Life-History Evolution in Uncertain Environments: Bet Hedging in Time

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1 vol. 168, no. 3 the american naturalist september 2006 Life-History Evolution in Uncertain Environments: Bet Hedging in Time Henry M. Wilbur * and Volker H. W. Rudolf Department of Biology, University of Virginia, Charlottesville, Virginia Submitted July 19, 2005; Accepted May 9, 2006; Electronically published July 28, 2006 abstract: Many vertebrates, forest herbs, and trees exhibit both variable age at maturity and iteroparity as adaptations to uncertain environments. We analyze a stochastic model that combines these two life-history adaptations with density-dependent fertility. Results for a model with only iteroparity are consistent with previous work; environmental uncertainty favors adult survival over juvenile survival. This holds true even if there is a moderately strong convex trade-off between adult survival and fecundity, but the direction of selection can depend on which life-history trait is considered a random variable. A life history with only developmental delay favors juvenile survival in uncertain environments, consistent with previous models of seed banks. When both developmental delay and iteroparity are included in the model, both adaptations are favored in uncertain environments. Our simulations show that selection is not necessarily a runaway process in which either developmental delay or iteroparity is favored, as recently proposed by Tuljapurkar and Wiener, but rather that selection can favor both mechanisms. Invasion analysis shows that selective pressure on life-history delays increases as environmental variation increases. Reproductive delay and adult survival can be either adaptations or constraints. Naturalhistory studies that estimate model parameters can resolve this uncertainty. Keywords: complex life cycles, delayed maturity, iteroparity, lifehistory evolution, polyphenism, random environment. Empirical and theoretical studies demonstrate that environmental uncertainty can strongly influence the evolution of life histories. Organisms adapt their life histories to temporally uncertain environments with life-history delays, such as seed banks, variable age at maturity, and * [email protected]. [email protected]. Am. Nat Vol. 168, pp by The University of Chicago /2006/ $ All rights reserved. iteroparity, and they adapt to spatially uncertain environments with dispersal. Such strategies were called spreading of risk by den Boer (1968), escape in time and space by Janzen (1971), and bet hedging by Slatkin (1974). Distributing reproduction in time has been visited by many studies since Cole (1954) coined the terms semelparity and iteroparity as he posed the paradox that iteroparity should not evolve in a constant environment. Much of the literature since then considers variable time of germination resulting in seed banks (developmental delay) or the trade-off between current reproduction and adult survival in iteroparous species. Many vertebrates, forest herbs, and trees exhibit both variable age at maturity and iteroparity, and most populations also experience density-dependent population regulation. However, to the best of our knowledge, no previous study has combined all three components in one model to analyze the evolution of developmental delay and iteroparity. Cole (1954) viewed iteroparity as a paradox because semelparity, a single bout of reproduction that exhausts the parent, should always be favored in a constant environment by the compounding nature of exponential growth. Cole s paradox has been resolved by numerous models demonstrating that variation in reproductive success favors iteroparity (Murphy 1968; Gadgil and Bossert 1970; Schaffer 1974; Wilbur et al. 1974; Bell 1976, 1980; Goodman 1984; Bulmer 1985; Orzack 1985; Bradshaw 1986; Roerdink 1987; Orzack and Tuljapurkar 1989, 2001; Fox 1993; Charlesworth 1994; Cooch and Ricklefs 1994; Erikstad et al. 1998; Benton and Grant 1999; Brommer et al. 2000; Ranta et al. 2000a, 2000b, 2002). Early models of semelparity and iteroparity suggested the rule of thumb that variation in adult survival favors semelparity with variable age at maturity, while variation in juvenile survival favors iteroparity, but some models predict mixed strategies (Haccou and Iwasa 1995; Sasaki and Ellner 1995, 1997; Sibly 1995, 1996; Ellner and Sasaki 1996; Van Dooren and Metz 1998; Neubert and Caswell 2000; Tuljapurkar and Wiener 2000; Orzack and Tuljapurkar 2001). Cohen s (1966, 1968) analysis of seed banks (Harper 1977) is the classic study of developmental delay. Most

2 Bet-Hedging in Uncertain Environments 399 Figure 1: Life cycle graph of the full model as represented in equation (4). The state variables are J p juveniles and A p adults. Parameters are sj p juvenile survival, sa p adult survival, b p probability of develop- mental delay, and F p fertility, which we expand to CAtexp ( aa t)sm, where C is clutch size, a is the strength of density dependence in the larval stage, and s M is density-independent survival from metamorphosis to 1 year of age. models of seed banks (Templeton and Levin 1979; Thompson and Grime 1979; MacDonald and Watkinson 1981; Bulmer 1984; Ellner 1985a, 1985b, 1987; Pacala 1986; Roerdink 1987; Tuljapurkar 1989, 1990; Kalisz and McPeek 1992, 1993; Charlesworth 1994) have followed Cohen s assumption of semelparity with variable age at maturity because of variation in the timing of germination or length of diapause of resting eggs of zooplankton (Tuljapurkar and Istock 1993; Ellner and Hairston 1994; Cáceras 1997; Ellner et al. 1998). Ellner et al. (1998) show that if the diapause and nondiapause options have variable payoffs (i.e., reproductive output), then the optimal life history is more sensitive to variation in the payoff for diapause and its covariation with the payoff for nondiapause than to the mean payoff for diapause. Similarly, Ellner (1985a) showed that density dependence in the payoff for nondiapause overturns the theory of Cohen (1966) that the amount of diapause should correlate positively with the frequency of bad years. This emphasizes the need to incorporate density dependence when analyzing the evolution of delayed development and iteroparity. Tuljapurkar and Wiener (2000) studied the evolution of developmental and reproductive delays in stochastic environments without density dependence. Benton and Grant (1999) analyzed the population-dynamic consequences of iteroparity with density dependence but without developmental delay. Neubert and Caswell (2000) analyzed both delays but considered only constant environments. We develop a model for a closed population that combines iteroparity, developmental delay, and their interaction in a stochastic environment with density dependence within a realistic framework that can be parameterized for species with two-stage life cycles, such as most amphibians and insects. We use invasion analysis following Benton and Grant (1999) and Caswell et al. (2004) to analyze the effect of increased environmental variation on the evolution of delayed development and iteroparity in populations with density-dependent vital rates. Our results confirm conventional wisdom that life-history delays can be bet hedges against temporal environmental uncertainty, but we expand this conclusion to include density-dependent population growth. Our model predicts that both developmental delay and iteroparity can increase within a life history as environmental stochasticity increases, rather than corroborating the runaway process of life-history adaptation found in the models of Tuljapurkar and Wiener (2000) in which either developmental delay or iteroparity dominates. Model Development We build our model (fig. 1) by progressively adding complexity to a general discrete-time model for densitydependent population dynamics, an N p RN e t t 1 t, (1) in which N t and N t 1 are the sizes of the population at time t and t 1, respectively; R is the finite rate of increase as density approaches 0; and a determines the strength of density dependence. The parameter R determines the stability properties of the model and therefore controls the structure of the bifurcation diagram (May 1973). This model mimics the density dependence of survival in experimental studies of amphibian and insect larvae (reviewed in Wilbur 1997). The general model can be expanded to include two lifehistory stages as developed more fully by Wilbur (1996): A Jt 1 p FA t, t 1 p sj, in which J is the density of juveniles and A is the density of adults. The new parameter s J is the probability of survival from being a juvenile at time t to becoming an adult at time t 1. The parameter F is the number of juveniles at time t 1 per adult at time t. We assume that F is a density-dependent function of the number of eggs laid, so the model becomes J t J p s CA e t t 1 M t, (2a) A t 1 p sj, J t (2b)

3 400 The American Naturalist where C is the mean clutch size per adult, and therefore CA t is number of eggs laid in the habitat of larvae, e t is the density-dependent probability of metamor- phosis, and s M is the expected density-independent survival of juveniles from metamorphosis to 1 year of age. This model is written in matrix notation as J 0 s Ce t t 1 M Jt [ A ] [ s 0 ][ A ] t 1 J t p. Note that the larval stage is assumed to take place within the first year of life, and so it is included in the fertility term. The age of first reproduction is 2 years. Generations are not overlapping in this core model, so we can look forward two time units, J p s Ce t 1 t 2 M A t 1, A t 2 p sj J t 1, and substitute J t 1 from equation (2a) into the equation for adults to obtain A p ss Ce t t 2 J M A. t If we let R p ss J MCand a p ac, this becomes a twoparameter model, a A A p RA e t, t 2 with the same dynamic behavior over two time steps as the standard model (eq. [1]) has over one time step. This is the model for a pure biennial plant without a seed bank. It has the peculiar property of cohort switching (Klinkhammer and de Jong 1983), in which one cohort is composed of juveniles in, say, even-numbered time steps and adults in odd-numbered time steps. The even-year and odd-year adult cohorts are dynamically independent and never interact. We know of no examples of a species that fits this model, but it is nevertheless a useful conceptual step in the development of a realistic model. As for biologically realistic models, cohorts can interact if individuals delay maturity for variable periods of time (polyphenism) or have postreproductive survival leading to iteroparity. Developmental Delay The general, two-age-class model (eq. [2]) modified to include developmental delay (b) without iteroparity is t J p sbj s C t t 1 J t M A t, A p s (1 b)j, t 1 J t where b is the probability that a juvenile that survives one time step remains in the habitat of juveniles and does not reproduce. The matrix representation of this model is J s b s Ce t t 1 J M Jt [ A ] [ s(1 b) 0 ][ A ] t 1 J t p. We assume that survival s J is the same for individuals that delay development and those that do not. This means that delay has a cost imposed by the risk of mortality during each time step that maturity is postponed. There is no benefit to delay except the possible one of bet hedging by having a female s offspring contribute to multiple future year classes. This can be advantageous in variable environments because females may reduce the variance in their expected fitness (Gillespie 1974). This model eliminates cohort switching because cohorts are now linked by the developmental delay of juveniles. This model would be appropriate for a so-called annual with a seed bank. Iteroparity If adults are iteroparous, adult survival (s A ) is added to the basic model (eq. [2]), which now becomes J p s Ce t t 1 M A t, (3a) A p sj sa, (3b) t 1 J t A t where adults have a probability s A of surviving from one time step to the next. An example would be a species that always matures at 2 years of age but has a variable life span as a breeding adult. Unlike in the model for pure biennials, initial cohorts are linked in this model because adults that survive after reproducing contribute to both odd-year and even-year juvenile cohorts. This system of two coupled equations with the four parameters of equation (3) can be represented in matrix notation as J 0 s Ce t t 1 M Jt [ A ] [ s s ][ A ] t 1 J A t p. In some studies, it is assumed that adult survival and reproductive output are negatively correlated, indicating a cost to adult survival. Takada (1995) explored effects of general linear, concave, and convex forms of the trade-off between fecundity and adult survival. We investigated all three shapes but will present results for only the convex

4 Bet-Hedging in Uncertain Environments function C p C m(1 qs A) because it is the form most likely to result in a single value of s A between 0 and 1 that represents an evolutionarily stable strategy (ESS). This function constrains adult survival to 0 s A! 1/ q to keep clutch size positive. Iteroparity and Delayed Maturation The general model with two life-history stages modified to include both iteroparity ( sa 1 0) and delayed maturity ( b 1 0; fig. 1) is which in matrix form is J p sbj s Ce t t 1 J t M A t, (4a) A p s (1 b)j sa, (4b) t 1 J t A t J s b s Ce t t 1 J M Jt [ A ] [ s(1 b) s ][ A ] t 1 J A t p. Here, odd-year and even-year cohorts exchange individuals through both adult survival and delayed maturity, so the population dynamics are those of an age-structured model with multiple juvenile and reproductive ages in which survival rates are constants within the two lifehistory stages. The trade-off between adult survival and clutch size is not included in our analysis of the full model in order to keep it as general as possible. We return to this decision in Discussion. The effects of the individual parameters on the equilibrium population size are 1 ss J MC(1 b)  p ln, ac (1 sb)(1 J s A) s J(1 s A) ss J MC Ĵ p ln. (5) ac (1 sb)(1 J s A) The equilibrium density of adults is increased by larger values of s J, s M, and s A and decreased by larger values of b and a. Increasing C initially increases  to a maximum and then decreases it asymptotically toward 0. Figure 2 shows the effects of developmental delay (b) and adult survival (s A ) on the stable equilibrium size of juveniles and adults. With increasing clutch size, there is a transition from a stable equilibrium point toward an orbital attractor of A t, J t. The life-history delays (b and s A ) stabilize dynamics. Larger values of s J and s M can destabilize the dynamics. The magnitude of the coefficient of density dependence a affects the position of the attractor but not its stability. Figure 2: Deterministic equilibrium population density of juveniles (top) and adults (bottom) in relation to the degree of developmental delay (b) and iteroparity (adult survival, s A ); sj p 1, C p 12, sm p 0.3, a p Introducing a Stochastic Environment It is important to consider the implications of how stochasticity is entered into a model. Our general model (eq. [4]) has six parameters. Three are the survival probabilities of juveniles (s J ), metamorphs (s M ), and adults (s A ); a is the coefficient of density dependence that determines survival during the larval stage ( e t ); b is the probability of de- velopmental delay; and C is clutch size. We chose to focus on the parameter a as a normally distributed random variable to simulate the effect of a stochastic environment because we believe external conditions, such as weather or food availability, often determine the intensity of competition among larvae (Wilbur 1997). Additional simulations showed that without the trade-off between adult survival and clutch size, our qualitative conclusions are robust and do not change when a, C, ors M is chosen as the random variable because they all determine fertility,

5 402 The American Naturalist F. The fertility term in the full model does not include b or the trade-off with s A, and thus variation in F has no direct effect on either s A or b. Our choice of introducing stochasticity into the densitydependent term means, however, that the effects of stochasticity cannot be separated from the direct effect of density dependence. To resolve this issue, we explored the effects of stochasticity in survival from metamorphosis to 1 year of age by making s M, rather than a, in equation (2) a normally distributed random variate truncated at 0 and 1. Assessing Dynamics in Deterministic and Stochastic Environments ESS analysis (Maynard Smith 1982), invasion analysis (Benton and Grant 1996, 1999; Grant 1997), and sensitivity analysis (Caswell 2001; Caswell et al. 2004) are powerful tools for the analysis of life-history evolution. Invasion exponents allow one to determine which mutant phenotypes can invade a population, and sensitivities are equal to the rate of evolutionary change, indicating the strength and direction of the selective pressure on each life-history parameter. The ESS cannot be invaded by any other strategy under the current constraints on adaptation. Positive sensitivities of the invasion exponent for a specific life-history parameter indicate that a mutant with a small increase in the parameter can invade the resident phenotype. Sensitivities in deterministic, density-independent, stage-structured population models are the partial derivative of the dominant eigenvalue l of the population projection matrix with respect to a given parameter (Caswell 2001). In a density-dependent population model, however, the fitness of an individual depends on the life histories of other individuals within the population. Thus, sensitivities cannot be estimated reliably based on changes in population growth rate, l, in density-dependent population models with nonequilibrium dynamics. In such density-dependent populations, the selective pressure is given by the invasion rate of the mutant type indicated by an invasion exponent c, which is the dominant Lya- punov exponent (Metz et al. 1992; Boyce et al. 2006). The sensitivities of the invasion exponent c to changes in the individual life-history parameters thus indicate the strength and direction of the evolution of the focal demographic parameters (Grant 1997; Grant and Benton 2000; Caswell 2001). Sensitivity of c with respect to a specific life-history parameter is given by the expression d exp (c) sj p, dx j where x j indicates any of the individual parameters or elements of the projection matrix and d denotes a small increment (Grant 1997; Grant and Benton 2000). In the density-dependent deterministic case with a stable equilibriumc p log l, the invasion exponent can be calcu- lated by taking the log of the dominant eigenvalue l of the projection matrix of the invading genotype at the equilibrium density of the resident type (Caswell 2001; Caswell et. al 2004), so that c p log l. In a stochastic environ- ment, c can be approximated numerically as d exp (c) exp(c) i exp (c) r exp(c) i 1 sj p p p, dx j (1 d)x j x j dx j with ci p invasion rate of the mutant invader (for cal- culation details, see Grant 1997; Grant and Benton 2000) if the dynamics of the deterministic case are stable. This method is not guaranteed to lead to convergence if the deterministic dynamics are unstable, and other methods must be used (Caswell 2001; Caswell et al. 2004). All our analyses are restricted to regions of the parameter space with stable population dynamics. The two methods give identical sensitivities in the deterministic case, demonstrating that the numerical differentiation method gives unbiased estimates. Caswell et al. (2004) present a general method of invasion analysis that can include different density-dependent functions in two or more life-history stages. Constraining density dependence to a single life-history stage satisfies a special case of Caswell et al. (2004) in which density dependence is a single linear combination of the densities of all life stages. In our case, only the adult age class is reflected in the fertility term. We first examined the behavior of each model in a deterministic environment to understand the effects of each parameter on the equilibrium values of the number of adults and juveniles and the stability of these equilibria. Then we chose parameter combinations for the stochastic analysis that always resulted in a stable equilibrium in the deterministic case. We used Monte Carlo simulations to estimate the sensitivities of c by iterating the resident genotype for 200 generations to avoid transient dynamics before invasion and then calculating the invasion exponents using the next 9,000 time steps. Sensitivities were calculated by averaging the means of 10 sets of 50 independent replicate invasions each to reduce the variation of estimates. We used an increment of 5% in the value of each life-history parameter for the sensitivity analysis because this amount of change yields accurate estimates of sensitivities in populations with stochastic parameters (Grant and Benton 2000). To avoid variation caused by different methods, we used the same numerical differentiation method and increment of change when calculating sensitivities for deterministic

6 Bet-Hedging in Uncertain Environments 403 models, but only one invasion attempt was needed. We also calculated separate sensitivities for b and s J for all life histories with b 1 0. Parameter values were chosen so that populations would be at a stable equilibrium under deterministic conditions for all possible combinations of the entire ranges of b and s A. We set s J equal to unity to allow for direct comparison between values of b and s A (fig. 1). This means that delayed maturity does not have a direct cost of increased mortality, but rather it has an indirect cost of missed opportunities for reproduction. The only possible benefit of b 1 0 in our model results from spreading the age of maturity of a female s offspring from a single bout of reproduction. We analyzed the complete model by investigating the effect of environmental variation on six different life histories, using combinations of three different degrees of developmental delay and iteroparity (values of b and sa p 0, 0.3, or 0.7). Life histories ranged from iteroparous without developmental delay ( b p 0) and with development delay but without adult survival ( sa p 0) to the four combi- nations of delay and adult survival, each having values of 0.3 and 0.7. When investigating the effect of stochasticity, we chose three different standard deviations (j) ofa, the strength of density dependence during the larval period, chosen so that the coefficient of variation (j/mean) was low (5%), moderate (15%), or high (25%). Because of the exponential effect of a, small variations in a can result in considerable changes in population size. We assumed in all models that vital rates are not correlated (except the analysis of a trade-off between clutch size and adult survival) and that there is no environmental autocorrelation. Results Developmental Delay ( sa p 0) The pattern of sensitivities in the deterministic scenario was similar for life histories with b p 0.3 and 0.7 (fig. 3A). Among the elements of the projection matrix, sensitivities were highest for the probability of remaining in the juvenile class (element 1,1; bs J ) and the probability of transition to the adult class (element 2,1; TA p [1 b]s J). Increasing environmental variation resulted in higher sensitivities for the probability of remaining in the juvenile class, while it decreased the sensitivities of F (the reproduction element 1,2 of the projection matrix) and T A (fig. 3B, 3C). Again, this trend was consistent for both life histories. These results demonstrate that increasing environmental variation selects for life histories with a high probability of delayed development. Because reproductive delay, b, and juvenile survival, s J, determine both the probability of remaining in the Figure 3: Deterministic sensitivities (A) and effect of increasing environmental stochasticity of the strength of density dependence (parameter a with coefficients of variation p 5%, 15%, and 25%) on changes in sensitivity with respect to the deterministic sensitivities in two life histories with different degrees of developmental delay (B, C; b p 0.3 inb and 0.7 in C). Parameter values on the abscissa are indexes of matrix elements for the model with only developmental delay: 1,1 p sb J ; 1,2 p smcexp ( t) ; 2,1p s J(1 b). For both values of the focal parameter b, the other parameters were held constant at sj p 1, C p 12, s p 0.3, and a p M juvenile class (bs J ) and the probability of a transition into the adult stage (T A ), we estimated the sensitivities for these parameters separately. There was no selection for increasing the probability of delaying maturity, b, inthe deterministic case (fig. 3A). Note that with s! 1, sen- J

7 404 The American Naturalist sitivities for b would become negative, indicating the disadvantage of delaying reproduction in constant environments. Increasing environmental variation, however, generally increased the sensitivities of b, indicating a positive selection pressure on the probability of delaying development (fig. 3B, 3C). Additional invasion analysis showed that this resulted in an ESS with b 1 0 even if there was a cost in mortality ( s J! 1) when remaining in the juvenile stage. The relative increase in sensitivity was higher for life histories with a low probability of developmental delay than for those with a high probability of developmental delay. The sensitivities of s J were in general high and showed a similar pattern of increase. The relative increase of sensitivities caused by higher environmental variation, however, was much higher in b than in s J for both life histories ( b p 0.3 and 0.7). This in- dicates that increasing environmental uncertainty favors a higher probability of delaying maturity over increasing juvenile survival. Iteroparity ( b p 0) The pattern of the deterministic sensitivities was similar for life histories with sa p 0.3 and 0.7 (fig. 4A). The tran- sition from juvenile to adult, TA p (1 b)s J, and the probability of adult survival, s A, showed similarly high sensitivities, while the sensitivities for fertility, F, and juvenile survival, s J, were low. As expected from the literature addressing Cole s paradox, greater stochastic variation increased sensitivities of s A and decreased sensitivities of F, T A, and s J (fig. 4B, 4C). This trend was consistent for both low and high adult survival, but the increase of the sensitivity of adult survival, s A, was higher for the life history with lower adult survival. This indicates that environmental uncertainty favors adult survival over fecundity and juvenile survival, and thus, stochasticity clearly favors iteroparity. This is consistent with results of Murphy (1968). These results suggest that without any cost to s A, at evolutionary stability, sa p 1, but that when the environ- ment is variable, the selection is even stronger to increase adult survival toward this limit. Clearly, selection will favor greater adult survival until some constraint prevents a further increase. We introduced the convex trade-off between 2 adult survival and clutch size, C p c m(1 qs A), to represent such a constraint. We used the change in the invasion exponent to evaluate the change in selective pressure on adult survival in a stochastic compared to a deterministic environment when there is a trade-off between adult survival and fecundity because the resulting correlation of vital rates does not permit calculation of sensitivity in stochastic, density-dependent models. In this density-dependent model, the evolutionarily stable value Figure 4: Deterministic sensitivities (A) and effect of increasing environmental stochasticity of the strength of density dependence (parameter a with coefficient of variation p 5%, 15%, or 25%; B, C) on changes in sensitivity with respect to the deterministic sensitivities in two life histories with different degrees of iteroparity ( sa p 0.3 in B and 0.7 in C). Parameter indexes on the abscissa are indexes of matrix elements for the model with only iteroparity ( 1,2 p smcexp [ t] ; 2,1p sj; 2,2 p sa). For both values of the focal parameter s A, the other parameters were held constant at s p 1, C p 12, s p 0.3, and a p J M of s A may be less than 1, given a large enough trade-off (q) between adult survival and fecundity. If stochasticity is introduced into the density-dependent term a, then increasing environmental variation decreases the adult survival, s A, further at evolutionary stability when the ESS

8 Bet-Hedging in Uncertain Environments 405 result in a greater adult survival at evolutionary stability (fig. 5). Both Developmental Delay and Iteroparity Deterministic sensitivities of all four matrix elements were identical in life histories where b p s A ; the effect of de- velopmental delay equaled that of adult survival (fig. 6). When b 1 s A, the sensitivities for the probability of re- maining in the juvenile class, bs J (matrix element 1,1), and moving to the adult class, T A (matrix element 2,1), were identical and higher than sensitivities for fertility, F (matrix element 1,2), and adult survival, s A (matrix element 2,2). The reverse was true for life histories with sa 1 b. In all life histories, sensitivities of b were 0, and sensitivities of s J increased with increasing degree of developmental delay and reduced adult survival. In general, sensitivities of adult survival, s A, and the probability of remaining in the juvenile stage, bs J, always increased with increasing environmental variation, while both fertility, F, and the probability of transition to the adult stage, T A, decreased. The magnitudes of changes in sensitivities, however, were dependent on the specific life history. In life histories in which sa p b, sensitivities in- creased to exactly the same extent in both s A and bs J. This increase was higher for life histories with low s A and b (fig. 7A) than for life histories with higher probabilities of both these parameters (fig. 7B). The selection pressure on adult survival, s A, and on the probability of remaining in the juvenile class, bs J, however, changed if these two probabilities differed. Here, sensitivities strongly increased al- Figure 5: Small inset shows the invasion exponent ( c) as a function of s A in a deterministic environment for life histories with only iteroparity in which the trade-off between adult survival, s A, and clutch size, C, is 2 a convex relationship, C p c m(1 qs A). Where this function crosses the X-axis (i.e., c p 0) indicates the value of s A at evolutionary stability (i.e., the evolutionarily stable strategy [ESS]). The larger panels are enlargements of the region of the ESS showing the effect of increasing environmental variation (solid line: coefficient of variation [CV] p 0% ; diamonds: CV p 5% ; circles: CV p 15% ; triangles: CV p 25% ) on the location of the evolutionarily stable value of s A. A, Environmental variation in the strength of the density-dependent parameter a; B, environmental variation in the density-independent metamorph survival, s M. Arrows indicate the s A value of the ESS under the respective environmental variation, which is indicated by the intersection of the curve with c p 0. For all simulations, sa p 0.3, sj p 1, C p 12, sm p 0.3, a p 0.001, and q p 1.5. under the deterministic trade-off results in s A! 1. How- ever, if the stochasticity is introduced in either metamorph survival, s M, or maximum clutch size, C m, then increasing environmental variation will have the opposite effect and Figure 6: Deterministic sensitivities for four life histories with different probabilities (0.3 and 0.7) of both iteroparity (s A ) and developmental delay (b). Parameter values on the abscissa are indexes of matrix elements for the full model (fig. 1; eq. [4]): 1,1 p sb J ; 1,2 p smcexp ( t) ; 2,1 p s J(1 b) ; 2,2p sa. For all combinations of iteroparity and developmental delay, the other parameters were held constant at sj p 1, C p 12, s p 0.3, and a p M

9 406 The American Naturalist Figure 7: Effect of increasing environmental variation in the strength of density dependence (parameter a with coefficients of variation p 5%, 15%, and 25%) on changes in sensitivities with respect to the deterministic sensitivities for four life histories with different probabilities (0.3 and 0.7) of both iteroparity and developmental delay. Parameter values on the abscissa are indexes of matrix elements for the full model (fig. 1; eq. [4]): 1,1 p sb J ; 1,2 p smcexp ( t) ; 2,1p s J(1 b) ; 2,2p sa. Changes in sensitivities are given as differences from deterministic sensitivity values (see fig. 6). Positive values indicate an increase and negative values a decrease in sensitivity. A, sa p b p 0.3; B, sa p b p 0.7; C, sa p 0.7, b p 0.3; D, sa p 0.3, b p 0.7. For all combinations of iteroparity and developmental delay, the other parameters were held constant at sj p 1, C p 12, s p 0.3, and a p M most exclusively in the larger matrix element (i.e., in s A if sa 1 bs J or in bs J if s A! bs J; fig. 7C, 7D). The increase in sensitivity was higher for s A (life history with sa 1 b) than for remaining in the juvenile stage (life history with s A! b). Next we isolated the sensitivities of developmental delay, b, from those when it is in combination with juvenile survival, s J. These sensitivities were changed from 0 in the deterministic scenario to positive values with the introduction of environmental uncertainty. Sensitivities of b generally increased with higher environmental variation, but the relative increase was dependent on the specific life history. The strongest increase was displayed in the life history with low adult survival, s A, and developmental delay, b, and the increase was lowest in life histories with sa 1 b. In contrast to sensitivities of b, sen- sitivities of juvenile survival, s J, alone increased only slightly in life histories with b 1 s A, and they were un- affected by increasing variation in life histories with equal probabilities of b and s A. The sensitivities even decreased in the life history with sa 1 b. This indicates that in gen- eral, higher environmental uncertainty favors increasing the delay of maturity rather than increasing juvenile survival. We repeated this analysis reducing juvenile survival, s J, to 0.9 and obtained the same patterns but with slightly reduced magnitudes. To test for the effect of density dependence on these results, we moved the stochastic term from the strength of density dependence, a, to the density-independent metamorph survival, s M, and compared simulations with and without density dependence. For the density-dependent model, the patterns followed the results of the simulations described above, with only a reduced magnitude of change in sensitivities. In the density-independent case,

10 Bet-Hedging in Uncertain Environments 407 Figure 8: Effect of increasing environmental variation in density-independent metamorph survival, s M (coefficients of variation p 5%, 15%, and 25%), on changes in sensitivities with respect to the deterministic sensitivities for four life histories with both iteroparity and developmental delay for model without density dependence. Parameter values on the abscissa are indexes of matrix elements for the density-independent model: 1,1 p sb J ; 1,2 p smcexp ( t) ; 2,1p s J(1 b) ; 2,2p sa. Changes in sensitivities are given as differences from deterministic sensitivity values (A). Positive values indicate an increase and negative values a decrease in elasticity. A, Deterministic sensitivities; B, sa p b p 0.3; C, sa p b p 0.7; D, sa p 0.7, b p 0.3; E, sa p 0.3, b p 0.7. For all combinations of iteroparity and developmental delay, the other parameters were held constant at s p 1, C p 12, and a p J increased variation in survival of metamorphs resulted in similar patterns of increased sensitivities of developmental delay, b, and adult survival, s A (fig. 8). The patterns of change in sensitivities of fertility, F, and juvenile survival, s J, were also similar to those in the densitydependent case, and only sensitivity of the transition between juvenile and adult stages, T A, increased instead of decreased with higher variation in s M. This indicates that

11 408 The American Naturalist the effects of environmental variation on b and s A hold generally true whether or not density dependence is included in the model. Discussion Recent studies of developmental delay (e.g., Neubert and Caswell 2000; Tuljapurkar and Wiener 2000) consider a two-stage life cycle in which individuals vary in whether they move to the adult stage at 1 year of age or remain as juveniles. Our model confirms the conventional wisdom that life-history delays can be bet hedges against temporal environmental uncertainty but expands this conclusion to models with density dependence and realistic parameters that can be estimated in field studies. Our model predicts that both developmental delay and iteroparity can increase within a life history as the effect of environmental stochasticity on fertility increases, rather than corroborating the runaway process of life-history adaptation found in the model of Tuljapurkar and Wiener (2000). They argue that their model encompasses iteroparity, but it includes only variation in timing of reproduction among members of a cohort in which individuals reproduce only once rather than repeatedly. Thus, individuals are in fact semelparous, but the authors consider the population to be iteroparous because there is variation in the age of reproduction. We believe this is a broader use of the term iteroparity than Cole (1954) and his successors have employed. We suggest distinguishing their effect as cohort iteroparity, as opposed to individual iteroparity, in which individuals reproduce repeatedly. In our view, a semelparous female contributes to multiple future year classes through the developmental delay of her offspring via cohort iteroparity, whereas an iteroparous female contributes to multiple year classes through her own repeated reproduction. The semelparous chinook salmon (Oncorhynchus tshawytscha), for example, achieves cohort iteroparity by juveniles maturing over a range of ages (Hill et al. 2003). Most trees and many vertebrates, on the other hand, exhibit individual iteroparity with a long adult reproductive life. The consequences to population dynamics of each process may be identical, but the distinction is critical to an understanding of life-history evolution. The analysis of our full model suggests that selection favors fast life cycles in constant environments, resulting in directional selection for no delay as the ESS. We confirm the literature on seed banks, showing that environmental uncertainty can change the direction of selection toward delaying reproduction by the mechanism of bet hedging, without any other benefit of reproductive delay, such as an increase in fertility if maturity is postponed in favor of growth. On the other hand, selection is directional for increased adult survival with the ESS of no adult mortality when adult survival has no cost. Environmental stochasticity increases the strength of this selection. Trade-offs between life-history traits can constrain selection. Selection for increased adult survival can be constrained if, for example, this gain comes at the cost of fertility. We have demonstrated that a convex trade-off between adult survival and clutch size can result in an ESS with reduced adult survival, even in a deterministic environment. The effect of a stochastic environment on adult survival depends on where in the life history environmental variation is introduced. We have shown that when the parameter determining the strength of density dependence is affected by environmental variation, the ESS will have a lower adult survival with increasing variation relative to the deterministic ESS. When the density-independent survival of metamorphs is affected by environmental variation, however, the ESS is shifted to higher adult survival. This suggests that it is important to identify which specific life-history parameter is subject to environmental variation. Only empirical studies can guide the application of lifehistory theory to particular habitats and organisms. For example, a negative correlation between adult survival and fecundity is expected to be a common constraint, but whether this relationship should be modeled as the realized clutch size as a function of adult survival or as adult survival as a function of clutch size depends on the physiological ecology of how resources are allocated between maintenance and reproduction. Takada (1995) showed how the shape of the function determines where the optimal trade-off will occur. This is also an empirical issue. In models with a trade-off resulting in correlations between life-history traits, it is important to understand the implications of these correlations to both density dependence and stochasticity. In our analysis of iteroparity, for example, the trade-off between fecundity and adult survival introduced a negative correlation between adult survival and fertility that changed the ESS and selection pressure on both traits. The use of invasion analysis permits an evolutionary interpretation of the implications of environmental stochasticity in a density-dependent model. Our analysis, however, begs the question of whether delayed maturity and iteroparity are adaptations or constraints at the level of an individual. The adaptive basis of iteroparity is easily argued, but skipping opportunities for reproduction also raises the question of adaptation or constraint. Are juveniles postponing maturity, or are adults foregoing a breeding season as a plastic response, as an adaptation to hedge their evolutionary bets, or because they have not yet obtained a state (e.g., sequestration of sufficient resources or attaining the necessary social status) that tips

12 Bet-Hedging in Uncertain Environments 409 the balance in favor of reproduction versus survival? An individual may delay or skip if it can predict reproductive success and reproduce only under auspicious circumstances. This is phenotypic plasticity. An individual may delay or skip because it has not obtained sufficient resources or social status to succeed. We call this a constraint. Use of the term bet hedging should be restricted to when individuals are spreading their risk because the outcome of a reproductive bout is uncertain. They are reducing the arithmetic mean fitness to maximize their geometric mean fitness under the theory of Gillespie (1974) in which selection reduces variance in individual fitness. Variation in age of maturity in vertebrates is generally explained by the greater reproductive success of older individuals resulting from the production of more or higherquality offspring than that of younger parents rather than as an adaptation to variation in mortality (Stearns and Crandall 1981, 1984; Curio 1983; Stearns and Koella 1986; Takada and Caswell 1997; Cam and Monnat 2000). Variation in age of maturity is viewed in these examples as a consequence of individual differences in the time required to obtain the phenotype (e.g., body size, dominance rank, or experience) necessary for successful reproduction. Hence, delayed maturity is considered a constraint rather than an adaptation. We would argue that the fundamental adaptation can be the delay rather than having a fixed age at maturity and semelparous mode of reproduction. The adaptive value of individual iteroparity can be enhanced by trade-offs between current investment in reproduction and survival leading to future opportunities for reproduction. Our sensitivity analysis of developmental delay showed that it may be misleading to focus only on the probability of remaining in the juvenile class because the two components of this element, juvenile survival and probability of delaying maturity, can be under selection individually. Our results demonstrate that this matrix element may have a low sensitivity even if the two components have appreciable sensitivities if they are in opposite directions. Selection can act on each component, so it is important to understand the individual life-history traits that contribute to each element of a population projection matrix. It is at this level that theoretical population biology confronts rigorously quantitative natural history. Acknowledgments We thank H. Caswell, L. Green, and S. 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