'FECUNDITY RELATIONSHIPS, GROWTH ... - ESA Journals

O R I G I NA L A RT I C L E doi:10.1111/j.1558-5646.2010.01112.x

SIZE-FECUNDITY RELATIONSHIPS, GROWTH TRAJECTORIES, AND THE TEMPERATURE-SIZE RULE FOR ECTOTHERMS Jeffrey D. Arendt1,2 1

Department of Biology, University of California at Riverside, Riverside, California 92521-6000 2

E-mail: [email protected]

Received November 21, 2009 Accepted August 10, 2010 Many ectotherms show crossing growth trajectories as a plastic response to rearing temperature. As a result, individuals growing up in cool conditions grow slower, mature later, but are larger at maturation than those growing up in warm conditions. To date, no entirely satisfactory explanation has been found for why this pattern, often called the temperature-size rule, should exist. Previous theoretical models have assumed that size-specific mortality rates were most likely to drive the pattern. Here, I extend one theoretical model to show that variation in size-fecundity relationships may also be important. Plasticity in the size-fecundity relationship has rarely been considered, but a number of studies show that fecundity increases more quickly with size in cold environments than it does in warm environments. The greater increase in fecundity offsets costs of delayed maturation in cold environments, favoring a larger size at maturation. This can explain many cases of crossing growth trajectories, not just in relation to temperature. KEY WORDS:

Adaptation, allometry, fecundity, life-history evolution, models/simulations, phenotypic plasticity.

Body size influences most aspects of biology including physiological performance, predator–prey interactions, life-history characters, ecological patterns, and evolutionary trajectories. Despite the importance of size, we know remarkably little about how a particular body size is achieved, especially the factors that influence how growth trajectories evolve. One might expect selection to usually favor larger size (e.g., Kingsolver and Pfennig 2004), because larger individuals typically have greater fecundity (Roff 2002), are more competitive (e.g., Oddie 2000; French and Smith 2005; Bashey 2008), and are less vulnerable to predators (e.g., Magnhagen and Heibo 2001; Craig et al. 2006) than smaller individuals. However, it takes time to grow large thus increasing the likelihood one will die before reproducing. In addition, it has long been known that early maturation confers a large fitness advantage (Cole 1954). As a consequence, one might expect any circumstance that promotes rapid growth (e.g., abundant resources, warm temperatures) to also result in a large size at maturation, because this minimizes the time costs of becoming large. The ex C

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pected result can be described as nested growth trajectories (sensu Arendt 2007), the slower growth trajectory being nested within the faster growth trajectory (Fig. 1A). There are, however, a number of circumstances that generate crossing growth trajectories in which fast-growers mature early but at a smaller size than slow-growers (Fig. 1B). One situation that may favor crossing trajectories is a highly seasonal life-history (Wiegmann et al. 1997; Angilletta et al. 2004a). For example, fast-growing Atlantic salmon (Salmo salar) initiate seaward migration at the end of their second year but slow-growing fish delay until the following year, probably because of a sizethreshold for surviving the transition to salt water (Okland et al. 1993). Given a third year of growth, slow-growing juveniles are larger at smolting than fast-growing fish (Metcalfe et al. 1988). Another circumstance that may result in crossing growth trajectories is protandry (males mature before females). Faster growth, earlier maturation, and a smaller size for males are known for a variety of butterfly species (e.g., Nylin et al. 1993; Fischer and

C 2010 The Society for the Study of Evolution. 2010 The Author(s). Evolution Evolution 65-1: 43–51

J E F F R E Y D. A R E N D T

Figure 1.

(A) Nested growth trajectories in which the faster growth is always associated with a large body size (dashed line). (B) Crossing growth trajectory in which faster growth results in earlier maturation but at a smaller size. In both cases growth is assumed to cease at maturation.

Fiedler 2000). However, the best known situation that generates crossing growth trajectories is temperature (i.e., thermal reaction norms). Atkinson (1994) found that individuals reared at warmer temperatures grew and developed faster but matured at a smaller size than those reared at cooler temperatures in just over 80% of studies reviewed. In the other cases, individuals mature at the same size or larger in the warmer environment. Attempts to explain the pattern as a constraint on either growth or development have been unsatisfactory because each constraint appears to be system specific (reviewed in Angilletta and Dunham 2003; Angilletta et al. 2004b) and can be reversed by selection (Kingsolver et al. 2007). Life-history models typically predict nested trajectories but several optimality models do predict crossing trajectories under some circumstances (reviewed by Angilletta 2009; see also Angilletta et al. 2004b). Briefly, one group of models requires that juvenile mortality must increase with temperature. Angilletta et al. (2004b) reviewed empirical data and found that temperature effects on mortality, when they occurred, were far below that needed to favor crossing trajectories. A second group of models predicts crossing trajectories when either growth is less efficient at warm temperatures or growth decreases with size in warm environments but increases in cool environment. Reviewing empirical data, Angilletta and Dunham (2003) rejected the former but find some support for the latter. Thus, most attempts to explain the temperature-size rule are either restricted to a few taxa or are restricted to a set of conditions that may occur only occasionally in nature. When theoretical models fall short of expectation it usually pays to examine the initial assumptions of the models. Most lifehistory theory is based on the “characteristic equation” (Roff 1992, 2002; Stearns 1992) (1) 1 = e−rx l(x)m(x).

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EVOLUTION JANUARY 2011

The first component of the equation, e−rx , describes how a population grows with e being the base of the natural log, r being the population growth constant, and x being either age or size. The next component, l(x), is age (or size) specific survival probability. The final component, m(x), is age (or size) specific fecundity. To date, theory trying to predict growth trajectories has only considered how mortality varies with environmental conditions (Angilletta et al. 2004b). However, age/size-specific fecundity may also vary with conditions. For example, Weetman and Atkinson (2004) showed that two species of waterfleas, Daphnia pulex and D. curvirostris, reared under cool conditions had a large increase in fecundity with size, but when reared under warmer conditions the increase was more modest (see also McCabe and Partridge 1997). Fecundity generally increases with body size in ectotherms (Roff 2002). What matters for predicting growth trajectories is the relative rate of increase. Intuitively, in situations in which there is a relatively large increase in fecundity with size, there will be a greater advantage to continued growth than in situations in which fecundity increases little with size, all else being equal. With regard to temperature, if fecundity increases with size faster under cool conditions than under warm conditions (as in the Daphnia of Weetman and Atkinson 2004), then the benefit in being large at maturation may outweigh the cost of added time needed to reach this size. I show this theoretically by adapting the model of Sibly and Atkinson (1994). Like prior authors, Sibly and Atkinson assumed that the size-fecundity relationship was fixed and the relevant parameter to consider was how the size-mortality relationship varied with environment. By contrast, I begin by assuming a fixed size-mortality relationship and allowing the size fecundity relationship to vary. This alone greatly expands the parameter space in which crossing growth trajectories are predicted to occur. Although the model is described in terms of cold and warm environments, it can be applied when comparing any situation in

S I Z E - F E C U N D I T Y R E L AT I O N S H I P S A N D T H E T E M P E R AT U R E - S I Z E RU L E

which growth rates differ (e.g., between environments, sexes, or species).

The Trade-off Between Growth Rate and Fecundity The model of Sibly and Atkinson (1994) assumes determinate growth (i.e., no growth after maturation) and a fixed neonate size. Simple relationships between two variables can take on three qualitatively distinct forms, concave up, linear, or concave down (Fig. 2A). To illustrate patterns, I initially assume linear relationships. Relevant parameters for equations are listed in Table 1. Sibly and Atkinson start with the assumption that adult size depends upon length of the juvenile stage (tj ) and that this relationship depends upon environmental conditions (i.e., individuals grow faster in one environment than they do in another; Fig. 2B). They also assume that fecundity (n) increases with body size (Wa ), but that this relationship is independent of environment. I relax this assumption, allowing fecundity to increase faster with size in one environment than it does in the other (Fig. 2C). To get crossing growth trajectories, it is necessary that fecundity increases faster in the environment where growth is slower (dashed lines in Fig. 2B, C). To determine the optimal size at maturation, we need to determine how fecundity relates to the length of the juvenile period as translated through the size-fecundity relationship. Assuming a linear relationship, adult size is directly proportional to juvenile period Wa = b1 + b2 × (t j ),

(2)

where b1 is the intercept and b2 is the slope. Similarly, the sizefecundity curve is n = b3 + b4 × (Wa ),

(3)

where b3 is the intercept and b4 is the slope of this line. To relate fecundity to length of the juvenile period we simply substitute equation (2) into (3) n = b3 + b4 × [b1 + b2 (t j )]

(4)

which can be rearranged to give t j = n(1/b4 × b2 ) − [(b3 + b4 × b1 )/b4 × b2 ].

(5)

This means that both the slope (1/b4 × b2 ) and the intercept [(b3 +b4 × b1 )/b4 × b2 ] of this relationship depend upon the slopes found in equations (2) and (3). Because individuals reared in cold environments take longer to mature (have smaller b2 ) than those reared in warm environments, they will have a steeper slope in (5) because b2 is found in the denominator. This relationship is

Figure 2.

Basic relationships assumed in the model. (A) Potential patterns between two variables can take on one of three shapes. (B) Relationships between size (Wa ) and age at maturation (tj = length of juvenile period) typically found for cold and warm rearing conditions, assuming a linear relationship. (C) Size (Wa ) versus fecundity (n) for the same environments, again assuming a linear relationship.


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Table 1.

Parameters used in equations.

Parameter

Biological interpretation

Wa tj ta n x b1 , b2

adult size length of juvenile period length of adult period fecundity ln(n) intercept and slope respectively of relationship between Wa and tj intercept and slope respectively of relationship between n and Wa rate of increase of an allele within a population juvenile mortality rate adult mortality rate reproductive value at birth

b3 , b4 F μj μa V

depicted in Figure 3A (see Fig. 3 of Sibly and Atkinson 1994). However, if individuals reared in a cold environment also have a steeper size-fecundity relationship (larger b4 ), as described for Daphnia by Weetman and Atkinson (2004), this will decrease the difference in slopes (Fig. 3B), and if b4 > b2 can even reverse the lines (Fig. 3C).

Optimal Size Under Spatial Heterogeneity Next we need to determine the point along each trade-off curve depicted in Figure 3 that optimizes fitness. Following Sibly and Atkinson, I replace n with x = ln(n) to simplify the analysis. Fitness is estimated in slightly different ways depending upon whether one is looking at spatial heterogeneity or temporal heterogeneity. For spatial heterogeneity, I assume that dispersal occurs just after fertilization and that offspring remain in their new habitat until death (Kawecki and Stearns 1993; Sibly and Atkinson 1994). In addition, assuming weak selection and that the proportion of alleles reaching a habitat is time invariant, the optimal plastic strategy under spatial heterogeneity is one that maximizes reproductive value at birth within each habitat (see Kawecki and Stearns 1993). The reproductive value at birth (V) is estimated as lifetime reproductive success discounted by e−Ft , where F is the overall rate of increase of an allele and t is age. For the life-cycle assumed here, Sibly and Atkinson (1994) show that this is V =

ne−(μ j +F)t j 1 − e−(μa +F)ta

(6)

where μj and μa are juvenile and adult mortality, respectively. Setting x = ln(n) and rearranging gives x = (μ j + F)t j = ln V 1 − e−(μa +F)ta .

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(7)

Figure 3.

Relationship between length of juvenile period and fe-

cundity under two different rearing conditions. (A) For a given juvenile period fecundity will be lower in the cool environment because adult size is smaller, assuming a fixed size-fecundity relationship. However, if the size-fecundity relationship is steeper in the cool environment, the difference may be narrower (panel B) are even reversed (panel C).


Solving for tj yields tj =

ln[V (1 − e−(μa +F)ta )] x − . (μ j + F) (μ j + F)

(8)

This means that on a plot of ln(fecundity) versus length of juvenile period (x vs. tj ) the V isoclines are straight lines with a slope of 1/(μj + F) with fitness increasing as the intercept decreases (to the bottom right in Fig. 4). Assuming a fixed size-fecundity relationship and replotting the trade-off curves in Figure 3 on these axes, the optimal plastic strategy is found where the slope of the trade-off curve equals that of the V isoclines (Fig. 4A), that is where dtj /dx = 1/(μj + F). In this situation, this value will always result in a larger x for the warm environment curve. Because the size-fecundity curve is fixed, a larger x can only be achieved by having a larger Wa (i.e., larger adult body size), and thus the model predicts that a larger size in the cold environment is never optimal. The advantage of Sibly and Atkinson’s approach is that this result holds no matter what growth curves or size-fecundity curves are used as long as the x by tj trade-off curves are concave up. Sibly and Atkinson argued that this was necessarily the case as either a linear or concave down curve will result in an optimal x either at 0 or infinity. I now extend the model by relaxing the assumption of a fixed size-fecundity curve. As noted above, it is now possible to have a trade-off curve for the cold environment that is both shallower and to the right of the warm-environment curve (Fig. 3B, C). This means the optimal x will be larger for the cold environment (Fig. 4B). However in this case a larger x does not necessarily mean a larger Wa (Fig. 2). It is relatively easy to calculate when the optimal Wa for the cold environment will be greater than

Figure 4.

the optimal Wa for the warm environment in our linear example. Because x = ln (n), this means n = ex . Equations 3 and 5 become Wa = (ex − b3 )/b4 and tj = (ex − b3 − b1 × b4 )/(b2 × b4 ), respectively. The optimal x is dtj /dx = (ex )/(b2 × b4 ) = 1/(μj + F) or ex = (b2 × b4 )/(μj + F). The optimal body size is thus b 2 × b4 − b3 (μ j + F) . Wa = b4

(9)

All we need to know is find when Wa cold > Wa warm, or b2 warm × b4 warm b2 cold × b4 cold − b3 cold − b3 warm (μ j + F) (μ j + F) > . b4 cold b4 warm (10) A little algebra yields b3 cold b2 war m − b2 cold b3 war m − > . b4 war m b4 cold (μ j + F)

(11)

That is, the optimal body size will be larger in the cold environment if the ratio b3/b4 for the cold environment is smaller than that in the warm environment by a value that exceeds the difference in growth rates (b2 warm-b2 cold) scaled by μj + F. The only datasets I have found that measure both growth rates and size-fecundity patterns at multiple temperatures are for Daphnia. These data cannot be used to test the model because Daphnia have indeterminate growth and the model assumes determinate growth, but they can be used to illustrate the calculations needed for the model. Weetman and Atkinson (2004) report mass

Determination of optimal fecundity under spatial heterogeneity. Fitness (V) isoclines are depicted by dotted lines with fitness

increase to the lower right. Each circle represents the point where slope of trade-off curve equals the slope of the V-isoclines and gives the optimal fecundity x. (A) For a fixed size-fecundity relationship, the trade-off curve for the warm environment is always shallower and to the right resulting in a larger optimal x and hence larger Wa . (B) If the size-fecundity relationship is steeper in the cool environment, the trade-off curve may be shallower and to the right of the warm trade-off curve. This means the optimal x is greater, but to determine optimal body size one must solve for the inequality given in the text.


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data for Daphnia, but growth in mass follows an exponential function and the mass-fecundity relationship is a power function. It is possible to extend this model to include nonlinear relationships (see below), but as growth in length is linear for Daphnia (Ranta et al. 1993), I illustrate equation (11) with data from Scheiner and Yampolsky (1998) who reared six clones of D. pulex at 17◦ C and 23◦ C. Using values from their Table 3, I was able to calculate average growth trajectories (length mature = initial length + b2 × age mature) of 1.7067 mm = 0.6173 mm + 0.0064 × 169.85 h at 17◦ C and 1.6283 mm = 0.6067 mm + 0.0093 × 109.32 h at 23◦ C. Scheiner and Yampolsky also provide size of the first brood. This gives a linear size-fecundity relationship (n = b3 + b4 × length) of 5.350 young = −7.9183 + 7.7744 × 1.7067 mm at 17◦ C and 4.367 young = −4.4752 + 5.4300 × 1.6283 mm at 23◦ C. Equation (11) thus yields 0.1943 > 0.0029/(μj + F). Because juvenile mortality (μj ) lies between zero and one and F = 1 for a stable population, the inequality holds. In fact, growth (b2 ) would have to be almost 20 times faster at 23◦ C than at 17◦ C to counter the size-fecundity effects. Thus, although this group does not fit the assumptions of the model, it is clear that biologically realistic values can easily account for the effects of temperature on size at maturation. To summarize the results for spatial heterogeneity, when the size-fecundity relationship is fixed, plasticity should always favor a larger body size in the environment where growth and development are faster (in this case, the warmer environment). However, it is possible to get a larger optimal body size in the cold environment if the costs of slower growth/development are offset by greater size-specific fecundity.

Optimal Size for Temporal Heterogeneity and Isolated Populations Sibly and Atkinson (1994) next considered the situation in which the environment varies temporally such that an individual spends its entire life within one habitat, but that habitat varies with time. Mathematically, this solution also predicts the optimal phenotype when comparing isolated populations (as with latitudinal clines in body size). Under spatial heterogeneity, the F used to calculate V isoclines was the same for all environments, because it represented overall contribution of an allele to the global population. Under temporal heterogeneity, F will vary among environments. Fitness of a given allele is thus 1 = ne−(μ j +F)t j + e−(μa +F)ta .

(12)

This can be rearranged as ln 1 − e−(μa −F)ta x tj = − . (μ j + F) (μ j + F)

(13)

As was the case for spatial heterogeneity, the fitness contours are straight lines with slope 1/(μj + F). However, because F varies, the slopes of these lines are no longer parallel as they were in Figure 4. Rather, as F gets larger, the fitness isoclines become less steep (Fig. 5). Under the assumption of a fixed size-fecundity relationship, this means the optimal x (and hence optimal Wa ) is larger in the environment with the shallower trade-off curve which, again, the warmer environment (Fig. 5A). However, unlike the case for spatial heterogeneity, under temporal heterogeneity

Figure 5.

Determination of optimal fecundity under temporal heterogeneity. In this case, fitness isoclines have different slopes but still increase to the lower right. (A) The shallower curve will always have a larger optimal x. (B) The lower curve may have a smaller optimal x. Under the assumption of fixed size-fecundity relationships, the warm curve will always be lower and usually less steep. With different

size-fecundity relationships, the cold curve may take this position. As with spatial heterogeneity, whether a larger x translates into a larger Wa depends upon the actual relationships between size and fecundity.

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the lower trade-off curve (which under these assumptions also occurs in the warm environment) can have a smaller optimal x (and Wa ) provided the curve is not too shallow compared to the higher curve (Fig. 5B). This is the one situation that generates crossing growth trajectories in the original model of Sibly and Atkinson. When we relax the assumption of a fixed size-fecundity curve, it is possible for the cold trade-off curve to be shallower or lower than the warm trade-off curve giving it a larger optimal x. Again, a larger optimal x does not necessarily mean a larger optimal Wa when size fecundity curves vary, so we have to calculate Wa separately for each environment. Conditions for a larger optimal Wa in the cold can be determined in the same manner as with spatial heterogeneity. The difference now is that F differs between environments: b2 warm b2 cold b3 warm b3 cold − > − . (14) b4 warm b4 cold (μ j + F warm) (μ j + F cold) The relationship is similar to that in equation (11). However, the trade-off curve with the larger optimal x will also necessarily have a larger F, this means F cold > F warm whenever Wa cold > Wa warm. As a result, the differences in the b3 /b4 ratio must be even greater than was the case for spatial heterogeneity. In summary, under the original assumptions of Sibly and Atkinson (1994), where size fecundity relationships do not depend upon the environment, one cannot get a larger optimal adult body size for the cold environment under spatial heterogeneity. However, under temporal heterogeneity or isolated populations, one can get a larger optimal adult size in the cold at least for some parameter values. The pattern is rather different when we allow the size-fecundity curve to vary with environment. If the size-fecundity curve is steeper in the environment where growth is slower (i.e., the cold environment), it is relatively easy to get a larger optimal size under spatial heterogeneity. In addition, one can get this pattern for a wider variety of trade-off curves under temporal heterogeneity than when the size-fecundity curve is fixed.

Extensions of the Model I have outlined the basic method for determining when a larger body size will be optimal in a cool environment. However, many organisms show a nonlinear relationship between size and fecundity. For example, salamanders (e.g., Tilley 1968), many fish (e.g., Fleming and Gross 1990), and many plants (e.g., Sugiyama and Bazzaz 1994) show a power relationship between size and fecundity. This model is easily extended to such a situation by simply replacing equation (3) with n = b3 W b4 a . For environmental heterogeneity, the inequality for optimal body size, Wa cold >

Wa warm, then becomes b4 cold × b2 cold × (b3 cold) /b4 cold (μ j + F cold) 1

>

b4 warm × b2 warm × (b3 warm)1/b4 warm (μ j + F warm)

(15)

Equation (15) holds for temporal heterogeneity. Under spatial heterogeneity F cold = F warm and the inequality reduces to the numerators. Thus far we have not even considered the influence of mortality rates on optimal body size. For the case of spatial heterogeneity, adult mortality (μa ) only alters the intercept of the V isoclines (eq. 8) and thus has no effect on optimal body size. Changes in juvenile mortality (μj ) alter the slope of the of the V isoclines. Because fitness is averaged for all environments, μj is the same for all life-history strategies if mortality is temperatureindependent. Greater juvenile mortality decreases the slope in equation (8), leading to an overall decrease in the right-hand side of the inequality of equation (11). As a result the difference, b3 warm/b4 warm − b3 cold/b4 cold, can also decrease making it more likely the optimal body size will be greater in the cold environment when juvenile mortality increases. For temporal heterogeneity (or distinct populations), juvenile mortality will differ in the different environments. Incorporating a greater juvenile mortality for the cold environment into equation (14) means the difference on the right-hand side gets larger. The difference on the left must also get larger meaning either b3 cold must be even smaller than b3 warm or else b4 cold must be even greater than b4 warm. This makes it less likely that a larger size in the cold environment will be favored, which makes intuitive sense because as juvenile mortality increases in this environment then the risk of death before reproducing increases. This would have to be compensated for by an even greater size advantage of delayed maturation. For temporal heterogeneity, adult mortality also affects the optimal size at maturation. An increase in μa results in a decrease in F, which leads to an increase in the slope of the F isoclines. This in turn translates into a larger optimal x and optimal Wa . When comparing two environments, one need to simply consider equation (14) again. If adult mortality is greater in the cold environment, this will decrease Fcold relative to Fwarm making the difference on the left of the inequality in (14) smaller. Adult mortality therefore has the opposite effect on size at maturation as does juvenile mortality. To summarize these extensions, almost any relationship between size and fecundity or size and length of juvenile period can be incorporated into this model. All one need do is find the inequality for optimal Wa in the two environments being compared. The effect of increased juvenile mortality in the spatial heterogeneity model makes it more likely that larger adult size will be optimal in the cold environment than the warm environment. The effect of juvenile mortality in the temporal heterogeneity model


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depends upon whether mortality is greater in the warm or the cold environment. As juvenile mortality increases the optimal adult size will decrease in that environment, as adult mortality increases the optimal adult size increases in that environment.

Discussion Incorporating variation in the size-fecundity relationship into theory changes predictions about how growth plasticity influences the optimal size and age at maturation. When the size fecundity curve has a steeper slope in a slow-growth environment, this can compensate for the costs of slow growth. This phenomenon may help explain the common occurrence for ectotherms having a larger size at maturation in cool conditions compared to warm conditions. This model greatly extends the parameter space over which such a pattern was found to be adaptive by Sibly and Atkinson (1994). In their model, a larger size in the cool environment only occurred with temporal heterogeneity (or discrete populations) and when the elevation of the x by tj curve was much higher in the cool than in the warm environment. Here, I show that when fecundity increases more with size in the cold environment than it does in the warm environment, a larger adult size may be adaptive under a much wider parameter space including spatial heterogeneity. Unfortunately, size-fecundity curves have rarely been measured for a single species in multiple environments (e.g., Brown and Shine 2007), so we do not know how often this assumption holds. There are other circumstances in which differences in sizespecific fecundity are known to vary. For example, the importance of size on reproduction often differs between the sexes. Male reproductive success depends upon access to females and female reproductive success depends upon the number of eggs she can produce. In the absence of male–male competition, fecundity usually increases little with size for males but greatly with size for females. It has been suggested that this can explain the crossing growth trajectories of some protandrous butterflies in which males grow faster as larvae but mature earlier and smaller than females (Nylin et al. 1993; Fischer and Fiedler 2000). Crossing growth-trajectories due to sexual differences in size fecundity may explain other cases of sexual-size dimorphism, although differences in age at maturation can also explain this pattern. At least one well-documented case of size-dimorphism due to crossing growth trajectories is found in Homo sapiens where females initiate the adolescent growth spurt earlier than males, but are smaller than males at maturation (e.g., Tanner et al. 1976). Changes in fecundity cannot account for all cases of crossing growth trajectories. For example, when Atlantic salmon migrate out to see the fastest growing juveniles (the upper-modal group) migrate a year earlier and at a smaller size than do slower growing juveniles (the lower modal group). The differences in these

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strategies cannot depend upon fecundity rates because they occur in juveniles. Instead, it is mortality rate that likely drives this pattern because salmon must reach a minimum size before they are capable of making the transition from fresh to salt water (Okland et al. 1993). In this case, the high mortality rate associated with the fresh to salt water transition is almost certainly high enough to fit the predictions of previous models (Stearns and Koella 1986; Sibly and Atkinson 1994). As Atkinson (1996) and Angilletta et al. (2004b) have argued, no single explanation will explain all cases of the temperature-size rule or its exceptions. We know that the relationship between size and fecundity varies among species (e.g., Tilley 1968; Wootton 1979; Aarssen and Taylor 1992) and among populations within species (e.g., Pinhorn 1984; France 1992; McGurk 2000). Only a few studies explicitly looked for plasticity in this relationship (e.g., Clauss and Aarssen 1994; Weetman and Atkinson 2004; Brown and Shine 2007). The model presented here shows that incorporating this variation into life-history theory can greatly modify our expectations for optimal size and age at maturation as well as the shapes of growth trajectories. In some cases, we can make a reasonable guess at how the size-fecundity relationship should vary. For example, a shallow size-fecundity slope seems likely under low resource conditions simply because all individuals, regardless of size, will have few resources to devote to reproduction. In addition, many species will produce larger offspring under low resource conditions (e.g., Gliwicz and Guisande 1992; Bashey 2008) so a shallow slope is also predicted based on a trade-off between offspring size and number (Smith and Fretwell 1974). Many ectotherms produce larger eggs under cool conditions (e.g., Kaplan 1987; Azevedo et al. 1996; Fox and Czesak 2000), so we might also expect shallower size-fecundity curves in comparison with warm conditions. However, larger offspring does not always result in fewer offspring (Fox and Czesak 2000; Weetman and Atkinson 2004). Aside from two datasets on Daphnia (Scheiner and Yampolsky 1998; Weetman and Atkinson 2004), there is also evidence that Drosophila melanogaster have a shallower sizefecundity slope at warm temperatures than at cool temperatures (McCabe and Partridge 1997). We are currently in need of good empirical studies measuring the effects of various rearing conditions, especially temperature, on growth trajectories as well as the within population relationships between body size, offspring size, and fecundity in general. ACKNOWLEDGMENTS J. Dudycha, S. Auer, and two anonymous reviewers provided many helpful comments on a previous version of this manuscript. Thanks to Weetman and Atkinson for inspiring this model. LITERATURE CITED Aarssen, L. W., and D. R. Taylor. 1992. Fecundity allocation in herbaceous plants. Oikos. 65:225–232.


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Associate Editor: B. Swalla


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