sign in sign up
<<
Home , Assembly rules, Evolution of fish

Evolutionary assembly rules for histories

Eric L Charnov, Henrik Gislason & John G Pope

SEDAR58-RD45

October 9, 2019

F I S H and F I S H E R I E S , 2013, 14, 213–224

Evolutionary assembly rules for fish life histories

Eric L Charnov1, Henrik Gislason2 & John G Pope3

1Department of Biology, The University of New Mexico, Albuquerque, NM 87131-0001, USA; 2National Institute of Aquatic Resources, Technical University of Denmark, Charlottenlund Castle, DK2920, Charlottenlund, Denmark; 3National Institute of Aquatic Resources, Technical University of Denmark, Charlottenlund Castle, DK2920, Charlottenlund, Denmark

Abstract Correspondence: We revisit the empirical equation of Gislason et al. (2010, Fish and 11:149– Eric L Charnov, Department of Biol- 158) for predicting natural mortality (M, year)1) of marine fish. We show it to be hi ogy, The University of 1:5 L )1 New Mexico, Albu- equivalent to M ¼ K, where L¥ (cm) and K (year ) are the von Bertalanffy L1 querque, NM 87131- growth equation (VBGE) parameters, and L (cm) is fish length along the growth 0001, USA dW E-mail: [email protected] trajectory within the . We then interpret K in terms of the VBGE in mass dT , and show that the previous equation is itself equivalent to a )⅓ power function rule Received 21 Nov 2011 between M and the mass at first reproduction (Wa); this new )⅓ power function Accepted 4 Mar 2012 emerges directly from the life history that maximizes Darwinian fitness in non- growing populations. We merge this M, Wa power function with other power functions to produce general across-species scaling rules for yearly reproductive allocation, reproductive effort and age at first reproduction in fish. We then suggest a new way to classify (or lifestyles) as to the life histories they should contain, and we contrast our scheme with the widely used Winemiller–Rose fish lifestyle classification.

Keywords fish growth, natural mortality, optimal life history

Introduction: what are assembly rules for fish life histories? 214 Introduction to natural mortality: two distinct ways for M to be size dependent 214 M vs. L, within and between species: the data 215 1=3 1=3 K is / W‘ / Wa 216

Body size (Wa) at first reproduction 217

The evolutionarily optimal size at first reproduction (Wa) in a non-growing population 217

Discussion: Ma/K » 1.8 implies a general Ma Æ a rule for indeterminate growth 219 Discussion: across-species life histories, power functions are not (always) allometries 219

Discussion: egg size, C1 220 Discussion: how shall we classify life histories, lifestyles and habitats? 220 Acknowledgements 222 References 222 Appendix 1: Some useful properties of the Bertalanffy growth equation 223 What Is A in the BG Equation? 223

Appendix 2: Fitness (R0) is a product 224

2012 Blackwell Publishing Ltd DOI: 10.1111/j.1467-2979.2012.00467.x 213 Natural mortality in fish life-history E L Charnov et al.

Introduction: what are assembly rules for fish Introduction to natural mortality: two distinct life histories? ways for M to be size dependent This study addresses life histories in female fish; Externally imposed mortality plays a key role in the what we call ‘assembly rules for fish life histories’ optimal size (age) for the initiation of reproduction, refers to the relationships between mortality rate, an so first we study natural mortality in fish and how it individual’s growth (production) rate, size (age) at relates to individual body weight and to body weight first reproduction, reproductive allocation thereafter at maturity both within and across species. and egg size. mediates these Opinion about the natural mortality (M) of fish is relationships and favours some very special ones polarized into those who believe M, at least near (Winemiller and Rose 1992; Charnov 1993). These adulthood, can be treated as an age/size-indepen- special ones are what we aim to predict, hence our dent constant within the species (e.g. Beverton and use of the term ‘evolutionary’. As a life history Holt 1957, 1959; Pauly 1980), and those who consists basically of a creature surviving and believe M often is a strong and well-defined negative growing to the optimal size to start offspring function of body size, up to and perhaps beyond the production, we concentrate on the size (age) of first length of first reproduction (La) (see Gislason et al. reproduction. Most fish species have indeterminate 2010 for a general overview). Both camps agree growth, where body size still increases after initia- that M is very high for larval (and other smaller) tion of reproduction/maturity. We will approximate fish, but they disagree as to whether M drops and fish growth and reproduction with a ‘determinate then remains approximately constant well before La growth approximation,’ where growth ceases at the or whether it shows some well-defined function of onset of adulthood. This approximation captures the length, Lx. Following the lead of Beverton and Holt essential power function forms we wish to study, (1959), Charnov (1979), Pauly (1980), Charnov even though the normalization constants generally (1993) and others (e.g. Cury and Pauly 2000; differ between determinate and indeterminate Griffiths and Harrod 2007; Andersen et al. 2009) growth. have shown that M near the age of first reproduc- Our evolutionary optimization scheme yields tion (a) shows the across-species rule of M = CÆK, relationships that are applicable across species where K is the Bertalanffy growth constant for the between mortality, individual growth rate, body species growth curve; a typical C value for fish size (age) at maturity and reproductive effort of »1.8. Charnov (1993) showed this rule with (Charnov 1993). The scheme suggests how we C » 1.5–1.7 in a large sample of indeterminate should classify habitats with respect to what life growing , snakes and lizards, so the rule histories they will contain, and we end up appears to apply to other organisms with indeter- proposing a new way to classify the interaction minate growth, as well; Charnov (1979) showed it of (or lifestyles) and life history. We will for marine shrimp. However, M is also known to contrast this to the widely used Winemiller–Rose decrease with asymptotic length (L¥) across species classification scheme, a variant of ‘r/k’ selection (e.g. Cury and Pauly 2000); presumably this M theory. refers to M near adulthood. But these M, L¥ plots Surprisingly, the size of an individual offspring are often quite noisy. (egg size) is decoupled from the previous evolu- In apparent opposition to this constant M idea, a tionary optimization, and we can only provide large amount of fish data demonstrate well-defined hints as to what sets optimal egg size (Smith and M effects of body size, both for factors like Fretwell 1974). Luckily, this decoupling means risk and total M at some Lx (e.g. McGurk 1986; that we need not solve the optimal egg size problem Lorenzen 1996). Indeed, models designed to simu- to predict the optimal size at maturity. This late the body size spectra of marine fish communi- decoupling happens because we use R0, the net ties (e.g. Andersen and Beyer 2006; Pope et al. reproductive rate, as an individual’s fitness mea- 2006) predict size-dependent scaling of M during sure (Charnov 1993, 1997; see also Appendix 2) ontogeny. and because we place all This literature intermingles two questions/ideas: among the very young (the standard kind of (i) How does M change with L over a growth curve spawner/recruit assumption in commonly used within a species? Are there general scaling rules ) fishery models). M L P?, and (ii) How does M, as some sort of

214 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 Natural mortality in fish life-history evolution E L Charnov et al.

average near adulthood (call it Ma), change with and 1.44, respectively) are not statistically different, body size (say, L¥) across species? Clearly, if we so it seems reasonable to fit the M equation with a ) know M near maturation size and M L P at other common exponent. This fit is just as good as sizes, we can know M at all sizes. But, why should Equation 1: M decline with L across species? a ¥ L In this study, we first reconcile the within-species logeðMÞ¼0:05 1:46 loge þ logeðKÞ L1 M vs. L with the between-species Ma vs. L¥ puzzle. r2 0:61; n 168 We will do this by means of life-history optimization. ð ¼ ¼ Þ First, the data. or, more simply, as the intercept is not different from 0, nor the exponent from 1.5, M vs. L, within and between species: the data L 1:5 M ¼ K ð3Þ In a modelling study of the North Sea fish L1 , Gislason et al. (2008) found that M should scale with both L¥ and L to generate spawner-for-spawner replacement, corresponding to a net reproductive rate R0 » 1, and hence coexistence, for large and small species of fish. (a) They also found that this could explain the strong negative relationship between the maximum slope of the stock- curve at the origin and L¥ observed by Denney et al. (2002). To test this further, Gislason et al. (2010) made a careful and critical review of empirical estimates of natural mortality, M (year)1), in marine (and brackish) water fish species and found that the observations could be fit by the following equation (length in cm):

logeðMÞ¼0:551:61logeðLÞþ1:44logeðL1Þ 2 ð1Þ þlogeðKÞðr ¼0:62;n¼168Þ (b)

The L¥ and K are from fitting the von Bertalanffy growth equation (VBGE) to size at age (x) data:

Kx L ¼ L1ð1 e Þð2Þ

Although Gislason et al. (2008) provide a biolog- ical justification for the terms of Equation 1, it is seemingly a mystery. In particular, the 1.44 scaling of M with L¥ seems puzzling as mortality rates are expected to decrease with body mass across species. There is one other puzzle here in that the authors also fit a version of Equation 1 with an added Arrhenius term (d/Ta, where d is a constant, and Ta is absolute ) and they expected a negative d coefficient, but the fit showed no signif- Figure 1 (a) Plot of M/K vs. L/L¥ in the Gislason et al. icant temperature effect when K was also in the 2010 data set.hi See text for discussion. (b) Plot of log M vs: log K L 1:5 yields a very useful predictive equation. e e L1 There are two things to note about this equation. equation for M; standard error of slope = 0.06 and

First, the coefficients of loge(L) and loge(L¥) (1.61 standard error of intercept = 0.07.

2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 215 Natural mortality in fish life-history evolution E L Charnov et al.

The form of Equation 3 is quite interesting; In Appendix 1, we derive several useful properties in Fig. 1a, we show the fitted relation of this growth/production equation. In particular, M L log ¼ðC0ÞþðC2Þlog . There is a lot we can derive W¥ (or L¥) and K from the A and B e K e L1 2 more scatter here (r = 0.48, n = 168), but C0 parameters. may be taken as zero (C0 = )0.06; 95% CI: )0.21 K ¼ B=3 ð5aÞ to 0.09) and C2 as )1.5. Surprisingly, the fit is reasonably good down to L/L¥ values of 0.15, 1=3 1=3 very small fish, about 20% the size of first repro- W1 ¼ A=B ¼ a L1 ð5bÞ duction ðLa=L1 2=3Þ. So, Equation 3 predicts the and hence that: M vs. ontogenic L at any fixed K value. We also fit Equation 3 using L/L data for only the post- ¥ A maturation period (only L/L > 0.65); this gave the K ¼ W1=3 ð5cÞ ¥ 3 1 same parameter values as the full L/L¥ range, so the M, L/L¥ rule extends into the adult size groups. So, in terms of A and B, we have two ways to Finally, we fit the equation with the 60% of the data answer what K is. First, (Equation 5a) 3 Æ K Æ W is the that came from unfished (or lightly fished) popula- term subtracted from a basic production term tions, and the answer was again the same. (A Æ W⅔); Charnov (1993, 2008) has argued that Second, Equation 1 turned into Equation 3 also 3 Æ K Æ W is really mostly a reproductive allocation. predicts the empirically observed Ma vs. K relation Equation 5c, however, shows that K is also a )⅓ across species. Setting L=L 2=3 , the expected ð 1 Þ power function of W¥. The intuition here is that if relative maturation size, gives ðMa=K 1:84Þ, quite K is related to reproductive allocation, higher K typical for indeterminate growers (i.e. fish, lizards, exhausts the productive capacity (A Æ W⅔) faster, and snakes; see Figs. 4.5 and 4.11, pp. 66 and 74 in and yields smaller asymptotic size (W¥). We will Charnov 1993). work with this power function form of K, using it in So, the Gislason et al. 2010 equation supports two ways. First, we shall write the fitted, empirical both the M vs. L (over the life history) position, and Equation 3 in terms of W¥ (or L¥). Second, we shall the Ma vs. K position. In Fig. 1b, we fit the log/log argue that the power function is really about the form of Equation 3, fixing the exponent at )1.5: size at first reproduction (La or Wa) and not the 1:5 asymptotic sizes, because W W . L a ¥ logeðMÞvs: loge K . L1 So, using Equation 5c, empirical Equation 3 for M can be written as: The fit is good (r2 = 0.61), and the residuals here are normally distributed, so this regression should 1=2 W A 1 M ¼ W 3 ð6aÞ be useful in prediction of M from K, L, L¥ data. 1 W1 3 or K is W1=3 W1=3 / ‘ / a L 1:5 1 A M L1; 6b ¼ 1=3 1 ð Þ To understand what Equation 3 means, we must first L1 a 3 understand what K is. scientists usually think of K as the curvature of the growth equation in length Beverton and Holt (1959) first showed that (see Equation 2); fish length-at-age data are usually La L¥ within many fish taxa with a central value ⅔ very well described by Equation 2. The production of » . While various taxa differ somewhat (e.g. ⅔ process underlying growth, however, is dW/dt, Clupeidae »0.8), is widely accepted and used in where W is mass (or more commonly just called the fishery literature. It seems less well appreciated 3 weight). W = aÆL3 for fish that grow isometrically. that La/L¥ = 2/3 means that Wa/W¥ = (2/3) = The differential equation for the VBGE is as 0.296, the inflection point for the VBGE in mass (see follows: Appendix 1). When we describe fish growth with equations like the VBGE, our fitting puts the mass at dW ¼ A W2=3 B W ð4Þ first reproduction (W ) approximately at the size of dt a the fastest growth. Why? The answer is probably as Solving Equation 4 for W(x) and transforming follows. Although the VBGE well describes fish to L(x) yields Equation 2. growth, Equation 4 is somewhat simplistic; the

216 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 Natural mortality in fish life-history evolution E L Charnov et al.

underlying growth process really has two phases: most fish) and ageing (senescence). If all personal before maturation, fish grow their self; after matu- production is devoted to reproduction at size Wa, ration, they grow slower because production also is growth ceases, and Wa is the adult size (called given to reproduction. Almost any body mass power determinate growth). If only some of the personal function for growth has dW/dt increasing with W at production is devoted to reproduction at size Wa, small W. Thus, if the growth rate decreases because growth (albeit slower) continues after age a (called of the initiation of allocation to reproduction (size indeterminate growth), generally leading to some W¥

La, Wa), the fastest growth – highest dW/dt – will size. The relation between Wa and W¥ depends just be at that size. If we fit a VBGE to fish data, upon the reproductive allocation schedule. Evolu- we will be putting La near the highest dW/dt. tionary models for optimal indeterminate growth And, the inflection point – the highest dW/dt in the are very complicated (e.g. Perrin and Sibly 1993;

VBGE – is at 0.296ÆW¥,orLa = 2/3 Æ L¥. Charnov et al. 2001; Thygesen et al. 2005; Quince Using these two rules allows us to write Equation et al. 2008), and most trade-off assumptions do not

6 in terms of Wa or La, the size at first reproduction. favour indeterminate growth as the optimal life

Substituting Wa/(0.296) for W¥ in Equation 6a history. These numerous cases favour all production yields: at Wa to be given to reproduction, resulting in Wa being the final adult size. The issue of determinate W 1=2 M ¼ 0:41 A W1=3 ð7Þ versus indeterminate growth is a complex one, and W a a we will not develop it here. What we will do is use life-history optimization to Body size (Wa) at first reproduction predict Wa in the face of a simple pre-reproductive Equation 7 is the most basic equation of all; set growth function and size dependence for pre-repro- ductive mortality. Let us derive the optimal adult W = Wa, then: body size, Wa, for a very simple, determinate growth 1=3 life history. Of course, fish are not this simple, but Ma ¼ 0:41 A Wa ð8Þ the model has several fundamental features in Remember that Equation 7 (or Equation 8) is just common with more complex, indeterminate growth the original fitted Equation 3 transformed to Wa cases, and this model is very simple to understand. under the most plausible of assumptions that Wa is Suppose juvenile growth follows the production law 0.296 W , the maximum of dW/dt, as represented dW 0:67 Æ ¥ dt ¼ A W . Suppose an individual grows until by the VBGE, an equation that well describes the some size Wa, when it then gives all of dW/dtto underlying two-part growth/reproduction process. offspring production and thus stops growing; Wa is

We call these Equations 7 and 8 the most basic the adult size. dWa/dt, the mass given to reproduc- of all because they predict M versus Wa across tion per year, is usually called the reproductive species, and adjust M to the relative size along the allocation (RA). growth trajectory W/Wa within a species. Writing What determines the optimal Wa? It’s mortality them this way focuses our attention on Wa, which and growth, of course. The mortality rate of young is the most basic of life-history size variables, when immatures is generally very high; suppose that, an individual begins to allocate production to after an initial burst of density-dependent that offspring and thus slows, if not stops, its growth. stabilizes recruitment, the mortality rate declines We will now show how life-history evolution sets with increasing immature body mass according to h Wa and how that relates to Ma. the rule M ¼ Di W i , where i refers to species i (this size rule may apply only after some small size is reached). We allow various species to have differing The evolutionary optimal size at first D and h variables, but all have some size dependence reproduction (W ) in a non-growing a (h = 0 is okay, too). Notice that this mortality is population imposed externally to the organism (its predators, Four main problems dominate the study of life- competitors, etc.); we leave unspecified where it history evolution (e.g. Stearns 1992): the size or age comes from. All that a growing organism can do is of first reproduction (Wa or a), the schedule for the decide at what size (age) to quit growing and begin allocation of personal production to reproduction reproducing (Wa). On the one hand, the longer after a, the size of an individual offspring (egg size in it delays initiating reproduction, the greater its

2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 217 Natural mortality in fish life-history evolution E L Charnov et al.

cumulative chance of dying without reproducing at dWa ð0:67 þ hÞ da all (a cost). On the other hand, however, the longer @logeR0=@a ¼ Ma ¼ 0; RWa it delays, the greater its reproductive ability, as it a d½ 0 MðxÞdx gives dW/dt to reproduction, its own growth as ¼ Ma 0.67 d capacity, which is A Æ W (a benefit). The optimal a W balances this cost (risk of death) and benefit and log A is just a constant. a e D C1 2=3 (reproduction rate). dWa 2=3 ð0:67þhÞAWa As ¼ A Wa we have M da Wa a Life-history theory (Stearns 1992) requires us ¼ 0, which gives to choose a measure of Darwinian fitness to carry 1=3 out this optimization. Numerous possibilities exist Ma ¼ð0:67 þ hÞA Wa ð12Þ (Stearns 1992; see also Metz et al. 2008), but the choice really mostly depends upon our population Equation 12 is equivalent in form to empirical dynamics assumptions (e.g. Charnov 2009). Non- Equation 8, excepting the constant multiplier of A, growing populations lead to the net reproductive rate which is (0.67 + h) here and 0.41 fitted for the (R ) being the appropriate fitness measure, and 0 indeterminate growing fish; our model assumption growing populations lead to the intrinsic rate of of determinate growth causes this numeric differ- increase, labelled ‘r,’ being the fitness measure; ence in the normalization constant. Equation 12 populations in highly variable environments may may, of course, be solved for Wa. require stochastic versions of R or ‘r. Here we 0 Notice that the only mortality (M) appearing in the assume non-growing populations and use R0 as final answer (Equation 12) is Ma; all the earlier fitness. R here has some very general and unusual 0 mortality, including density-dependent mortality multiplicative properties, which we derive and discuss very early in life, has gone away. This is a generic in Appendix 2. For the simple determinate growth life feature of R0 maximization, which sets up this special history modelled here, R0 takes a very simple form: R relation between Ma and Wa (e.g. Charnov 1993). a MðxÞdx However, we can recover M values at some earlier 0 b e )h R0 ¼ ð9Þ ages/sizes through the M = D Æ W function. If we Ma pick some other W, say Wy, we have as follows:

h where the exponential is the chance of living to My ¼ D Wy and reproduce at age a (often called S(a)), the 1/Ma is M ¼ D Wh giving the average adult lifespan, and b is the egg a a production per unit of time while an adult. h h My Wy Wy 0:67 ¼ or My ¼ Ma A W Ma Wa Wa b ¼ a ; ð10aÞ C 1 or, from Equation 12: because offspring production is just diverted self- W h growth and C is the mass of each egg. Of course, y 1=3 1 My ¼ð0:67 þ hÞ A Wa ð13Þ Wa h Ma ¼ D Wa ð10bÞ (dropping the i subscript). Equation 13 is the same form as empirical Equation 7; we predict My at sizes other than Wa by multiplying R0 can now be written as: h R Wy a the M equation by , the relative mass ratio a Wa 0:67 MðxÞdx A W e 0 R a 11a raised to the power –h, the exponent of M vs. W along 0 ¼ h ð Þ C1 D Wa the growth trajectory. For determinate growers, this or applies to the smaller W. A similar principle for adjustment of M to W would apply for other A y y loge R0 ¼loge þð0:67þhÞlogeðWaÞ M ¼ gW functional forms. D C y y Z 1 a ð11bÞ Suppose A is greatly affected by temperature (or, MðxÞdx: say, diet or mode); then that A will simply 0 match Wa to the prevailing M function. In a basic

The optimal Wa is where @loge R0=@a¼0. sense, temperature or diet effects on the optimal Wa This is when are already accounted for in A (Charnov and

218 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 Natural mortality in fish life-history evolution E L Charnov et al.

Gillooly 2004). The temperature term was not is the necessary result to predict the correct Ma/K significant when added to Equation 1, most likely number. This is worth remembering. because temperature was already present when K was in the equation recall; K ¼ A W1=3 . 3 1 Discussion: across-species life histories, Mortality (M) need not have a particular size-based power functions are not (always) allometries form during growth, and our optimal life-history scheme works just as well if the M is independent of In this section, we further use the determinate dW 0:67 size; simply set h = 0 in Equation 13. So, the theory growth approximation for fish. Let dT ¼ A W . does not predict h, and it simply takes it as an input At maturation size Wa, the organism will be giving dWa 2=3 from the environment. For each species, natural dT ¼ A Wa mass to reproduction per unit of selection matches Wa to the environmentally given time (its reproductive allocation, RA). Reproductive Ma (or M schedule) based on A, the height of the effort (RE) is usually defined as RA divided by body production function. Thus, Ma and Wa are brought mass (e.g. Stearns 1992; Charnov et al. 2007), dWa 1=3 into a very special relation with each other (Equation so RE ¼ ¼ A Wa . Integrating the determi- WadT 12). Both A and ‘h’ may vary among species/habitats, nate growth equation from W =0atT = 0 shows 1=3 and this will introduce between-species variation into that½ 3=A Wa ¼ a, where a is the age of first 1=3 the multiplier of Wa in Equations. 12 and 13. We reproduction; Appendix 1 shows that solving the suspect that pooling species with various (h)ina VBGE for the age at first reproduction when 1=3 Wa single data fit will reduce the correlation and simply Wa=W1 ¼ 0:296 gives a similar result: a / A . predict some sort of average h. So, we have power functions for three key life-

Expanding this fitness optimization model to full history variables (a, Ma, RE): indeterminate growth is a formidable task, and we 1=3 will not do it here. Charnov et al. (2001) and Ma / A Wa ð14aÞ Charnov and Gillooly (2004) show that A enters the 1=3 RE ¼ A Wa ð14bÞ Ma, Wa power function in exactly the same way as in Equation 12 in a model that yields indeterminate 3 a ¼ W1=3 ð14cÞ growth as the optimum. What is expected to differ A a between determinate and indeterminate growth is 1=3 Equations 14a, b and c are three of the four the multiplier of A Wa , the ‘non-A’ part of the central variables used to describe fish life histories: a normalization constant. is just the accumulated growth function, RE is just

the turning of personal growth at Wa into produc-

Discussion: Ma/K » 1.8 implies a general MaÆa tion of offspring and Ma is because of the optimi- rule for indeterminate growers zation of Wa in the face of mortality. To obtain yearly offspring production (call it b), just divide Charnov (1993, p. 61) noted the following RA ¼ A W2=3 by the mass of an egg, C : identity for the VBGE, where age of first reproduc- a 1 L tion, a, is measured from size zero: a ¼ 1 2=3 L1 A Wa eKa ¼ 1 eðÞðK=Ma aMaÞ. Any life-history optimiza- b ¼ ð14dÞ C1 tion model where the resulting lifetime growth is adequately described by the VBGE (thus setting Each of the four power functions that sit at the La 2=3) and which sets a Æ Ma » 2, the typically centre of how we describe life histories generalizes to L1 observed fish (and ) value (Charnov 1993, indeterminate growth life histories (with generally Chapter 4), makes the following prediction: different multipliers of A). Every one of them contains A. Thus, they will form across-species ⅓, K ða M Þ ¼ 1:1 ⅔ allometries only if all the species in the data set a M a have the same A. If the species differ a lot in A, or across-species plots may not look much like ()⅓, ⅓, M 2 ⅔ ⅔ a ¼ ¼ 1:82 ) power functions; the scaling of yearly clutch K 1:1 size also requires that egg size be similar across

species (or at least uncorrelated with Wa across We may not understand just what trade-offs species). In all cases, we expect good allometries favour indeterminate growth in fish, but a Æ Ma » 2 (across-species ⅓, ⅔ power functions) if we correct

2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 219 Natural mortality in fish life-history evolution E L Charnov et al.

a, RE, Ma,orb by dividing (for a multiplying) by A. splendid attempt to predict optimal egg size for

Of course, if A is itself a power function of Wa, pelagic marine fishes using these principles. Having across-species plots will be allometries that combine said that, we do know many things about egg sizes the two exponents. The plotting of Ma vs. K is in ; for example, they are small for pelagic equivalent to correcting Ma by A across species, fish, larger in freshwater, larger with male parental which is why plots of Ma vs. K are so much tighter care, sometimes changes seasonally and so forth than plots of Ma vs. L¥ (or W¥) (e.g. Cury and Pauly (excellent review in Freedman and Noakes 2002). 2000), where A varies among the data points. Generally, across species, they do not correlate with Clearly, understanding A is a central question in adult size except when adult size itself correlates the study of fish life histories; fortunately, we can with the spawning/larval development environment 1=3 estimate A as 3 K W1 (Equation 5c) whenever (e.g. spawning season), and the same rules probably we describe fish growth with the VBGE. We should apply to egg size vs. body size within a species plot A versus foraging mode, food type and temper- (Fleming and Gross 1990). We have every reason to ature; we should determine whether A varies believe that natural selection can easily change egg consistently with other habitat features, body size size to match a local optimum (e.g. Fleming and itself or perhaps phylogeny. Various scaling rules Gross 1990; for ). Downhower and Charnov 1=3 are known for how K varies with L1 or W1 (1998) found that the average egg size in Gambusia within pelagic and demersal fish (Gislason et al. hubbsi in the Bahamas was a constant with adult 2008, 2010), and we will develop these in relation size within populations, but the average egg size to A in a later study. Perhaps A and M are (volume, mass) varied by a multiplier of five themselves interrelated; if higher A often requires between isolated populations – remarkable size more foraging movement, then perhaps M will be variation that is yet unexplained. We are lucky driven up too. that the decoupling of egg size from evolutionary

Pauly (1981), reviewed in Cury and Pauly 2000) optimization of Wa allows us to treat egg size as a dðÞ log R suggests that O2 is the limiting growth in e 0 constant that drops out when we set da ¼ 0. water-breathing animals, and thus, surface area and function determines ‘A’. While we think this Discussion: how shall we classify life unlikely as a universal limiting factor, perhaps the histories, lifestyles and habitats? of can be used to estimate A, as gill structure is likely to be adjusted to ensure The mostly widely used scheme to classify habitats delivery of the needed quantity of oxygen for (and lifestyles) with respect to fish life histories is production, independent of what actually limits that of Winemiller and Rose (1992), a variant of r/k production. selection theory. All such classification schemes map the habitat to the expected life history through assumptions about how natural selection operates C Discussion: egg size, 1 there to favour that particular life history. Our life-

Egg size (individual offspring size, C1, in mass) was history assembly rules in Equation 14a–c suggest dropped out of the optimization of R0 argument, that to map the environment to the life history, we leading to Equation 12. This is why we could ignore need to know A, the height of the individual’s egg size in this evolutionary scheme; egg size and production function, and M, the externally imposed

{a, Ma or reproductive effort (RE)} are effectively mortality risk. As M (or more precisely Ma) must be uncoupled. What determines C1 is very complicated. very hard to know, it is not too surprising that it is The key trade-off is clearly that of egg size and egg not generally included as a feature of the habitat; it number; larger eggs mean fewer of them (Smith and should be. Fretwell 1974). We know how to solve for the Equations 14 suggest a classification scheme with optimum (Smith and Fretwell 1974), and we know M on one axis and A on the other, illustrated in that optimal C1 is expected to be usually indepen- Fig. 2a. Each ray from the origin corresponds to a 1=3 dent of the total resources being devoted to eggs. fixed value of Wa , as Equation 14a has 1=3 A Wa / ; this is the optimal W in the face of A What we do not know is how the trade-off structure Ma a for individual performance vs. egg size maps to the & Ma. Notice that for fish of the same body shape, 1=3 environment in which the eggs and young juveniles Wa / La, so body length at first reproduction is develop, although Wootton (1994) has made a the same along each ray. And, of course, steeper

220 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 Natural mortality in fish life-history evolution E L Charnov et al.

1=3 (a) have similar size ranges for Wa ð¼ LaÞ, but all the reproductive efforts will be low in Habitat 1 and high in Habitat 2. Clearly, the distribution of life histories we see depends upon how A and M are correlated with each other across habitats; Fig. 2b illustrates a positive A and M correlation. The scheme of Equations 14a–c and Fig. 2 has one very unusual feature; suppose the average length of the adult life span is 1 , then (repro- Ma ductive effort) (average adult life span) is RE/Ma and, by Equations 14a and b, is predicted to be approximately constant for all fish, independent of

Wa or A. Lifetime reproductive effort (LRE) obeys the rule (approximately) among and lizards (Charnov et al. 2007). Gunderson (1997) found the (b) same relation in a diverse collection of 28 marine fish species, although his numeric LRE value is different from the mammals and reptiles. Of course, Fig. 2 is incomplete to fully classify fish life histories; there is a third axis, extending into the plane to plot the size of an egg (or newborn). But without a better understanding of the egg and larval environment, that axis/dimension is the most diffi- cult of all to predict using natural selection argu- ments. Charnov et al. (2001) aggregated the yearly offspring production in the Winemiller and Rose (1992) data set to estimate the average yearly mass

given to reproduction as a function of Wa across species (essentially reproductive allocation, RA). The Figure 2. A scheme to classify life histories. (a) This data plot was uncontrolled for variation in A among 2 square plots height of the growth curve (A) on the y axis the species, yet it showed a very strong (r = 0.74, n = 139) power function form with a slope of 0.8. and mortality near adulthood (Ma) on the x axis. Lines » 1=3 of equal body size Wa ¼ La are just rays from the origin. Thus, the species viewed as very different by

Reproductive effort is proportional to Ma, independent of Winemiller and Rose (1992) satisfy a RA vs. Wa

A or Wa. (b) Hypothetical plot of two ‘habitats’ and scaling rule similar to that predicted by Equation 14; on the A, Ma plane. Because both span the same A/Ma the species differ mainly in how the RA is divided range, the two habitats will contain similar body sizes. But among spawning batches, and of course, egg size. all the reproductive efforts will be high in and low in . We agree with W–R that large-bodied species usually have long lifespans and much reproductive 1=3 rays correspond to bigger La Wa . Equations 14a allocation when compared with small-bodied spe- and 14b together imply that reproductive effort, RE, cies and that offspring size relates to many identified is Ma independent of A or Wa. Thus, lines of equal features of the parental care/larval development RE are just vertical lines (parallel to the A axis); environment. We believe that much of this high Ma means high RE, and low Ma means low RE. between-species variation can be best understood Fig. 2a is a schematic of this classification scheme. in the context of Fig. 2: production rates (A) in the

Different environments/habitats/foraging modes face of mortality rates (M) determine Wa, and then, may impose different ranges for A and M. Fig. 2b the RA divided by individual offspring size sets the shows how this scheme might work between yearly offspring number. Species only grow to large habitats to predict life histories. In this hypothetical body size if external mortality rates are low; hence, example, Habitat 1 has low A and low M, while their lifespan is long. All the variables in this study Habitat 2 has high A and high M. Both habitats will can be treated as ‘average annual’ values, and thus,

2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 221 Natural mortality in fish life-history evolution E L Charnov et al.

they smooth over the annual cycle of production Cury, P. and Pauly, D. (2000) Patterns and propensities in and mortality. Perhaps it is not necessary to suggest reproduction and growth of marine fishes. Ecological that small-bodied species are selected for fast intrin- Research 15, 101–106. sic rates of increase, and maybe the non-growing Denney, N.H., Jennings, S. and Reynolds, J.D. (2002) Life history correlates of maximum population growth rates population assumption, mandating R0 as a fitness in marine fishes. Proceedings of the Royal Society B– measure, is adequate for all body sizes. Biological Sciences 269, 2229–2237. Downhower, J.F. and Charnov, E.L. (1998) A resource Acknowledgements range invariance rule for optimal offspring size predicts patterns of variability in parental phenotypes. Proceed- Henrik Gislason was supported by grants from the ings of the National Academy of Sciences USA 95, 6208– Danish Research Council to the research network 6211. Fishnet and by the Otto Mønsted Foundation. ELC Fleming, I.A. and Gross, M.R. (1990) Latitudinal clines – a thanks Kirk Winemiller for a very useful discussion. trade-off between egg number and size in Pacific salmon. 71, 1–11. Freedman, J.A. and Noakes, D.L.G. (2002) Why are there References no really big bony fishes? A point-of-view on maximum body size in and elasmobranchs. Reviews in Fish Andersen, K.H. and Beyer, J.E. (2006) Asymptotic size Biology and Fisheries 12, 403–416. determines species in the marine size spec- Gislason, H., Pope, J.G., Rice, J.C. and Daan, N. (2008) trum. American Naturalist, 168, 54–61. Coexistence in North Sea fish communities: implications Andersen, K.H., Farnsworth, K.D., Pedersen, M., Gislason, for growth and natural mortality. ICES Journal of Marine H. and Beyer, J.E. (2009) How community ecology links Science 65, 514–530. natural mortality, growth, and production of fish Gislason, H., Daan, N., Rice, J.C. and Pope, J.G. (2010) populations. ICES Journal of Marine Science, 66, 1978– Size, growth, temperature, and the natural mortality of 1984. marine fish. Fish and Fisheries 11, 149–158. Beverton, R.J.H. and Holt, S.J. (1957) On the dynamics of Griffiths, D. and Harrod, C. (2007) Natural mortality, exploited fish populations. Ministry of Agriculture, growth parameters, and environmental temperature in Fisheries and Food. Fishery Investigations, London, 2, fishes revisited. Canadian Journal of Fisheries and Aquatic 19–533. Sciences 64, 249–255. Beverton, R.J.H. and Holt, S.J. (1959) A review of the Gunderson, D.R. (1997) Trade-off between reproductive lifespans and mortality rates of fish in nature, and their effort and adult survival in oviparous and viviparous relation to growth and other physiological characteris- fish. Canadian Journal of Fisheries and Aquatic Systems 54, tics. CIBA Foundation Colloquia on Ageing 5, 142–180. 990–998. Charnov, E.L. (1979) Natural selection and sex change in Lorenzen, K. (1996) The relationship between body weight Pandalid shrimp: test of a life history theory. American and natural mortality in juvenile and adult fish: a Naturalist 113, 715–734. comparison of natural and aquaculture. Charnov, E.L. (1993) Life History Invariants: Some Explo- Journal of Fish Biology 49, 627–647. rations of Symmetry in . McGurk, M.D. (1986) Natural mortality of marine pelagic University Press, Oxford, UK. fish eggs and larvae: role of spatial patchiness. Marine Charnov, E.L. (1997) Trade-off-invariant rules for evolu- Ecology Progress Series 34, 227–242. tionarily stable life histories. Nature 387, 393–394. Metz, J.A.J., Mylius, S.D. and Diekmann, O. (2008) What Charnov, E.L. (2008) Fish growth: Bertalanffy K is does evolution optimize? Evolutionary Ecology Research proportional to reproductive effort. Environmental Biol- 10, 629–654. ogy of 83, 185–187. Pauly, D. (1980) On the interrelationships between natural Charnov, E.L. (2009) Optimal (plastic) life histories in mortality, growth parameters and mean environmental growing versus non-growing stable populations. Evolu- temperature in 175 fish stocks. Journal du Conseil Interna- tionary Ecology Research 11, 983–987. tional pour l’Exploration de la Mer 39, 175–192. Charnov, E.L. and Gillooly, J.F. (2004) Size and temper- Pauly, D. (1981) The relationship between gill surface and ature in the evolution of fish life histories. Integrative and growth performance in fish: a generalization of von Comparative Biology 44, 494–497. Bertalanffy’s theory of growth. Berichte der Deutschen Charnov, E.L., Warne, R. and Moses, M. (2007) Lifetime Wissenschaftlichen Kommission fu¨r Meeresforschung 28, reproductive effort. American Naturalist 170, E129–E142. 251–282. Charnov, E.L., Turner, T.F. and Winemiller, K.O. (2001) Perrin, N. and Sibly, R.M. (1993) Dynamic models of Reproductive constraints and the evolution of life energy allocation and investment. Annual Review of histories with indeterminate growth. Proceedings of the Ecology and 24, 379–410. National Academy of Sciences USA 98, 9460–9464.

222 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 Natural mortality in fish life-history evolution E L Charnov et al.

Pope, J.G., Rice, J.C., Daan, N., Jennings, S. and Gislason, Thygesen, U.H., Farnsworth, K.D., Andersen, K.H. and H. (2006) Modeling an exploited marine fish community Beyer, J.E. (2005) How optimal life history changes with 15 parameters – results from a simple size-based with the community size-spectrum. Proceedings of model. ICES Journal of Marine Science, 63, 1029–1044. the Royal Society B–Biological Sciences 272, 1323– Quince, C., Abrams, P.A., Shuter, B.J. and Lester, N.P. 1331. (2008) Biphasic growth in fish 1: theoretical founda- Winemiller, K.O. and Rose, K.A. (1992) Patterns in life- tions. Journal of Theoretical Biology 254, 197–206. history diversification in North American fishes: implica- Smith, C.C. and Fretwell, S.D. (1974) The optimal balance tions for population regulation. Canadian Journal of Fish- between size and number of offspring. American Natu- eries and Aquatic Systems 49, 2196–2218. ralist 108, 499–506. Wootton, R.J. (1994) Life histories as sampling devices: Stearns, S.C. (1992) The Evolution of Life Histories. Oxford optimum egg size in pelagic fishes. Journal of Fish Biology University Press, Oxford. 45, 1067–1077.

Appendix 1: Some useful properties of the We can use Equation A2 to transform W¥ to L¥ : Bertalanffy growth equation A 1 K ¼ L1 The Bertalanffy growth (BG) equation is the most 3 a1=3 1 ðA5Þ widely used descriptor of body size growth for A 1 log K ¼ log log a log L fish (and other indeterminate growers). It’s usual e e 3 3 e e 1 integral form (two parameters) is where L = )KÆX A, of course, is estimated as A ¼ 3 K W1=3.We L¥(1 ) e ) length, L¥ = asymptotic length, X is 1 age and K is the growth coefficient. In weight (W), can estimate A from the K, L¥ data if we know the )KÆX 3 shape coefficient of Equation A2. the equation is W = W¥(1 ) e ) . In this appendix, we review and point out some of the What Is A in the BG Equation? properties of the BG equation, beginning with the differential equation form for weight (W). We As A of dW/dt is a key parameter, it seems begin with dW/dt, as production is a mass-based worthwhile to ask what it means. In von Berta- process, even if fishery scientists usually work in lanffy’s original derivation, the A Æ W2/3 term was terms of length. anabolism, the building of new tissue. It is probably The differential equation form of the BG equation reasonably interpreted as reflecting the intake of is as follows: nutrition (food), which is assumed to scale with W⅔. The BW term is catabolism, the breakdown of dW ¼ A W2=3 B W ðA1Þ tissue (but B Æ W also must include reproductive dt allocation; Charnov 2008). While fisheries scientists no longer accept the simple physiological interpre- Length (L) is related to weight by the rule: tation of von Bertalanffy, it may well be useful to W ¼ a L3: ðA2Þ treat the A Æ W2/3 term as the scaling of new tissue production. The asymptotic weight W is where dW 0, thus ¥ dt ¼ There is another way to interpret the A coefficient. A Rewrite Equation A1 as W1=3 ¼ ; ðA3Þ 1 B dW 2=3 B 1=3 1=3 ¼ A W 1 W ; W1 dt A or L1 ¼ a .

Notice that Equation A2 allows us to write dW but by Equation A3, dt ¼ 2 dL dL dW 1 "# a 3L or ¼ 2 . Now, combining this dt dt dt a3L dW W 1=3 with Equations A1, A2 and A3, we can show that ¼ A W2=3 1 : ðA6Þ dt W dL B B 1 dt ¼ 3 ½¼L1 L KL½1 L ; so that K ¼ 3. This is the differential equation for length. ⅔ Thus, dW is proportional to W at any fixed dt B 1=3 A A 1=3 W/W¥ value with A (and the W/W¥ term) As K ¼ 3 and W1 ¼ B ; K ¼ 3 W1 so that determining the ‘height’ of the ⅔ power function. A log K ¼ log 1=3 log W : ðA4Þ For example, at very small body size, W/W¥ » 0 e e 1 ⅔ 3 and Equation A6 becomes dW/dt » AW ; A is the

2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224 223 Natural mortality in fish life-history evolution E L Charnov et al.

2/3 height of the power function (W ) growth curve uðyÞ R e dy at small W. 1 uðyÞ 0 e dy A interpreted this way is also true for the size at fastest growth. The max of dW is where @ðÞdW=dt ¼ 0, Now, multiply Equation (B2) by: dt @W R which implies from Equation A1 that 1 /ðÞy 0 e dy 2 1=3 R W A B ¼ 0. We combine this with Equation 1 /ðÞy 3 0 e dy A3 for W¥ to show that W/W¥ = 0.296 (which 3 dW to yield: "# = (2/3) ) at the max of dt . Putting this into Equation Z Z 1 euðyÞ 1 A6 shows that dW ¼ A W2=3 at the size of fastest R ¼ SðaÞ bðyÞR dy euðyÞdy dt 3 0 1 euðyÞdy growth; thus, A/3 is the height of this ⅔ power 0 0 0 dW ðB3Þ function for dt at fastest growth. Table A1 shows a dW simple way to estimate dt at fastest growth (W = 0.296 Æ W¥). S(a) is the chance of living to reproduce at age a, while the term in square brackets is simply b, the average rate of production of offspring over the reproductive adult life, and the term in curved Table A1 Max dW dt brackets is simply E(a), the expectation of further life dW 2=3 dW at age a, the average length of the adult lifespan. So, As dt ¼ A=3 W (Equation A7), we have dt ¼ 2=3 1=3 Equation (B3) is really: A=3ð0:296 W1Þ at fastest growth. But K ¼ A=3 W1 dW 2=3 (Equation A4), so that max ¼ð0:293Þ K W1 ¼ dt R0 ¼ SðaÞb EðaÞðB4Þ 0:44 K W1; the maximum growth rate is estimated by

0.44 of the product of K times W¥. Equation (B4) applies to any age-structured life

history; R0 is the simple product of three aggregated terms, each an average. For Equation (B4) to be

Appendix 2: Fitness (R0) is a product used as a fitness measure, the population must not (derivation from Charnov 1997) be growing. This makes R0 » 1 for typical individ- uals, owing to density dependence. But mutant The ‘net reproductive rate,’ R , is defined as R 0 individuals may have their own R „ 1, and it is R ¼ 1 lðxÞbðxÞdx (Equation B1), and calculates 0 0 0 fitness and trade-offs for mutants, which are the average number of daughters produced over a discussed here. Thus, we use R , Equation (B4), as female’s lifespan. (l(x) is the probability of being 0 a fitness measure, with the condition that it must alive at age x; b(x) is the daughters produced at equal unity at the optimum, when mutants are the age x that are alive at independence from mother; same as typical (wild type). b > 0 only if x > a, the age at first birth, measured Evolutionary ecologists often prefer to work with from independence.) Now, write bxðÞ¼byðÞfor a version of Equation A4 where byðÞ¼RyðÞ; R(y)is y ¼ x a and denote l(x) for x > a as l(x)= m0 the mass allocated to reproduction at age y, while S(a) Æ e)u(x)a) = S(a) Æ e)u(y). (Notice that u(y) is m is the size of each offspring. Then Equation B4 zero at y = 0 and is increasing with y; 0 becomes @ð log lðxÞÞ=@y ¼ @uðyÞ=@y,so¶u/¶y is the instantaneous mortality rate at age y.) S(a) is the SðaÞ R0 ¼ R EðaÞðB5Þ chance of living from independence to a. R0 can be m0 written for this general life history as: Z A further common step is to make R(y) body-size 1 uðyÞ dependent. R0 ¼ SðaÞ bðyÞe dy ðB2Þ 0 Notice that if the adult instantaneous mortality

rate (M) is an age-independent constant (Ma), E(a) Recall from the stable age distribution theory that is simply 1/Ma; when combined b ¼ b, this yields the proportion of the breeding lifespan spent bSðaÞ R0 ¼ , Equation 9 in the text. See Metz et al. between ages y and y +dy (the probability density Ma function for the adult ages) is given by: (2008) for a discussion of fitness measures.

224 2012 Blackwell Publishing Ltd, F I S H and F I S H E R I E S , 14, 213–224