 0.051
Table 7.2: Table of parameter estimates using mixed-effects model
Value approx. S.E. p-value L 0.425 0.028 < 1e−4 L2 -0.004 0.001 < 1e−4 D -1.049 0.180 < 1e−4 SEX : L2 0.001 3.01×10−4 0.007
Model df AIC BIC Test p-value 1 9 1905.473 1938.594 2 6 1904.756 1926.837 1 vs 2 0.152
Figure 7.9 shows the observed values versus the fitted values, which suggests that the linear mixed effects model explains the lobster growth curves. The observed moult increments are mainly scattered around the fitted values, except for a number of outlying observations.
Figure 7.10 clearly shows that randomness in the lobster moult increment is greater among males than among females. Nevertheless, the variability within this group is more likely to be constant except as shown in the outliers for female lobsters, 29, 32, 33, 39, 60, and 124, and as shown in the male lobsters 28, 30, 40, 42, 48, 54, 56, 61, and 123. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 131
Figure 7.9: Observed versus fitted values plot for moult increment
For the sampling errors, we can assess the assumption of normality with the quantile of standard normal by the qqnorm plot of the within-group residuals. Figure 7.11 produces normal plots of fixed effects and random effects respectively. The normal plots indicate that the within-group errors distribute with a heavy tail under normality, but they are also symmetric around the origin.
The outlying points for the fixed effects were much more than in the random effects component. The normal plot in the random effect component has three departures from normality. The dataset shows a considerable deviation from normality. Overall, the assumption of normality seems to be not applicable in this case. However, the normality may not be the most crucial part of the assumptions that we made. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 132
Figure 7.10: Scatter plots of standardized residuals versus fitted values by sex-specified lobsters
Figure 7.11: Quantile for standard normal 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 133
Figure 7.12 displays both female and male growth increment predictions according to different densi- ties. Growth increments tend to be greater when lower population density exists.
Figure 7.12: Predicted moult increment based on cluster (density)
7.1.2 Modelling Moult Increments With Subordinator-based Mod- els
Time Transformation (TT) Models
To better understand growth of crustacean species, individual variability as well as environmental perturbation were incorporated into the aforementioned model in §4.1.2. The parameter estimates are sex-specified, showing k and L∞ in relation to different set of models. From Table 7.3, when f follows a gamma density function, the growth parameters of random L∞ are comparatively more consistent than the fixed L∞. When f follows a gamma density function, L∞ has the largest estimates when the asymptotic length is constant, whereas when the growth parameters of random L∞ and f follow an inverse Gaussian, they are almost the same in both models. The growth rate k varied in the range of 0.2-0.4 mm/year. Overall, males have greater L∞ than female lobsters. A realisation of the moult increment lengths of various models is presented from Figure 7.13 to Figure 7.21. To check individual lobster central tendency and variability, a 95% confidence interval boundary was added to 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 134 the growth curve to compare among the candidate models. Apparently, the beta process exhibits the least deviation from the moult increments similar to Figure 7.20 and Figure 7.21.
Female Male
Model f(·) l∞ k l∞ k l∞ 1 Gamma Fixed 0.30 175.34 0.21 258.65 2 Inverse Gaussian Fixed 0.27 189.95 0.32 196.27 Female Male
Model f(·) L∞ k E(L∞) k E(L∞) 3 (GG) Gamma Gamma 0.33 167.27 0.39 168.08 4 (GL) Gamma Log normal 0.25 169.98 0.29 171.95 5 (IGG) Inverse Gaussian Gamma 0.22 161.43 0.25 165.21 6 (IL) Inverse Gaussian Log normal 0.22 171.35 0.25 172.13
Table 7.3: Parameter estimates for different processes in male and female lobsters
Figure 7.13 illustrates moult increments in females and male lobsters with +/- 2 standard-deviation limit. For females, the growth trajectories show size increments below 10 mm respectively. Neverthe- less, males possess higher increment sizes rising to a peak of 15 mm and slowing down after 80 mm carapace length. This implies that males have larger moult increments than females, rendering males to have a greater asymptotic length compared to females.
Additionally, one may consider the inverse Gaussian process to describe the MI as in §4.1.2. There is an intersection point at 130mm carapace length, implying that both females and males have a similar growth rate by the end of the study. Additionally, variance of inverse Gaussian is comparatively greater than variance in the gamma process. Since there is no large discrepancy in both features, either one is flexible enough to be used for modelling growth in this dataset. A model selection can be determined to identify which of the model is best fitted to our dataset.
The moult increment estimation above deals with a fixed L∞. We assume L∞ to be a variable following either gamma or lognormal function (prior distribution), and we compare the posterior distribution among the different models with the gamma as well as the inverse Gaussian process.
Among the models, model 3 seems to well describe the growth data of rock lobsters, while models 5 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 135
Figure 7.13: Estimated mean (Gamma increments and fixed L∞) with error bounds
Figure 7.14: Estimated mean (inverse Gaussian increments and fixed L∞) with error bounds and 6 are slightly lower than average in terms of the moult increment lengths. Overall, males have greater increments and thus a larger L∞ compared to females.
Extended Time Transformation Models
Based on the assumption that moult increments follow a beta distribution in §5.3, the estimated value of overdispersion ζ in females indicates a presence of greater variability compared to males (Table 7.4). The growth rate of females k = 0.303 is slower than males k = 0.385. Hence, the asymptotic length of females L∞ = 175.4 is greater than males L∞ = 174.3. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 136
Figure 7.15: Estimated mean (gamma increments and gamma L∞) with error bounds
Figure 7.16: Estimated mean (gamma increments and lognormal L∞) with error bounds
Once the parameters are estimated, the population mean of moult increments and variance for both males and females can be obtained from Eqns (5.4) and (5.5). For moult increments, the growth length increases initially and then decreases over time. Again, we can notice that the females were relatively lower than male lobsters in terms of the increment length. (Figure 7.13 - Figure 7.21.) 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 137
Figure 7.17: Estimated mean (inverse Gaussian increments and gamma L∞) with error bounds
Figure 7.18: Estimated mean (inverse Gaussian increments and lognormal L∞) with error bounds
A comparison of different models as in §5.2 is featured in Figure 7.22 and Figure 7.23, for both female and male tank lobsters. Note that IL and BG models are superimposed with respect to the estimated length of male lobsters. There are 39 stepwise growth functions generated for females and 36 for males that yield a population mean growth curve in a monotonic increasing pattern. The other six regressions are models embedded with random asymptotic lengths where individual heterogeneity is taken into account in modelling the rock lobsters’ growth.
To evaluate the quality of each model, the AIC will be taken into account. Since we use negative log likelihood, ˆl for the estimation, the value of AIC is defined as 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 138
Female Male
Model f(·) L∞ k ζ E(L∞) k ζ E(L∞) 7 Beta Fixed 0.275 73.538 186.23 0.247 102.025 228.04 8 (BG) Beta Gamma 0.303 68.537 175.39 0.385 62.470 174.32 9 (BL) Beta Lognormal 0.283 75.055 183.87 0.351 70.449 184.77
Table 7.4: Process with beta increments and L∞ (gamma distribution)
Figure 7.19: Estimated mean (beta increments and fixed L∞) with error bounds
AIC = 2ˆl + 2k,
where k is the number of parameters in a model. A set of candidate models is listed below, presenting the AIC values to assess the fitness of the underlying models (Table 7.5).
From the resulting AICs, the model 6 is preferred, as it has the lowest value among all. Additionally, it is consistent with our findings earlier where the beta process produces the least deviation from the underlying model; moreover, the beta process constrains the growth paths from exceeding the terminal length of a lobster. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 139
Figure 7.20: Estimated mean (beta increments and gamma L∞) with error bounds
Figure 7.21: Estimated mean (beta increments and lognormal L∞) with error bounds
Female Male
Model f(·) L∞ AIC AIC 1 Gamma Gamma 1304.69 1588.64 2 Gamma Lognormal 1867.17 2260.85 3 Inverse Gaussian Gamma 408.62 562.73 4 Inverse Gaussian Lognormal 408.62 562.73 5 Beta Gamma 1573.60 1888.55 6 Beta Lognormal 110.44 176.57
Table 7.5: Comparison of AIC measures among different models for both female and male tank lobsters. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 140
Figure 7.22: Comparison between individual growth paths and estimated lengths of females. The population mean growth curve shows in red line.
Dol´eanExponential (DE) Model
Recall that Eqn (4.4), one of the solutions is using an interpolation approach (Russo et al., 2009a) to estimate the growth parameters. A more appropriate statistical approach will be considered in our study to quantify the growth through the MLE technique. We summarise the estimated parameters in Table 7.6, where the variable L∞ follows a gamma function with a mean of E(L∞).
Figure 7.24 displays the population mean curves of carapace length over a period of time based on two different approaches, while the segmented lines indicate growth trajectories. The growth curve estimation using the DE model (blue line) seems to have a relatively larger variability compared to the time transformation, TT model (red line). 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 141
Figure 7.23: Comparison between observed trajectories and estimated lengths of males. The population mean growth curve shows in red line.
Females Males γ 0.01 0.01 α 5.67 7.20 λ 19.69 16.08 δ 28.45 12.44 η 0.15 0.06
Table 7.6: Table of the parameter estimates for Dol´eanexponential (DE) model
Hence, we have k = γ + α log((λ + 1)/λ) = 0.291, E(L∞) = δ/η = 189.67 for females, while k = 0.444 −1 year and E(L∞)=207.33 mm for males.
Figure 7.25 shows a nondecreasing function where constant growths followed by instantaneous growths 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 142
Figure 7.24: Comparing the mean of DE (blue) and TT (red), with observed mean (dashed).
Figure 7.25: Simulated growth trajectories with the subordinator be a gamma process.
form the jumps for twenty simulated data. We can see that the highest deviation occurred approxi- mately around 2 years old. It is obvious that male lobsters exhibit a greater L∞ and smaller individual variability than females. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 143
7.1.3 Modelling Intermoult Periods With Cox Proportional Hazard Model
Besides parametric approach, the individual growth can be characterized using semiparametric ap- proach as discussed in §4.2.1. If the baseline hazard function h0(t) is classified into male and female strata, there are two survival curves that can be characterised by sex-specific traits in Figure 7.26. The plot displays the probability of moulting processes changed over a period of time (in days). From day 100 onwards, female lobsters were generally more likely to moult than males.
Figure 7.26: Survival curves for both sexes
Table 7.7 summarises the estimated parameters together with the p-values. All the parameters show significant results on the intermoult period at a significant level of 0.05. A negative coefficient decreases the risk and so increases the expected intermoult period.
Model checking: Data Diagnostics 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 144
Parameter Estimates S.E. p-value
−16 β1 -0.05 0.01 < 2 × 10 −04 −05 −14 β2 2.59 × 10 5.54 × 10 6.07 × 10
β3 0.07 0.05 0.04 −05 β4 0.08 0.03 7.46 × 10
Table 7.7: Table of the parameter estimates and p-values
The proportional-hazards assumption can be tested based on scaled Schoenfeld residuals by cox.zph function in R (Tableman & Kim, 2003). Overall, the computation for each covariate is associated with the global test for the model as shown in Table 7.8 below.
χ2 p L 0.555 0.456 L2 0.174 0.676 T 0.046 0.830 D 0.219 0.640 GLOBAL 0.938 0.919
Table 7.8: Table of the χ2 and p-values for model 4.5
Note that the tests are sensitive to linearity in the hazard. The global test (for 4 degrees of freedom) is not statistically significant, similarly to the rest of the covariates, as p-values are larger than the significant level of 5%. Hence, the dataset is well fitted by the Cox model in Eqn (4.5).
Figure 7.27 is a plot of scaled Schoenfeld residuals over time. The smooth straight line indicates the fitted values to the plot, while the dash lines represent ±2 standard error around the fit. Any systematic departures from horizontal lines are indicative of non-proportional hazards (Schoenfeld, 1982); therefore, the underlying model is likely to be a true proportional hazards assumption for all the covariates. From the findings, we can conclude that premoult length, water temperature and density are factors in determining the growth of lobsters. We consider a method of dfbeta, which is used to evaluate if there are any outliers and to ensure if there is any influential data existing on the estimated coefficients. Dfbeta is defined by the differences in beta, dfbeta divided by s.e.(βˆs). 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 145
Figure 7.27: Plots of scaled Schoenfeld residuals against time (days) for each covariate
It measures how much impact each observation has on a particular covariate. Figure 7.28 shows the differences between the greatest value and the fitted line for each covariate, implying that none of these observations have extreme values.
Subsequently, we test the linearity by plotting the martingale residuals. Figure 7.29 appears to almost coincide with the whole. In this case, a linear trend is plausible for our assumption of this model. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 146
Figure 7.28: Dfbeta based on index for each predictor
Figure 7.30 shows the observed intermoult duration against the premoult length. The curves show the estimated intermoult periods of both male and female lobsters. At the early stage, females required longer moulting periods than males. However, when the carapace length of both sexes exceeded 100mm, the moulting time remained the same in males and females.
Figure 7.31 displays the growth curves for female and male lobsters. Almost all the observations are linear at each temperature respectively, except those lobsters having only been observed once. As a 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 147
Figure 7.29: Martingale residuals to assess the linearity assumption
result, water temperature is positively correlated to carapace length.
7.1.4 Modelling Intermoult Periods With Generalised Linear Mod- els (GLM)
Using the model from §4.2.2, the underlying growth parameters were estimated for both lognormal (Table 7.9) and gamma (Table 7.10) distributions through maximum likelihood approach. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 148
Figure 7.30: Estimating intermoult periods against premoult length
α β σ S.E. Female 0.102 0.015 0.637 1.63 × 10−2 Male 0.107 0.0140 0.585 1.62 × 10−2
Table 7.9: Parameter estimates for intermoult periods of lognormal function.
GLM Using Lognormal Distribution
It should be noticed that the estimated intermoult period for both males and females were identical from the beginning up to a carapace length of about 75 mm (Figure 7.32). The deviation of growth rates is caused by the stage of maturity and energy allocation. Females grew at a much slower rate to reach an average maximal length of above 150 mm, while males grew to approximately 140 mm in maximal length. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 149
Figure 7.31: Individual growth curves at each temperature 25 − 30oC
GLM Using Gamma Distribution
β0 β1 γ S.E. Female -1.845 0.005 12.659 0.021 Male -2.265 0.012 35.865 0.013
Table 7.10: Parameter estimates for intermoult periods of gamma function. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 150
Figure 7.32: The expected intermoult periods versus premoult length
The intermoult periods were assumed to follow a gamma distribution. Figure 7.33 exhibits the mean of intermoult periods for both males and females between 45 to 260 days (less than a year).
Female lobsters grew from initial intermoult periods of 48.95 days to 253.07 days with a maximum carapace length of 154.4 mm.
In male lobsters, the intermoult periods began at 53.68 days and increased steadily to the longest carapace length of 137 mm.
Once the carapace length of lobsters grew up to 70 mm, both males and females shared an equivalent growth rate. The beginning of sexual maturity was explicit in females. From carapace length 100 mm onwards, male lobsters overtook females to reach an asymptotic length. The passage time of females indicates that energy was reserved for reproduction. Female lobsters grew slower, but they managed to reach a longer asymptotic length than males.
The intermoult period that follows a gamma function as in Figure 7.33 has a closer estimation in comparison with the realisation of growth from Figure 7.32 between both males and females. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 151
Figure 7.33: The expected intermoult periods versus premoult length
7.1.5 Population Mean Growth Curve via Monte Carlo Simulation
The purpose of this simulation study is to assess the outcome of the proposed model and method with respect to its parameter estimation. All the simulation studies are implemented in R environment. The integration of the two stochastic processes (moult increment and intermoult periods) is as follows:
(1) We simulated 1000 trials or iterations for a subordinator-based model with each sample size of 100 males and females. Individually, we denote L as the premoult lengths and T as the intermoult periods for each lobster. Essentially, for moult increments, two scenarios are constructed: fixed L∞ and random L∞ that follows a lognormal distribution. The initial time T0 is randomly sampled in a −kT range of (10,150) days, while the initial carapace length is presumed to be VBGF, L0 ∼ L∞(1 − e ) given that parameter estimates of k and L∞ are known.
(i) For fixed l∞, we presume the increments follow a beta distribution with (l∞ − L)(shape1 = [1 − exp(−kT )](ζ −1), shape2=exp(−kT )(ζ −1)) given k, ζ and l∞ as parameters to be estimated through a maximum likelihood approach; both parameters in males and females were estimated, respectively.
For females, k = 0.303, ζ = 68.54, l∞ = 175.44, whereas for males, k = 0.385, ζ = 62.47, l∞ = 174.32 , as previously shown in §7.1.2. 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 152
(ii) For random L∞, L∞ follows a gamma distribution. Each specimen has an individual maximal length. We generate 100 realisation of L∞ ∼ Γ (shape = 306.5, scale = 1/1.747). Once obtaining the parameters, the estimated l∞ can be used to describe a population mean of individuals with asymptotic lengths, and we can now sample the process with beta increment as discussed in (i) above.
(2) Since moult increments I and intermoult periods T are inter-correlated, I can be used to generate 2 the intermoult period of T ∼ log N (meanlog=log(α) − σ /2 + βL, sdlog=σ), where L1 = L0 + I and α, σ, β are constants. Growth parameter estimates were obtained from §7.1.4.
(3) By incorporating both components of I and T , the stochastic growth trajectories for individuals can be characterised mathematically.
Eventually, the individual growth features in Figure 7.34 display a monotonically increasing pattern with respect to a fixed and random L∞.
Initially, both females and males seemed to moult in an monotonic increment, but the variability of males became more extensive than females for fixed L∞. Overall, both males and females converged to a common L∞ when time approached infinity.
In comparison to the fixed L∞, a model of random L∞ seems to be more realistic and plausible as the variation is much greater than for fixed L∞. Due to intrinsic variability, each specimen has a different terminal length. Both males and females possess different L∞ when growing indefinitely for the case of random L∞, as shown in Figure 7.34 (bottom panel). 7.1 Reared Ornate Rock Lobsters (Panulirus Ornatus) from Laboratory Experiments 153
Figure 7.34: Growth with fixed L∞ (top), random L∞ (bottom) and the population mean (blue). 7.2 Data Analysis of the Single Recapture Data from Field Experiments 154
7.2 Data Analysis of the Single Recapture Data from Field Experiments
7.2.1 IP Estimation Using Spiny Lobsters (Panulirus Argus)
With respect to §4.2.2, wWe can infer the lognormal intermoult period of individuals following the dataset given by the authors. We considered a tagging dataset from Little (1972), where 68 spiny lobsters (panulirus argus) were used to illustrate the growth paths of tagging animals under the single recapture data scenario. The log-likelihood of the model was optimized using R package, and the 2 maximum likelihood estimate of the lognormal intermoult period is exp(log(α) − σ /2 + βL1). A local maxima was found with respect to (ˆα, β,ˆ σˆ) =(1.003, 0.0323, 0.227), with log-likelihood value of -19.137 (Figure 7.35).
Figure 7.35: Log-likelihood for log(σ) of the lognormal intermoult period model.
The resulting estimated intermoult periods from the local maximum can be performed as Figure 7.36. The relationship between time at liberty and carapace length indicates that the greater sizes attained by the lobsters results in longer time needed for them to moult. 7.2 Data Analysis of the Single Recapture Data from Field Experiments 155 Estimated intermolt periods (years) 0.00 0.04 0.08 0.12 50 60 70 80 90 100 110 carapace length
Figure 7.36: Estimated intermoult periods from the lognormal model.
7.2.2 Simulation Study
Simulated data were generated to test the efficiency of the specified algorithm or approach. The simulation was performed 100 times with each run including 100 lobsters with single recapture. The first and last captures were measured (see Appendix A.1.1).
−kT Firstly, we generated individual growth trajectories by assuming the initial length, L0 ∼ L∞(1−e 0 ), p 2 2 where k = 0.2, L∞ ∼ log N (log(l∞), log(1 + σ∞/l∞)) and where l∞ = 150, while T0 is the initial age that ranging from 1 month to 6 months. Suppose there are seven repeated measurements in each sample. Since a moulting process comprises moult increment, I, and intermoult period, T , −kT we assume that in an individual i at jth measurement, Iij = (L∞ − Lij−1)(1 − e ij ), in which 2 2 Tij ∼ log N (log(α) − σ /2 + βLij−1, σ ) given that α = 0.0028, β = 0.0326 and σ = 0.23. This procedure is iterated 100 times to yield 100 stepwise trajectories of which length at tagging, L1, and length at recapture, L2, will be identified, provided the age at tagging, A1 and age and recapture, A2, is known.
Secondly, we recovered the pairwise data (L1,A1) and (L2,A2) for each sample by assuming that
A1 ∼ U(1, 2)years (juvenile lobsters have high mortality at young age, they are unlikely to be tagged at early phrase) while the time at liberty, ρ ∼ U(0.3, 1.91)years. Knowing that A2 = A1 + ρ and A1, 7.2 Data Analysis of the Single Recapture Data from Field Experiments 156
we can obtain L1 and L2 based on the individual growth trajectories that we produced earlier. Once we retrieved L1 and L2, the single recapture data was molded into the proposed model for parameter estimation.
Each set of simulations is based on different combination of (σ∞, σ). The measurement errors, , in carapace length were assumed to follow N(0, 1.52). The parameter estimates corresponding to lognor- mal intermoult period and deterministic model (σ = 0) are shown in Table 7.11.
Lognormal approach Deterministic approach
σ∞ α β σ α β 2 σ = 0 0 ME 1.218 0.001 0.993 1.778 0.004 SE 0.049 0.003 0.020 0.066 0.009 2 σ = 2.25 0.2 ME 1.202 0.002 0.993 1.837 0.003 SE 0.048 0.003 0.020 0.059 0.009
Table 7.11: GLM with mean estimates (ME) and standard errors (SE) for different σ∞ (The true parameters are (α, β)=(1.1, 0.001))
The lognormal method is more efficient than the deterministic model since it has less standard errors and considered heterogeneity to minimise the biases in the analysis. Comparing measured data and simulated results, the growth pattern provides validation and verification to the proposed model.
Millar and Hoenig (1997) claim that the deterministic model in Eqn (6.1) is responsive to those which have not moulted for a long term period. For this reason, the modelling approach may need to be modified, for example, to consider whether the intermoult period is longer than the time at liberty for non-moulters in the growth model.
7.2.3 Maximum Likelihood Estimation for Nephrops norvegicus
The field experiments of tagged N. norvegicus were conducted in Upper Loch Torridon (west coast of Scotland, Aberdeen) between the year 1986-1989. All the works were implemented at a field station of 7.2 Data Analysis of the Single Recapture Data from Field Experiments 157 the Scottish Office of Agriculture and Fisheries Marine Laboratory. From the tagged population, 136 specimens were obtained from within the bay, while another 213 were tagged from outside the bay. The steps associated with tagging experiments include: sexing the captured population, recording lengths (±0.1mm accuracy) and release date, determining the moult status using a microscopic examination followed by Aiken’s (1973) approach.
There were 349 specimens released into a 20-25m deep mud bay located next to the field station between September 9-18, 1986. Once released into the bay, they remained in the wild for three different durations, ranging from one to three years (1987 up to 1989). For the three separate recapturing phases, tagged specimens were caught by local fisherman who were fishing outside the bay. Recorded information includes the identity of each specimen, carapace length at recovery, moult status and time at liberty. Cysts were stored in Baker’s Formol Calcium and transformed to Leicester for dehydration, which were later used to assess the moult stage based on the thickness of the layers in a cyst.
Using the joint likelihood f(LG,LR) from §6.1.2, we optimised the probability function to yield the parameters as displayed in Table 7.12. Since both asymptotic length and age at tagging are considered 2 to be individual variabilities, we assume L∞ ∼ N(µ, σ ) and A ∼ Γ(α, β). Given the resulting param- eter estimates, we can quantify the rate of growth, k, by maximising the aforementioned likelihood function. In addition, we present the parameter estimates for k and l∞ obtained from MLE (random
L∞ and A) and Fabens (constant L∞) methods (Table 7.13).
µ σ α β k(year−1) E(A) 60.268 9.317 8.645 1.353 0.205 6.388
Table 7.12: MLE approach with parameter estimates from tagged Nephrops norvegicus.
Notice that the parameter µ = 60.268 mm is equivalent to the mean of L∞ while α/β = 6.388 years for the mean of A.
Maximum likelihood Fabens method
k l∞ k l∞ 0.205 60.268 0.167 63.696
Table 7.13: Comparison of parameter estimates using MLE and Fabens methods. 7.2 Data Analysis of the Single Recapture Data from Field Experiments 158
7.2.4 Simulation Study
Two procedures are required to generate a tag-recapture dataset. First, we implemented 1000 simula- tions for producing 100 individual stepwise growth paths with the following distributional assumptions:
p 2 2 −kT0 L∞ ∼ log N (log(l∞), log[1 + (σ∞/l∞)] ), initial length, L0 ∼ L∞(1 − e ) with T0 ranges between − −kT 2 − 1-month and 6-month period, I ∼ Γ(λ(L∞−L )(1−e ), λ) where T ∼ log N (log(a)−σ /2+bL , σ), given that (λ, σ, a, b, k) = (0.7, 0.24, 0.0028, 0.053, 0.2). Two different conditions will be considered where the measurement errors, σ = 0, 1.5, hence we have a set of combination (σ∞, σ)=(0,0) and
(0.2, 1.5). We set (k, l∞) = (0.2, 150) to be the true parameter value. For 100 samples, the measure- ments of I and T will be iterated seven times to reach their asymptotic length, respectively.
Second, we randomly selected an age at tagging, A1, as well as an age at recapture, A2, based on the aforementioned growth curve. Since both A1 and A2 are random variables, considering A1 ∼ U(1, 2) years and ρ ∼ U(100, 700) days results in A2 = A1 + ρ. Knowing the age of individual lobsters allowed us to retrieve information of LG and LR easily. Therefore, we have 100 set of combinations of (A1,LG) and (A2,LR).
0 0 Subsequently, we suppose L∞ ∼ N (µ , σ ) and A ∼ Γ(α, β) and yield the parameter estimates through optimisation. Further, we employ the MLE method to quantify the growth rate k of the simulated tagging data and compare the results with the traditional Fabens method (see Table 7.14).
Maximum likelihood Fabens method
0 0 0 σ∞ µ σ α β k l∞ k l∞ λ 2 σ = 0 0 ME 149.80 285.31 3.54 1.80 0.238 149.80 0.166 149.12 0.236 SE 0.254 0.508 0.096 0.070 0.014 0.254 0.008 0.100 0.024 2 σ = 2.25 0.2 ME 149.95 303.34 2.999 1.52 0.242 149.95 0.168 147.95 0.180 SE 0.277 0.559 0.111 0.075 0.016 0.277 0.012 0.263 0.020
Table 7.14: Maximum likelihood approach versus Fabens method with mean estimates (ME) and
standard errors (SE) for different σ∞. The true parameters are (k, l∞)=(0.2,150).
The outcome in Table 7.14 shows that the maximum likelihood method is more robust compared to 7.2 Data Analysis of the Single Recapture Data from Field Experiments 159 the Fabens approach. When the individual heterogeneity increases, the bias in the estimation will also increase. For computational simplicity, we assumed the values of both s and v were zero and results in SE estimates in the MLE approach were larger than than the SE values using the Fabens method. Overall, the results in Table 7.14 are consistent with the simulated studies.
7.2.5 Unbiased Estimation Using Regression Approach for Panulirus ornatus
The rock lobsters panulirus ornatus were collected between the year of 1980 and 1985 from the Gulf of Carpentaria, Australia. Initially all individuals were captured, tagged, and recorded respectively before releasing to the ocean. Juvenile lobsters do not usually moult in short periods of time (e.g., weekly), thus we only consider the first recapture given that the lobsters had to have been staying in the wild for a period of 30 days and above to eliminate potential biases.
−kT Referring to § 6.1.3, we employ Eqn (6.6) wherein I ∼ Γ[λ{l∞ − βE(LG) + (β − 1)LG}(1 − e ), λ] to quantify growth estimation through optimisation. The parameters were estimated using improved Fabens method and the Fabens methods as performed in Table 7.15.
Improved Fabens method Fabens method
β k l∞ k l∞ 0.562 0.737 135.92 0.441 167.19
Table 7.15: Comparison of parameter estimates using improved Fabens and Fabens methods. 7.2 Data Analysis of the Single Recapture Data from Field Experiments 160
Note that the values of β deviate far from zero in both models, indicating high individual variability in the growth of rock lobsters.
7.2.6 Simulation Study
To examine the performance of the proposed model, simulated tag-recapture data were generated following two basic procedures. Firstly, we implemented 1000 simulations for 100 samples wherein seven iterations were carried out in each sample. We aimed to yield 100 individual discontinuous growth curves with distributional assumptions. Suppose the initial time T0 ranges from a month to 6-month old while the initial length follows a form of the VBGF. The moult increments of an individual, I, follow a gamma distribution while the mean, E(I), follows a VBGF. Therefore, I ∼ − −kT 2 − Γ(λ(L∞ − L )(1 − e ), λ), where T ∼ log N (log(a) − σ /2 + bL ), given that (a, b, λ, σ, k) = −kT (0.0028, 0.053, 0.7, 0.24, 0.2). We have L ∼ L∞(1 − e ) in which k = 0.2, l∞ = 150, L∞ ∼ p 2 2 log N (log(l∞), 1 + σ∞/l∞). Additionally, we considered two different values of measurement er- rors, σ = (0, 1.5), and asymptotic length variability, σ∞ = (0, 0.2).
Secondly, we recovered length at tagging, LG, and length at recapture, LR, on the given age at tagging,
A1 and age at recapture, A2, from the resulting growth curves above. Let A1 ∼ U(1, 2) years and time at liberty, ρ ∼ U(10, 700) days results in A2 =1 +ρ. We retrieved both (LG,LR) once the age information was obtained.
Using the regression approach to obtain an unbiased estimation, the parameter estimates obtained from the proposed method were performed together with the parameter estimates using the primary Fabens approach included in Table 7.16. In this case, the moult increments given in Eqn (6.6) are −kT assumed to follow a gamma distribution Γ{λ[l∞ − βE(LG) + (β − 1)LG][1 − e ], λ}.
Significant growth variabilities occur when the measurement error increases. Furthermore, we notice the standard errors increase due to an increasing variability of L∞ and . Overall, the proposed model is slightly more efficient than the Fabens method due to lower standard errors in growth. 7.3 Data Analysis of the Multiple Recapture Data 161
Improved Fabens Fabens method
σ∞ β k l∞ k l∞ 2 σ = 0 0 ME -0.340 0.168 149.83 0.171 148.11 SE 0.075 0.009 0.140 0.011 0.239 2 σ = 2.25 0.2 ME 0.852 0.171 149.94 0.174 149.82 SE 0.051 0.010 0.142 0.009 0.245
Table 7.16: Unbiased regression approach and Fabens method with mean estimates (ME) and
standard errors (SE) in different (σ∞, σ) (The true parameters are (k, l∞)=(0.2,150))
7.3 Data Analysis of the Multiple Recapture Data
7.3.1 Slipper Lobsters (Scyllarides Latus) from Field Experiments
The specimens, Mediterranean slipper lobsters (Scyllarides latus), were tagged, released and recap- tured in Sicilian waters (Bianchini et al., 2001). They were sex-specified and measured in carapace length, CL (mm), carapace width, CW, as well as antennae length, AL. During the three-year experi- ments (1994-1997), 288 Sicilian slipper lobsters were tagged, and after antibiotic treatment was given, the specimens were allocated in special tanks with food provided for at least a month before they were released, one by one, by SCUBA divers to the three selected sites off the western coasts of Sicily: the artificial barriers in Castellammare del Golfo, a region to west of Mazara del Vallo and five artificial caves on Linosa island (Mediterranean sea). Subsequently, the divers monitored the animals through underwater visual census. The exact location and whereabouts of each specimen was recorded.
A common way to uncover growth in the wild is to monitor the captive specimens in aquaria for an extended period of time first in order to determine their moult increments (Castro et al., 2003). The tagged specimens in captivity were reared in various types of aquariums:
(1) Messina An indoor 1000 liter fiberglass tank with direct flow of water from the sea. The slipper lobsters were monitored several times per day. 7.3 Data Analysis of the Multiple Recapture Data 162
(2) Mazara A glass aquarium situated in semi-darkness with an aeration system and two cylindrical concrete tanks about 1000 liters of sea water.
(3) Biotecno Two 500 liter fiberglass square tanks filled with “filtered” sea water.
Aquaria were checked daily except Sundays. Hence, the moult status was assigned to each animal based on each observed time. Some of the data may have gone missing due to mortality, while others may have been released to the ocean (migration) during the moult cycle.
Among the slipper lobsters in captivity, observation was carried out from two weeks, a few months and more than one year. The increment length and intermoult periods during a moulting process resulted in discontinuous trajectories for each individual.
There was a total of 154 moults in captivity (68 females and 86 males), of which some were measured between three to five times, and there was one male with a number of 8 times measurement. Assuming a common point of origin for all animals, there were 86 male lobsters that showed a stepwise growth pattern over a period of time with both positive and negative jumps (Figure 7.37).
The falling and rising in length for some of the individual lobsters may be due to measurement error or caused by a sudden change of environmental factors. Nevertheless, many of them stopped moulting after the third measurements. A more biologically comprehensive representation of growth can be obtained if moulting times are recorded in a long run. Overall, males do not increase much in CL where the average size is about 100mm. In average, males seem to have a lower terminal growth length compared to females CL. Figure 7.38 displays CL of 68 female lobsters during moulting. Females are slightly larger on average compared to males (approximately 110mm versus 100mm). There is not much growth for females though there were some gains in length during the first few months of the study. The expected CL decreases and begins to level off for the subsequent years, due to insufficient information such as disappearance or mortality.
Male lobsters increase to 3mm for the first three months and gradually decline in size until day 500 (Figure 7.39). The moulting process does not end at this stage, as the males increase in size slightly during the following year. 7.3 Data Analysis of the Multiple Recapture Data 163
Length at time for males Carapace length(mm) Carapace 60 80 100 140 0 100 200 300 400 500 600 700 time (days)
Figure 7.37: Carapace length over time for male tagged lobsters
During the first three months, 76 females showed a mean increment of 5mm CL. However, Figure 7.40 exhibits a downward growth trend for females, and eventually the moulting process ceased after a year. In general, females possess a higher growth rate and a larger asymptotic length compared to males for the three-year study.
Of the 154 captive animals, the number of moults for males and females in captivity are summarised in Table 7.17. From the third measurement, lobster growth declined from 40% to 50% in males and females, respectively. During the fourth observation, the number of moults did not deviate much between males and females (27 moults and 20 moults) until the next moulting period. Ten females and nineteen males moulted four times. A drastic decrease occurred in the subsequent measurements where the number of moults dropped to less than 10% for the entire population. One male managed to moult as much as seven times; however, no moults were detected in females at this stage. Overall, the moulting frequency reduces as the observed time increases. 7.3 Data Analysis of the Multiple Recapture Data 164
Length at time for females Carapace length(mm) Carapace 60 80 100 140 0 100 200 300 400 500 600 700 time (days)
Figure 7.38: Carapace length over time for female tagged lobsters
Number of Moulting times individuals 1 2 3 4 5 6 7 Males 86 35 27 19 6 2 1 Females 68 31 20 10 5 3 0
Table 7.17: The number of individuals with different moulting times.
As Figure 7.37 shows length over intermoult periods by assuming that all individual lobsters initiated from the origin of time 0, Figure 7.41 presents lobster length based on real time measurement. Each individual moulting time can be referred with respect to months and years throughout the study, resulting in the determination of when the specimen will moult and the interval length prior to the onset of the next moult for the population. We notice that most of the male specimens reached their length measurement around August 1995 and ceased moulting after February 1996. The first measurement 7.3 Data Analysis of the Multiple Recapture Data 165
Males Increment(mm) 0 2 4 6 8 10
100 200 300 400 500 600 700 time(days)
Figure 7.39: Increments(mm) over time for male tagged lobsters
began in October 1994, and the final measurement ended before January 1997. Likewise for females, the beginning and ending time of measurement were similar to males (Figure 7.42). Moreover, the majority of females shared a common measurement date around August 1995. However, the total number of moults among female specimens dropped abruptly thereafter. Overall, the growth of both males and females monotonically increased though there was no substantial moult increments for the entire moult cycle.
7.3.2 Generalised Estimating Equation Approach for Scyllarides Latus
Following the description in §7.3.1, a total of eight measurements were carried out in each lobster for a period of three years (1994-1997) in Sicilian water. The information we have for Scyllarides Latus is basically the CL in each recapture and the time at recapture in which they were classified into males 7.3 Data Analysis of the Multiple Recapture Data 166
Females Increment(mm) 0 5 10 15
0 100 200 300 400 500 600 700 time(days)
Figure 7.40: Increments(mm) over time for female tagged lobsters
and females category and the time difference between two consecutive recaptures of an individual lobster is equivalent to the time at liberty.
Through the application of an optimal estimating equation, we obtained parameter estimates as ex- hibited in Table 7.18. Overall, both models shows that the tagged Scyllarides Latus, on average, spent a 5-month period moulting.
GEE Improved Fabens
k l∞ k l∞ 0.449 94.75 0.418 136.22
Table 7.18: Comparison of parameter estimates using GEE and improved Fabens methods. 7.3 Data Analysis of the Multiple Recapture Data 167
7.3.3 Simulated data
To assess the performance of the candidate model, simulated multiple recapture data were used based on two procedures (see Appendix A.1.2). First, we considered a population in which there were 100 samples that were executed with 100 simulations. For each sample, seven iterations were carried out to ensure an individual lobster has grown for a duration of approximately twenty years. In this context, our aim was to produce 100 individual discontinuous growth curves with the following distributional assumptions. Suppose the initial time, T0, ranges from a month to six months while the initial length follows a form of the VBGF. The moult increments of an individual, I, follows a gamma − −kT distribution, while the mean, E(I), follows a VBGF. Therefore, I ∼ Γ(λ(L∞ − L )(1 − e ), λ), where T ∼ log N (log(a)−σ2/2+bL−), given that (a, b, λ, σ, k) = (0.0028, 0.053, 0.7, 0.24, 0.2). We have
− −kT p 2 2 L ∼ L∞(1−e ) in which k = 0.2, l∞ = 150, L∞ ∼ log N (log(l∞), 1 + σ∞/l∞). Additionally, we considered two different values of measurement errors, σ = (0, 1.5) and asymptotic length variability,
σ∞ = (0, 0.2).
Second, we recovered length at tagging, LG, and a series of length at recaptures, LRn , where n is the number of recaptures of a sample, AG is the condition on the given age at tagging, and An is the age at recapture. Let AG ∼ U(1, 2) years and the time at liberty, ρ ∼ U(10, 700) days results in
An = AG + ρ. We randomly sampled the ρ for n different measurements. We retrieved n sets of the pairwise (LG,LRn ) once the age information was obtained.
Once we produced the growth curves and identified the tag-recapture data, we fit a model into the resulting dataset to estimate the growth parameters. To facilitate the analysis, we illustrate a sample growth trajectory and multiple recapture data in Figure 7.43. In plotting the stepwise function, the length at tagging, 55.28mm, was identified provided the age at tag was 1.96 years old. In this case, we assumed that there were five recaptures in the sample which had a CL of 111.37mm when it was 4.62 years old during the fifth recapture.
For ease of calculation, we presumed the covariance Ψi to be an identity matrix based on the GEE approach in §6.2.1. The growth parameters (k, l∞) can be quantified and compared with the parameter estimates from the improved Fabens approach as performed in Table 7.19. In this case, the moult increments given in Eqn (6.6) are assumed to follow a gamma distribution Γ{λ[l∞ − βE(LG) + (β − −kT 1)LG][1 − e ], λ}. 7.3 Data Analysis of the Multiple Recapture Data 168
Apparently, the consistency of the improved Fabens method is slightly better than the GEE approach (smaller standard error). Since we do not take into account the correlation between the variables
(L∞,LG), the resulting parameter estimates may not be at best in terms of getting an unbiased esti- mation.
GEE Improved Fabens
σ∞ k l∞ k l∞ 2 σ = 0 0 ME 0.191 158.12 0.175 145.41 SE 0.004 5.091 0.002 0.692 2 σ = 2.25 0.2 ME 0.187 152.40 0.192 146.86 SE 0.003 4.674 0.001 0.533
Table 7.19: Parameter estimates using GEE and improved Fabens methods with true parameters
(k, l∞= 0.2, 150). 7.3 Data Analysis of the Multiple Recapture Data 169 1997 − Figure 7.41: Carapace lengths for males, 1994 7.3 Data Analysis of the Multiple Recapture Data 170 1997 − Figure 7.42: Carapace lengths for females, 1994 7.3 Data Analysis of the Multiple Recapture Data 171
Figure 7.43: A randomly selected sample with multiple recaptures over time(years). Chapter 8
Conclusion
8.1 Major Findings and Remarks
Lobsters are widely harvested as a delicacy and raised worldwide in the fishery industry. Populations of commercial fishing targets, such as spiny and slipper lobsters, are prone to decline due to increasing global demand. To develop sustainable fisheries, mathematical modelling in aquaculture forms the base for all stock assessment and monitoring of fish farming to prevent overfishing.
The main objective of this thesis is the characterisation of the discontinuous growth paths of crus- taceans by adding stochasticity, such as individual and environmental variability, into the growth models. A set of continuous population growth curves produced from data on tank lobsters through a moult increment model based on various distributional assumptions has been proposed. The results obtained from the observed data analysis were comparable with those obtained from simulated data, indicating a consistency of the modelling approach to the dataset.
Most of these studies are conducted using tag-recapture experiments. Nonetheless, the time between capture and recapture can only be inferred for estimating intermoult periods as the exact moulting time is not known. Furthermore, we generalised single recapture data points to a multiple-recapture dataset through a simulation approach. Alternatively, I implemented another quantitative method for modelling growth of crustacean species: laboratory experiments. The moult increments and intermoult periods can be observed directly from tank data. However, this setting is only suitable for juveniles; 8.1 Major Findings and Remarks 173 the duration of moulting for an adult crustacean can be considerable (more than a year) when it grows larger in size.
Two conditionally independent functions are formed by de-convoluting two intercorrelated stochastic processes: moult increments and intermoult periods. When both probability functions are assumed to be independent of each other, the parameters of both moult increment and intermoult period can be estimated separately. Once the estimated values are obtained, we integrate both main processes through a Monte Carlo technique to form discontinuous growth curves, and subsequently used a preferred model to produce a population growth curve in a continuous fashion.
Russo et al. (2009a) introduced a subordinator in which the L∞ in the model was assumed to be a parameter. However, all individuals may not approach the same terminal length due to the intrinsic variability within a population. A biased estimation would result in errors in modelling individual growth trajectories, and thus the information provided for stock management could be misleading. To this end, a novel contribution in our model is the consideration of both individual heterogeneity and environmental perturbation data based on tank data and tag-recapture data.
To quantify moult increments, we reformulated the subordinator-based model by considering L∞ as a variable, which is more biologically realistic in comparison with the candidate model. A joint likelihood probability was introduced, which considers individual asymptotic length to eliminate biases in parameter estimation. A simulation was conducted to investigate the structural properties of moult increments and intermoult periods; however, we encountered a problem wherein the current length L exceeded the asymptotic length L∞ for simulated data. Therefore, we developed a novel − approach in which the fraction of (L∞ − L ) is constrained to a boundary of (0,1), as a solution to the aforementioned problems.
For modelling intermoult periods, a more sensible model should be taken into account based on the function of g(Tij|Lij−1,Tij−1) as obtained in §5.5.2. The estimation of the intermoult period considers merely the premoult length regardless of the premoult interval periods, where the premoult interval period is one of the factors affecting the next onset of moult. Therefore, we can rewrite the estimation function as
− − − β1 − β2 E(T |L ,T ) = β0(L ) (T ) , 8.1 Major Findings and Remarks 174
− − where L is the premoult length, T is the premoult interval period and β0, β1 and β2 are the parameters.
Our study revealed that inclusion of random L∞ in a growth model increases consistency (lower value in standard error) in parameter estimation. As a result, effects of individual heterogeneity were shown to have a great impact on persistence in quantitative growth methods.
Furthermore, environmental perturbation also shown to contribute significantly to the variability in individuals’ growth rates. More specifically, our findings in modelling moult increments using a linear mixed effects approach reveal both factors: water temperature and population density in the tank, influence the growing process of lobsters. It is observed that individual carapace length increases when water temperatures range between 25 and 300C. Under these conditions, a negative relationship between moult increments and the population density of tank lobsters was also observed. In other words, the greater the density of the individuals in aquaria, the smaller the incremental growth of crustaceans.
Tag-recapture experiments are ubiquitous for measuring growth parameters because they are less labour intensive than laboratory experiments. In most ecological tag-recapture studies, single re- capture is the most commonly used approach. Nevertheless, in the absence of heterogeneity, the information provided can lead to biases in sampling probabilities. Hence, we generalized single re- capture data points to a multiple-moult dataset, considering the correlation between two consecutive moults in a long run.
In tag-recapture, a plausible and reliable outcome relies on the number of moults and the probability of the next recapture. Moreover, the use of hidden variables is necessary as we do not know with certainly the number of moults in between captures, only that it has moulted at least once if there was an observable increment in size. Therefore, given the real number of moults, a likelihood approach has been suggested, as in Hoenig and Restrepo (1989). Furthermore, the GEE approach was used to quantify k and L∞ for multiple-recapture data in comparison with an improved Fabens approach. From the implementation of the simulation, the improved Fabens method is preferable because it shows a lower standard error in parameter estimations. 8.2 Future Directions 175
8.2 Future Directions
The Cox model we described earlier can sometimes produce inconsistent estimations if the proportional hazard is not assumed appropriately; thus, the accelerated failure time model (AFT) may prove to be an attractive alternative.
Normally, the maximum likelihood method is applied to estimate the parameters in the model. Unfor- tunately, this approach is highly sensitive to outliers. In general, the rank estimation method is likely to be robust and quite stable to outliers. Tsiatis (1990) used the rank estimation method to estimate the regression parameters; however, the underlying density for error is unknown. Subsequently, Jin et al. (2003, 2006) developed the Gehan-rank estimation method and the least square estimation method. In addition, Zhou (2005) proposed an empirical likelihood estimation for a univariate AFT model, but it is computationally difficult with respect to the EM algorithm.
Alternatively, a rank test regresses the logarithm of the failure time to the covariates based on the AFT model. However, the rank estimating functions are discontinuous in nature, which leads to inconsistency in the analysis of growth models. The covariance matrix is unlikely to be estimated due to the step function; however, an induced smoothing mechanism for the AFT model makes it possible to resolve the difficulties (Brown & Wang, 2007). Subsequent research has extended the induced smoothing procedure to clustered failure time data (Johnson & Strawderman, 2009); however, this independent model does not consider within-cluster effects in the analysis.
Wang and Fu (2011) worked on the within-cluster correlation by resolving the Gehan-weighted rank regression into the between- and within-cluster estimating functions and reintegrating the estimators to improve the cluster effects. Meanwhile, the induced smooth function is included to enable the estimation of model parameters and covariance matrices.
Apart from using parametric methods to quantify parameters from the growth model, an alternative can be accounted for by looking into the conditional approach on equilibrium distribution. Bhat- tacharya et al. (2011) employed a classical variational matrix approach through the conditions for the equilibrium distribution of multiple populations and their moments. One of the advantages of using this approach is that the assumption of the density function is not needed. Furthermore, numerical computation with respect to moments is not required for higher dimensional variables.
There might be misspecification in the existing models with respect to the effect on growth parameters. 8.2 Future Directions 176
In practice, we cannot say with certainty what kind of distribution of a random variable is appropriate. A conditional moment approach could be an alternative because it is much less restricted to any assumption of a distribution. Therefore, this framework is robust to departures from the distributional assumptions that are not being evaluated.
We have investigated the effects of environmental variability in tank data analysis; the underlying covariates can be further incorporated in tag-recapture studies. Evaluations of heterogeneity and en- vironmental randomness in modelling crustacean growth could provide valuable insights for managing ecosystem population dynamics, either in natural or controlled marine environments. References
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The codes of capture-recapture
A.1 The analytical method
To generate capture-recapture data, we require Monte Carlo simulation following two steps:
(i) information of the length at tagging, L1 and age at tagging A1 for each of the individuals.
(ii) information of the length at recapture, L2, age at recapture, A2 for the tagged animals. n <- 100 linf<-150 siginf<-30 sdlog<- sqrt(log(1+(siginf^2/linf^2))) T0<-0.1+0.5*runif(n) k<-0.2 L0<-Linf*(1-exp(-k*T0)) Linf<-rlnorm(n,meanlog=log(linfty),sdlog=sdlog) a <- 0.002792338 b <- 0.0532552241 s <- 0.238912565 lam <- 0.7 para=function(T0,Linf,L0,k,lam) { A.1 The analytical method 194
LL=list() TT=list() L_no_error=list() for(i in 1:n) { L<-c(L0[i],rep(NA,ni[i])) I<-c(NA,rep(NA,ni[i])) E<-c(0, rnorm(ni[i],0,1.5)) T<-c(T0[i],rep(NA,ni[i]))
for(j in 1:ni[i]) { if((Linf[i]-L[j])<=0) I[j]=0 else I[j] <- rgamma(1,shape=lam*(Linf[i]-L[j])*(1-exp(-k*T[j])),scale=1/lam) L[j+1]<- L[1] + sum(I[1:j]) #without error
T[j+1]<- T[j] + rlnorm(1,meanlog=log(a)-((s^2)/2)+(b*L[j]),sdlog=s) } L_no_error[[i]]<-L TT[[i]]<-cumsum(T) LL[[i]] <- L+E } return(list(len=LL,time=TT,L=L_no_error)) } res=para(T0,Linf,L0,k,lam)
A.1.1 For single-recapture data no.recap=1 tag_len=function(k=0.2, LL, TT, no.recap) { A.1 The analytical method 195
match.time <- function(tt, TT, LL) { ind <- which.min(abs(TT - tt)) return(list(time=TT[ind], len=LL[ind], index=ind)) }
n = length(LL) L1 <- L2 <- array(NA, c(n,no.recap))
A1<-runif(n,1,2) time_liberty <- array(runif(n*no.recap, 100/365.25, 700/365.25),c(n,no.recap)) A2 <- numeric() for(i in 1:n) L1[i] <- match.time(A1[i], TT[[i]], LL[[i]])$len for(no in 1:no.recap) { A2 <- cbind(A2, A1 + rowSums(matrix(time_liberty[,1:no],n,no))) for(i in 1:n) L2[i,no] <- match.time(A2[i,no], TT[[i]], LL[[i]])$len } return(list("L1"=age, "A2"=age_recap, "L1"=L1, "L2"=L2, "time_liberty"=time_liberty)) } tag.res = tag_len(k=0.2,res$len, res$time,no.recap)
A.1.2 For multiple-recapture data
To generate multiple-recapture data, we repeat the steps above (section A.1) following by the coding of: no.recapture <- 5 tag_len=function(k=0.2, LL, TT, no.recap=no.recapture) { match.time <- function(tt, TT, LL) { A.1 The analytical method 196
ind <- which.min(abs(TT - tt)) return(list(time=TT[ind], len=LL[ind], index=ind)) }
n = length(LL) L1 <- L2 <- array(NA, c(n,no.recap))
A1<-runif(n,1,2) time_liberty <- array(runif(n*no.recap, 100/365.25, 700/365.25),c(n,no.recap)) A2 <- numeric() for(i in 1:n) L1[i] <- match.time(A1[i], TT[[i]], LL[[i]])$len for(no in 1:no.recap) { A2 <- cbind(A2, A1 + rowSums(matrix(time_liberty[,1:no],n,no))) for(i in 1:n) L2[i,no] <- match.time(A2[i,no], TT[[i]], LL[[i]])$len } return(list("L1"=age, "A2"=age_recap, "L1"=L1, "L2"=L2, "time_liberty"=time_liberty)) } tag.res <- tag_len(k=0.2,res$len, res$time, no.recap=no.recapture)