Integrative and Comparative Biology

Integrative and Comparative Biology 2004 44(6):494-497; doi:10.1093/icb/44.6.494
© 2004 by The Society for Integrative and Comparative Biology

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Size and Temperature in the Evolution of Fish Life Histories¹

Eric L. Charnov^2,1 and James F. Gillooly^1
1 Department of Biology, The University of New Mexico, Albuquerque, New Mexico 87131-0001

	SYNOPSIS

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Body size and temperature are the two most important variables affecting nearly all biological rates and times, especially individual growth or production rates. By favoring an optimal maturation age and reproductive allocation, natural selection links individual growth to the mortality schedule. A recent model for evolution of life histories for species with indeterminate growth, which includes most fish, successfully predicts the numeric values of two key dimensionless numbers and the allometry of the average reproductive allocation versus maturation size across species. Here we use this new model to predict the relationships of age-at-maturity, adult mortality and reproductive effort to environmental temperature and maturation size across species. Age-at-maturity, adult mortality and the proportion of the body mass given to reproduction per year are predicted to show ±0.25 power allometries with mass at maturity, and an exponential (Boltzmann) temperature dependence. Temperature is assumed to affect only body size growth, so the temperature linkages of maturation, mortality and reproductive effort are indirect via life history optimization; this is briefly contrasted with the idea that (for example) temperature directly affects mortality.

	INTRODUCTION

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Fitness optimization and game theory (ESS) models have long been used to understand the evolutionary adjustment of life-history parameters in fish. Examples include age-at-sex-change in sequential hermaphrodites (Warner et al., 1975), frequencies of alternative reproductive strategies among males (Gross and Charnov, 1979), egg size (Wootton, 1994; Charnov and Downhower, 1995), age and size at maturity (Roff, 1984; Schaffer and Elson, 1975; Stearns, 1992; Jensen, 1996) and reproductive effort (Reznick, 1983, 1996; Gunderson, 1997). Life-history theorists (e.g., Stearns, 1992) often talk about the "General Life History Problem" by which they usually mean the co-adjustment by natural selection (=fitness optimization) of the age (size)-of-first-breeding and the reproductive allocation thereafter. Indeterminate or continued growth after the initiation of reproduction is thought to result from not-all resources being diverted to reproduction, and the resulting body-size growth curve is due to the individual's productive capacity blended with the reproductive allocation decisions. This paper reviews a recent fitness optimization model for the general life-history problem in fish (Charnov et al., 2001), and then extends the model to make predictions about the relations between body size, environmental temperature, mortality rates, reproductive effort, and age-at-maturity.

	OPTIMAL LIFE HISTORY

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Individual production (growth) prior to the initiation of reproduction is assumed to follow the differential equation proposed by West et al. (2001),

where m is mass and b is the maintenance metabolic rate per existing cell divided by the cost of building a new cell. To add reproduction to this equation, Charnov et al. (2001) suggested that after the onset of reproduction (at age , size m), the quantity c·m mass is given to reproduction per unit of time; so, prior to maturation, growth follows eq. 1, and after maturation, growth follows eq. 1 minus c·m. This is illustrated in Figure 1, where growth follows the hatched area. Charnov et al. (2001) further developed the idea that b and c were linked together in a trade-off: higher c required higher b, so that c/b = q. For non-growing populations, a female's fitness is given by the net reproductive rate (R₀), the average number of offspring produced over a female's life span, or R₀ = L_t·b_t dt, where L_t is the chance of being alive at age t and f_t are the offspring produced at that age. For a fixed offspring (egg) size (I), f_t = c·m_t/I or f_t c·m_t; thus,

We assume (Fig. 2 in Charnov et al., 2001) that the instantaneous mortality rate is high (and density dependent) for small t, but drops to some constant value (Z) prior to feasible ages of maturation. Equation 2 is then written straightforwardly in terms of , survival to age , Z, q, b, a, and body size at age t (m_t). The equation is very complicated and so is the search for the optimal life history, which is assumed to set R₀/ = 0 and R₀/c = 0. Thus, the optimal life history optimizes fitness with respect to age at first reproduction, , and reproductive allocation, c (see Charnov et al., 2001 for the details).

View larger version (20K): [in this window] [in a new window]   FIG. 1. Schematic diagram of the growth model (dm/dt, m = mass). Growth prior to maturation (m) follows the West et al. (2001) production relation (thick line). After maturation, c·m of production is given to reproduction, so growth then follows the thick line minus c·m. Overall growth follows the hatched area. Natural selection is assumed to adjust (or m) and c to maximize R0 (from Charnov et al., 2001) (text eq. 2)

 

View larger version (14K): [in this window] [in a new window]   FIG. 2. The optimal life history sets the value of two dimensionless numbers, ·Z and b/Z, as a function of q, the assumed trade-off between c and b (c = q·b); thus, the optimal life history makes c Z (from Charnov et al., 2001)

 
The optimal life history takes a very simple form, shown in Figure 2. The value of q determines the optimal values of the two dimensionless numbers, ·Z (=C₁) and b/Z (=C₃), thus setting the number b·q/Z = c/Z (=C₂). These optimal dimensionless numbers are independent of the value of a, the coefficient of the first term in the production equation (eq. 1, Fig. 1). ·Z has been estimated to be an invariant value of 2 for several fish data sets (discussed in Charnov, 1993), and Gunderson (1997) has estimated that c is indeed proportional to Z (there called M), so that c/Z 0.6. Remarkably, Figure 2 shows that ·Z near 2 predicts c/Z to be about 0.6, the observed value.

Charnov et al. (2001) also show that if is the average mass given to reproduction per unit of time by the adults, the optimal life history can be expressed as:

where (·) is a dimensionless function of q, ·Z and b/Z. Since these dimensionless numbers are predicted (Fig. 2) to be constants across species, we have a 0.75 scaling of average reproduction allocation () on maturation size (m) for a collection of species with the same (similar) a values. A large data set (139 temperate zone species) showed just such a log linear relation with a high correlation (r = 0.86) and a slope (exponent) not different from the predicted 0.75 (Charnov et al., 2001).

	OPTIMAL RELATIONSHIP BETWEEN , C, Z, AND M

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Equation 1 may be integrated from time zero (size m₀ 0) to time (size m) to give (West et al., 2001):

Since (a/b)⁴ is the asymptotic size implied by Eqn. 1 (m where eq. 1 = 0), we may define the quantity µ as

µ is expected to be an invariant since ·b is an invariant (=C₁·C₃), as shown in Figure 2.

Notice that eqs. 3 and 4 combined with the invariance rules of Figure 2 (for ·Z = C₁, b/Z = C₃ and c/ Z = C₂) imply:

So, , reproductive effort (c), and adult mortality (Z) are all predicted to scale as ±0.25 with mass at maturity (m) for species with the same a value. Since C₁, C₂ and C₃ are predicted numerically (Fig. 2) at the optimal life history (e.g., ·Z = C₁ 2 implies C₂ 0.6 and C₃ 0.15), the intercepts of the scaling lines are predicted if a is known.

	BUT A DEPENDS ON ENVIRONMENTAL TEMPERATURE

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Gillooly et al. (2001, 2002) gave theoretical argument and showed much data on metabolic rates and developmental times supporting the idea that a = A·e^–E/k·T, where T is degrees Kelvin, k is Boltzmann's constant, and E is the average activation energy for the rate-limiting biochemical reactions of metabolism fueling growth (E 0.6 eV). Here A is a proportionality factor expected to be approximately constant within a taxa like bony fish. Some variation in A as been shown in zooplankton (Gillooly et al., 2002) to covary with resource availability (i.e., stoichiometry) of limiting nutrients (e.g., phosphorus).

Notice that if we substitute A·e^–E/k·T for a in Equation 5, in addition to the quarter-power mass dependence, we predict an exponential temperature dependence for , c and Z, even though temperature only directly affects a in this theory. This leads to:

Since the dimensionless numbers (Fig. 2) C₁, C₂ and C₃ are independent of a, they are thus predicted to be independent of temperature. ₁, ₂ and ₃ are the intercepts of the temperature/body size relations for Z, and c: since C₁, C₂ and C₃ are predictable from life-history theory, fitting (discussed below) one of the three relations suffices to predict A, and thus predicts the intercepts for the remaining two relations.

For example, consider eq. 6a for Z. We can rewrite it in one of two ways:

In words, eq. 7a means that the natural log of the size-corrected mortality rate should be linearly related to the temperature function 1/k·T with a slope equal to minus the activation energy (–E), which should be about –0.6 (Gillooly et al., 2001, 2002; Savage et al., 2004). Eq. 7b means the natural log of the temperature-corrected mortality rate should show a –0.25 allometry with size at maturity (m). Eqs. 6b and 6c may be treated similarly. In the following section, we provide a preliminary test of eqn. 7a for the temperature and body-size dependence of Z.

	PAULY'S DATA COMPILATION

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Pauly (1980) compiled data on Z (there called M), mean annual environmental temperature (°C, easily converted to degree Kelvin) and adult maximum size (mass = m_max) for 175 populations (stocks) of 84 species of fish. No data are provided on maturation size, so here we assume that m m_max, known to be approximately true for fish (Beverton and Holt, 1959; Charnov, 1993). Figure 3A shows that the log of the size-corrected mortality (eq. 7a) is indeed linearly related to the temperature function with an estimated slope of –0.45, not too different from the expected –0.6. Figure 3B shows eq. 7b; the log of temperature-corrected (E = 0.45 is used here) mortality scales as –0.23 with the log of body mass. While the correlations are not particularly strong, the slopes are close to those that are theoretically predicted.

View larger version (19K): [in this window] [in a new window]   FIG. 3. (A) The relationship between mass-corrected instantaneous mortality rate and the yearly average environmental temperature function (1/k·T; k = Boltzmann's constant; T = degrees Kelvin). From text eq. 7, we expect a negative, linear relation with a slope near 0.6. (B) The relationship between the temperature-corrected instantaneous mortality rate (E = 0.45, as estimated in Fig. 3A) and body mass, estimated as asymptotic mass (mmax). We expect, by text eq. 7, a linear relation with a slope of –0.25 (if m mmax). Lines fit with type 1 linear regression. Data from Pauly, 1980

 

	DISCUSSION

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
Many biological rates (units of 1/time) such as Z show a temperature/body-size dependence proportional to e^–E/k·T·m^–0.25, while biological times such as show a dependence proportional to e^E/k·T·m^0.25 (e.g., Gillooly et al., 2001, 2002; Savage et al., 2004). The present life-history model predicts the temperature and size dependence of Z, c, b, /m and entirely through a mechanism of life-history optimization. The 0.25-power size dependence follows from the optimization shown in eq. 2, and holds for a collection of species with the same a value. The exponential temperature dependence described by the Boltzmann factor follows only from the effects of temperature on a, the height of the growth curve in eq. 1. Consider the mortality rate Z. The life-history theory allows Z to vary or not vary directly with temperature and still satisfy the temperature- and body-size dependence discussed above. This is because life-history optimization would adjust m to the a at any temperature matched to any value of Z. Thus, Z determines m via natural selection in the face of the growth/production function (eq. 1; Charnov, 1993).

Note, however, some have argued that natural mortality rates are intrinsically determined through properties of organisms controlled by metabolic rate (i.e., free-radical-caused cellular damage). Laboratory data showing that maximum life span in the absence of predation follows a Boltzmann-like temperature dependence appears to support these intrinsic causal factors (Gillooly et al., 2001). Yet, the theory presented here may also predict this temperature dependence of maximum life span. This follows provided the investment in cellular maintenance that sets maximum life span is closely related to average adult life span (Z^–1) in the field. There is, in fact, evidence that supports this relationship. Beverton and Holt (1959) showed that Z^–1 was correlated strongly with field maximum life spans in fish, while Charnov (1993, 2004) showed the same for mammals. Further research, including better data to estimate the intercepts of the scaling rules of eq. 6, is needed to address more fully whether mortality is somehow internally determined, or can be primarily considered external, as is done in the life-history theory developed here.

	ACKNOWLEDGMENTS

 
We have benefitted from discussions with James H. Brown and Donald Gunderson. ELC was supported by NSF. JFG was supported by a Packard Interdisciplinary Science Grant.

	FOOTNOTES

 
¹ From the Symposium Evolution of Thermal Reaction Norms for Growth Rate and Body Size in Ectotherms presented at the Annual Meeting of the Society for Integrative and Comparative Biology, 5– 9 January 2004, at New Orleans, Louisiana.

² E-mail: rlc{at}unm.edu

	References

TOP SYNOPSIS INTRODUCTION OPTIMAL LIFE HISTORY OPTIMAL RELATIONSHIP BETWEEN... BUT A DEPENDS ON... PAULY'S DATA COMPILATION DISCUSSION References

 
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Charnov, E. L. 1993. Life history invariants. Oxford University Press, Oxford.

Charnov, E. L., and J. F. Downhower. 1995. A trade-off-invariant life-history rule for optimal offspring size. Nature, 376:418-419.[CrossRef][Medline]

Charnov, E. L., T. F. Turner, and K. O. Winemiller. 2001. Reproductive constraints and the evolution of life histories with indeterminant growth. Proc. Natl. Acad. Sci. U.S.A, 98:9460-9464.[Abstract/Free Full Text]

Charnov, E. L. 2004. The optimal balance between growth rate and survival in mammals. Evol. Ecol. Res, 6:307-313.

Gillooly, J. F., J. H. Brown, G. B. West, V. M. Savage, and E. L. Charnov. 2001. Effects of size and temperature on metabolic rate. Science, 293:2248-2251.[Abstract/Free Full Text]

Gillooly, J. F., E. L. Charnov, G. B. West, V. M. Savage, and J. H. Brown. 2002. Effects of size and temperature on developmental time. Nature, 417:70-73.[CrossRef][Medline]

Gross, M. R., and E. L. Charnov. 1979. Alternative male life histories in bluegill sunfish. Proc. Natl. Acad. Sci. U.S.A, 77:6937-6940.

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Pauly, D. 1980. On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. J. Cons. Int. Explor. Mer, 39:175-192.

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Reznick, D. 1996. Life history evolution in guppies: A model system for the empirical study of adaptation. Neth. J. Zool, 46:172-190.

Roff, D. A. 1984. The evolution of life history parameters in teleosts. Can. J. Fish. Aquat. Sci, 41:989-1000.

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