Matthew Spencer, Department of Biochemistry, University of Cambridge, Tennis Court Road, Cambridge CB2 1QW, UK. Tel.: +44 1223 333687; fax: +44 1223 333345; e-mail: firstname.lastname@example.org
Many organisms survive unfavourable seasons as resting stages, some of which hatch each favourable season. Hatching fraction and timing of resting stage production are important life history variables. We model life cycles of freshwater invertebrates in temporary pools, with various combinations of uncertain season length and density-dependent fecundity. In deterministic density-independent conditions, resting stage production begins suddenly. With uncertain season length and density independence, resting stage production begins earlier and gradually. A high energetic cost of resting stages favours later resting stage production and a lower hatching fraction. Deterministic environments with density dependence allow sets of coexisting strategies, dominated by pairs, each switching suddenly to resting stage production on a different date, usually earlier than without density dependence. Uncertain season length and density dependence allow a single evolutionarily stable strategy, around which we observe many mixed strategies with negatively associated yield (resting stages per initial active stage) and optimal hatching fraction.
There are two key life-history variables for organisms inhabiting seasonal environments: (1) the proportion of resting stages that hatch (or germinate) at the start of the favourable season; and (2) the time at which active stages begin to produce resting stages (or to enter diapause) as the end of the favourable season approaches. There has been extensive theoretical work on both hatching fractions (e.g. Cohen, 1966, 1967; Venable & Lawlor, 1980; Bulmer, 1984; Ellner, 1985a,b) and the timing of resting stage production (e.g. Cohen, 1970; Taylor, 1980; Hairston & Munns, 1984; McNamara, 1994). However, these models each consider only one of the two key life history variables. All the above models of hatching fractions assume that the yield (new resting stages per initial active stage) in a given year is a stochastic variable, while the models of the timing of resting stage production assume that the hatching fraction is fixed.
In general, we expect negative relationships between different traits that tend to reduce the impact of environmental stochasticity (Brown & Venable, 1986; Venable & Brown, 1988; Ellner et al., 1998). For example, plants with a long-lived adult stage or efficient dispersal mechanisms show lower rates of seed dormancy than plants with short-lived adult stages and inefficient dispersal (Rees, 1993, 1994). Increasing variability in season length is likely to select for earlier resting stage production (Hairston & Munns, 1984). Density dependence is also likely to favour earlier resting stage production by reducing the reproductive value of active stages as the season progresses (Taylor, 1980), and by making the per-capita yield less predictable (Bulmer, 1984; Ellner, 1985b). However, reducing the hatching fraction is an alternative strategy for coping with variance in yield. By spreading hatching over several seasons, the risk that all offspring will hatch in a year with unfavourable conditions is reduced. This maximizes the long-term probability of persistence of a phenotype in unpredictable conditions, at the expense of short-term growth rate in favourable conditions (Cohen, 1966). Because the timing of resting stage production and the hatching fraction can both alter the influence of environmental stochasticity, understanding life histories in stochastic environments requires that we study both at once.
We describe simple models for the complete life cycle of an organism inhabiting a seasonal environment, in which the timing of resting stage production and the hatching fraction evolve together. A similar problem was addressed by Ellner et al. (1998), using a different approach. Ellner et al. produced a general model for trade-offs between diapause and bet-hedging during active stages. They assumed the existence of a quantitative trait expressed during the active stage, with a general fitness function relating the fitness of a phenotype to the state of the environment. With random variation in environmental state among years and a fitness function such that an increase in mean fitness is associated with an increase in the variance of fitness, they showed that geometric mean fitness can be maximized at an intermediate value of the quantitative trait, and that there is likely to be a negative within-population relationship between the hatching fraction and the riskiness of the active-stage trait. They did not explicitly consider the population dynamics of the active stages and did not allow mixed strategies. We take a more mechanistic approach in which we explicitly consider the population dynamics of active stages, and show how the interaction between hatching fractions and active-stage traits emerges from these dynamics. We also allow mixed strategies, and consider the relative cost of producing active and resting stages. We were motivated by the life histories of invertebrates inhabiting temporary freshwater pools, although our results should apply to other seasonal environments. The length of the season in temporary pools is unpredictable because it depends on local weather conditions. Density dependence in active stages is likely because of the high fecundity of the organisms (typically small crustaceans, rotifers, etc.) and the small size of the habitat. We address the following questions: How does the timing of resting stage production change as the season length becomes less predictable or the importance of density dependence increases? Do the timing of resting stage production and the fraction of resting stages that hatch interact with each other? Can the cost of producing resting stages relative to active stages affect the timing of resting stage production? We answer these questions using models with deterministic and stochastic season lengths, with and without density dependence in active stage population dynamics.
Consider an environment having alternating periods of favourable and unfavourable conditions. We refer to the favourable period as the ‘season’. Organisms have two life stages. Active stages grow and reproduce in a density-independent fashion when conditions are favourable, but immediately die when conditions become unfavourable. Reproduction by active stages can produce either more active stages or resting stages. Resting stages are the result of reproduction, and are analogous to the diapausing eggs of cladocerans or the seeds of plants. The active stages of some organisms (including tardigrades and some copepods) are able to encyst directly. Although we do not specifically address such cases, the general results should be similar. The number of resting stages at the end of the season per initial active stage is the yield Y. Resting stages are able to survive unfavourable conditions, and a fraction G hatch synchronously to produce active stages when conditions become favourable. Crustacean resting stages usually hatch synchronously at the start of the season (e.g. Brown & Carpelan, 1971; Hairston & Olds, 1987), although subsequent changes in physicochemical conditions sometimes trigger additional hatching (Brendonck, 1996). We assume that resting stages have no information on the environment they will experience if they hatch. This is unlikely to be strictly true (Hansson, 1996; Pake & Venable, 1996), but is a reasonable approximation if the major environmental factors (such as the amount of rainfall) are difficult to predict over a whole season. A fraction D of all resting stages die between one season and the next. We assume that no mortality of resting stages occurs during the season. This is reasonable if the annual mortality of resting stages is low (Hairston et al., 1995), and the favourable season is short compared to the intervening unfavourable period. We ignore dispersal between habitats. Genetic evidence suggests that this is reasonable for freshwater invertebrates with passive dispersal (e.g. Hebert & Payne, 1985; Weider & Hebert, 1987). Given these assumptions, the expected long-term growth rate r of the population of resting stages of a given phenotype (measured just before the season begins) is:
where Pi is the probability of yield Yi. This is Cohen’s (1966) model, except that we assume mortality of resting stages occurs outside rather than during the season. There is always a single optimum hatching fraction Gmax which will maximize r (Cohen, 1966), although Gmax cannot usually be found analytically if there are more than two different possible yields. Similar models have motivated several empirical studies of hatching fractions in crustaceans (e.g. De Stasio, 1989; Simovich & Hathaway, 1997). Following Cohen and many others, we use r as a measure of fitness. A phenotype with higher r will increase in frequency relative to a phenotype with lower r.
The set of possible yields depends on the duration of the season and the population dynamics and reproductive strategy of the active stages. We divide the season into discrete time steps from 0 to tmax (the maximum possible duration of the season, although the season may end earlier than tmax). A fraction v of active stages survive from one time step to the next. At any time t, a fraction ut of the per capita reproductive output L of active stages is devoted to the production of more active stages, and a fraction (1 – ut) is devoted to the production of resting stages. We assume that mixed strategies in which 0 < ut < 1 are possible. Individual crustaceans cannot simultaneously produce both resting and active stages (e.g. Hairston & Olds, 1984; Hobaek & Larsson, 1990), but a mixed strategy requires only that the individual has a nonzero probability of producing either kind of offspring. The relative cost of a resting stage compared to an active stage is c. We will refer to the ordered set (u0, u1, …, utmax) as a reproductive allocation strategy. The dynamics of the populations of active stages A and newly produced resting stages R are described by
The initial population of newly produced resting stages (R0) is zero by definition. In the absence of density dependence, the initial active stage population A0 can be set to any positive value without affecting the relative fitness of different strategies or hatching fractions. Then if the probability P of the season ending at any given time step t is (P0, P1, …, Ptmax) with sum 1 (the probability mass function describing the distribution of discrete season durations), the corresponding yield Y is (R0/A0, R1/A0, …, Rtmax/A0).
Density dependent cases
We assume that density dependence reduces reproductive output but not survival of active stages. This is a reasonable approximation in some freshwater crustaceans. For example, starving Daphnia do not reproduce, but may survive for some time (Bradley et al., 1991). We also assume that density has no effect on resting stages other than on the number produced. Resting stages produced by crowded organisms might be less well-provisioned and thus have lower survival than resting stages produced by organisms at low density, but we do not know of any data on this. We model the density-dependent reduction in reproductive output by replacing the constant L in eqns 2 and 3 with
where β is the per capita effect of the active stage population on reproductive output and Lmax is the maximum per capita reproductive output. If there is density dependence, the set of yields will depend on the initial population of active stages A0, which will in turn depend on the hatching fraction G. The exact form of density dependence is not important, as we obtained qualitatively similar results with a discrete-time logistic density dependence function.
For density-dependent cases, there is no longer a constant growth rate that can be used as a measure of fitness. Instead, we use boundary growth rate (r when a phenotype is rare and another phenotype is common) as a measure of fitness, following Ellner et al. (1998) and many others. A phenotype with positive boundary growth rate will increase in frequency when rare, and a phenotype with negative boundary growth rate will decrease in frequency when rare.
1A. Deterministic season length, no density dependence
With a deterministic season length >0, there is an analytical solution to eqns 2 and 3 that maximizes fitness. The probability P of the season ending is 1 for time tmax and 0 for all other times. We solve for the reproductive allocation strategy that maximizes the yield Rtmax/A0 by backwards induction (Bulmer, 1994, pp. 83–86). The optimal value of ut is either 1 or 0, and we obtain the condition for ut=1 (otherwise ut=0):
If the LHS and RHS of eqn 5 are equal, ut=1 and ut=0 are equally good. Analogously, insects should enter diapause when the remaining time allows each female to produce one female offspring that will enter diapause (Taylor, 1980).
From the derivative of eqn 1 with respect to G, the optimal hatching fraction Gmax is 1 if the yield from the optimal reproductive allocation strategy is greater than 1. Otherwise Gmax is 0 (and the phenotype cannot have a positive growth rate). The relative cost of resting vs. active stages (c) has no effect on the optimal reproductive allocation strategy in the deterministic case as it affects all strategies equally. However, increasing c may change the optimal hatching fraction from 1 to 0 by reducing the yield. For given values of v and L, the switch from active to resting stage production occurs a fixed number of time steps before the end of the season. However, the total length of the season (tmax) does affect the yield, the optimal hatching fraction and the population growth rate of the phenotype.
1B. Stochastic season length, no density dependence
With stochastic season length, we cannot solve eqns 2 and 3 analytically. We use simulations (Appendix 1) to find the hatching fraction and reproductive allocation strategy that maximize the expected long-term population growth rate r of a phenotype for a given distribution of season durations, assuming that the parameters v, L, c and D are fixed by environmental or physiological constraints. These simulations converge on the analytical solutions in deterministic cases. The set of probabilities Pi of the season ending on step i (eqn 1) describes the uncertainty in season length. We consider a set of scenarios having the same mean season length, but different variances in season length. For simplicity, we assume that the season is equally likely to end on any step within a given range. We compare four ranges: step 20 (deterministic), steps 15–25 (low uncertainty), steps 10–30 (medium uncertainty) and steps 0–40 (high uncertainty). As the uncertainty in season length increases, resting stage production begins earlier relative to the maximum season length (and relative to the constant mean season length) under the optimal reproductive allocation strategy, but the transition to producing only resting stages occurs more gradually (Fig. 1).
Cohen (1966) showed that the optimal hatching fraction is less than 1 (giving a long-term resting stage bank) if the harmonic mean yield is less than 1 (in our model; in Cohen’s original model the condition is that the harmonic mean yield is less than (1 – D)). This is true if the yield may be zero (the high uncertainty case in Fig. 1: the optimal hatching fraction is 0.71), and may also be true if the yield is certainly greater than zero (although this does not occur in our example). In low and medium uncertainty cases, there is a temporary trough in the allocation to active stages just before the first date on which the season might end (Fig. 1). This ensures that enough new resting stages are produced if the season ends immediately. Subsequently, effort can be switched back to active stage production because resting stages that have already been produced contribute to the yield.
The optimum allocation strategy may also depend on the relative cost of resting and active stages (c). For example, consider a system with a maximum season length of 20 steps, the season ending either on step 1 or step 20 with equal probability, and the parameters L (reproductive output)=0.5, v (survival of active stages)=0.9, A0 (initial number of active stages)=10 and D (mortality of resting stages between seasons)=0.1. With c=5 (resting stages five times as expensive as active stages) the strategy that maximizes the population growth rate of the phenotype has u=0 on steps 17, 18 and 19, and u=1 on all other steps. In years when the season ends on step 1, no new resting stages will be produced. To compensate for this risk, the optimal hatching fraction is 0.49. However, with c=1.3 (resting stages not much more expensive than active stages), the strategy maximizing long-term population growth rate of the phenotype has u=0 rather than u=1 on step 0. New resting stages will be produced in every year, and the optimal hatching fraction is 0.81. When resting stage production is more expensive, it may pay to invest later in resting stages if season length is uncertain, and compensate for the lower yield by reducing the hatching fraction. Although c acts as a scaling factor on all reproductive allocation strategies equally, the absolute rather than relative number of resting stages produced becomes important when compared with the fitness resulting from not hatching.
2A. Deterministic season length with density dependence
We use the algorithm described in Appendix 2 (generating rare mutants and determining whether they were able to invade) to search for the set of combinations of a reproductive allocation strategy and hatching fraction that can coexist and are resistant to invasion by any further combinations. With density dependence and deterministic season length, we usually find pairs of allocation strategies that are able to coexist. Each member of the pair is a pure strategy, switching suddenly and on a different date from producing only active stages to producing only resting stages (Fig. 2a). Although the population is always dominated by either one or two pure strategies (Fig. 2b), we also find very rare mixed strategies (relative abundance no more than 4 × 10–6) that invade the set of common, pure strategies (boundary growth rates no more than 3 × 10–8). If the season is very short, we find only a single pure strategy (Table 1) and no mixed strategies.
Table 1. Effects of season length on switch date (the first date on which all reproductive output is devoted to resting stages) and last successful invasion attempt (out of 50) in models with density-dependent active stage reproductive output and deterministic season length. At season lengths 20 and 30, there are pairs of pure strategies with different switch dates. Very rare mixed strategies also occur at season lengths 20 and 30 (relative abundance no more than 4 × 10−6), but reproductive allocation fractions are always either zero or one outside the range of switch dates given here. Parameters: Lmax (maximum reproductive output) = 0.5, v (survival of active stages) = 0.9, c (relative cost of producing a resting stage) = 5, D (mortality of resting stages between seasons) = 0.1, β (per capita effect of active stage density on reproductive output) = 0.01. A0 (the initial active stage population) is not a parameter in the density-dependent case (see text ‘Density dependent cases’).
We allow 50 invasion attempts in each simulation. Most later invasion attempts do not find a mutant with positive boundary growth rate. For those that do, the mutant boundary growth rates are extremely small (Fig. 2c). The total abundance of resting stages also changes little after the first few invasions (Fig. 2d). Ellner & Hairston (1994) suggest that there should be an invasion-resistant set of coexisting strategies in this situation. We think that continued invasions occur because many invasion attempts are necessary to approach the optimal solution to the problem. Successful invasions may continue for longer when the season is longer (Table 1) because it is more difficult to find an optimal strategy over a long than a short season.
In a deterministic environment without density dependence, the switch from active stage to resting stage production occurs a fixed number of time steps before the end of the season, no matter what the absolute length of the season (other things being equal: eqn 5). This is because the reproductive value of active stages is affected only by the length of time remaining in the season (Taylor, 1980). When active stage density has a negative effect on fecundity, the absolute length of the season becomes important. The switch to producing resting stages occurs earlier relative to season length as the season length increases (Table 1), because active stage density begins to limit reproductive output in long seasons. Ultimately, further increases in season length will have no effect on absolute switch date, although we do not reach this point with the parameters we examine.
As in the case of deterministic season length without density dependence, there is no stochasticity in yield. The optimal hatching fraction for a rare mutant is therefore always either 1 (if yield is greater than one, in which case the mutant increases when rare) or zero (if yield is less than one, in which case the mutant decreases when rare). We confirmed by simulation that β (which sets the strength of density dependence) does not affect the set of common strategies able to coexist. This is because, at equilibrium, abundance increases until the per capita yield is just sufficient to maintain the population no matter what the value of β. If no other parameter has also changed, the distribution of yields across alternative reproductive allocation strategies at equilibrium is identical for different values of β.
2B. Stochastic season length with density dependence
With stochastic season length and density dependence, mixed strategies are favoured (Fig. 3a: for maximum season length of 40, with the season equally likely to end on each time step), as in the case of stochastic season length without density dependence (Fig. 1). In the stochastic density-independent case (Fig. 1) the optimal strategy is not always strictly nonincreasing, but changes much more smoothly over the season than the set of strategies in the density-dependent case (Fig. 3a). All of the 40 strategies with positive abundance after 50 invasion attempts have very similar hatching fractions (range 0.3412–0.3507, median 0.3460, for the parameters used in Fig. 3). Most strategies are rare (Fig. 3b), but the distribution of relative abundances is much more even than in the case of deterministic season length with density dependence (Fig. 2b). Mutants with positive boundary growth rate appear even after many invasion attempts (Fig. 3c). Although the boundary growth rates of later invaders are much lower than those of earlier invaders, the rate of decrease in boundary growth rate with each successive invasion is very low after many successful invasions. The total population size changes little after the first few invasions (Fig. 3d).
Ellner & Hairston (1994) suggest that any coexisting set of strategies can be invaded by another mixed strategy in a stochastic environment, when the yields from different strategies are not always identical. Our simulations show invasions that continue to succeed even after many previous invasions (Fig. 3c), but with low boundary growth rates. This suggests a single evolutionarily stable strategy surrounded by a flat fitness surface. We expect to see many different genotypes with small differences in fitness in the population at any time. Allowing only pure strategies (cf. Hairston & Munns, 1984; Ellner et al., 1998) does not substantially affect these results.
Increasing the uncertainty in season length increases the amount of variation in reproductive allocation fractions at a given time (Fig. 4). Resting stage production begins earlier relative to the maximum season length as uncertainty increases (Fig. 4), as in the density-independent case. Relative to the constant mean season length, resting stage production begins earlier as uncertainty in season length increases from no to medium uncertainty. However, the first date of resting stage production is the same in the high and low uncertainty cases (Fig. 4). This is because the optimal hatching fractions in the high uncertainty case (median 0.3460, see above) are much lower than in the medium uncertainty case (median 0.6615, range 0.6511–0.6994), allowing more risky reproductive allocation strategies. For the deterministic and low uncertainty cases, the optimal hatching fraction is 1. Higher uncertainty in season length does not tend to increase the number of strategies with positive abundance after 50 invasion attempts, although there are many fewer strategies when season length is deterministic (deterministic, six strategies; low uncertainty, 45 strategies; medium uncertainty, 39 strategies; high uncertainty, 40 strategies). The proportion of reproductive effort allocated to active stages at a given time is rarely greater with than without density dependence (with the exceptions occurring at high uncertainty: Fig. 4d). Density dependence usually leads to earlier production of resting stages, as in the deterministic case.
Changing the density dependence parameter β has little effect on the set of strategies with positive abundance at the end of the simulation (for the same reason as in the deterministic density-dependent case, described above). The individual strategies are not identical, but the (unweighted) mean reproductive allocation over all strategies is similar between simulations with an order-of-magnitude difference in β (Fig. 5). Although there is little variation in hatching fractions where they are less than 1, there is a negative association between the among-year variance in yield (over all possible values of season length) and the hatching fraction, for strategies having positive abundances after 50 invasion attempts (high uncertainty case: Kendall’s τb=–0.96, N=40).
Life history traits can interact to reduce the impact of environmental variability (Brown & Venable, 1986; Venable & Brown, 1988; Rees, 1994; Ellner et al., 1998). We illustrated this using a simple model with explicit description of the dynamics of both active and resting stages. Life history constraints prevent organisms simultaneously maximizing their fitness under all conditions (Brown & Venable, 1986). In our model, these constraints follow from our assumption that a resting stage has no information on the environmental conditions it will experience if it hatches, and from our specification of the population dynamics of active stages. Our model thus has a strong mechanistic basis. Ellner et al. (1998) reached similar conclusions using a very different approach, in which they assumed a positive relationship between the mean and variance of fitness for a phenotype.
Effects of stochastic season length
Our models and others (Cohen, 1970; Hairston & Munns, 1984; Hairston et al., 1985; McNamara, 1994) predict that increasing uncertainty in season length favours an earlier and more gradual shift to resting stage production, if organisms have no warning of the approaching end of the season. If organisms have some warning of the end of the season (e.g. Denver et al., 1998), their subjective uncertainty in season length is reduced but the qualitative relationship between uncertainty and resting stage production should be unaffected. Some freshwater crustaceans do show earlier and more gradual shifts to resting stage production in the presence of more uncertain season length (Green, 1966; Frey, 1982; Hairston & Olds, 1984; Hairston & Olds, 1987; Innes, 1997). However, because it is difficult to estimate season length, it will be easier to test predictions about the abruptness of the shift than its timing.
Effects of density dependence
With density dependence in active stage fecundity, we find sets of apparently coexisting reproductive allocation strategies. This agrees with analytical theory in deterministic environments (Ellner & Hairston, 1994). However, in stochastic environments there is a single optimum mixed strategy (Ellner & Hairston, 1994; Sasaki & Ellner, 1995). In our model, we observe many reproductive allocation strategies, because the rate at which the optimum strategy replaces all others is very low. For the same reason, we expect many strategies in a natural population. One caveat is that we model time discretely. In continuous time, the same qualitative pattern should hold (coexistence in deterministic environments, a single optimum in stochastic environments: Ellner & Hairston, 1994). However, convergence to the optimum strategy might be more rapid in stochastic environments, giving fewer strategies in the population at any time. In both deterministic and stochastic environments, density dependence usually results in earlier resting stage production than in the density-independent case. This is because the reproductive value of future active stages decreases as population density increases through the season (Taylor, 1980).
Interactions between stochasticity and density dependence
In models with stochasticity and density dependence, a high variance in the yield of new resting stages per initial active stage is associated with a low hatching fraction, and vice versa, as predicted by previous theoretical (Brown & Venable, 1986; Venable & Brown, 1988; Ellner et al., 1998) and empirical (Hairston et al., 1996; Pake & Venable, 1996) work. Early production of resting stages and a low hatching fraction both reduce the variance of population growth rate of a phenotype, and are alternative ways of dealing with environmental stochasticity. However, the relationship may be difficult to detect in natural populations unless there are relatively discrete reproductive allocation strategies with large differences in variance in yield.
Effects of the cost of resting stage production
With uncertain season length, the optimal allocation of reproductive effort to active and resting stages in crustaceans inhabiting temporary pools can depend on the relative costs of producing resting and active stages. When the relative cost of producing resting stages is high, the optimal allocation strategy may result in no new resting stages being produced in some years, and the hatching fraction is reduced to compensate. Thus there are two reasons why a population might sometimes fail to produce new resting stages. Either the season sometimes ends too early for resting egg production to occur under any strategy (the ‘constraint hypothesis’), or because many crustaceans are choosing risky reproductive allocation strategies (the ‘risky strategy’ hypothesis). The latter explanation is more likely when resting stages are much more expensive to produce than active stages. This prediction has not been tested as far as we know.
We are grateful to Steve Ellner and Nelson Hairston, Jr for helpful discussions, and to two anonymous reviewers for constructive criticism. This work was partially supported by United States–Israel Binational Science Foundation grant #95/0035 awarded to Leon Blaustein and Joel E. Cohen.
*Present address: Department of Zoology, Oklahoma State University, USA.
Appendices. Algorithms used in simulations
We implemented all algorithms in Matlab 4.2d for Macintosh (The Mathworks, Inc., Natick, MA, USA).
Appendix 1. Algorithm for Case 1B (stochastic season length, no density dependence)
1 Generate a random reproductive allocation strategy.
2 Use eqns 2 and 3 to find the distribution of yields for this strategy given v, L, c, D, the distribution of season durations, and an arbitrary positive value of A0.
3 Given this distribution of yields and season durations, numerically minimize –r (given by eqn 1) to find the optimum hatching fraction Gmax. The expected long-term growth rate r at Gmax is the fitness of this combination of hatching fraction and reproductive allocation strategy.
4 Change a randomly chosen element of the reproductive allocation strategy by a small amount Δ, and repeat 2 and 3.
5 If the new reproductive allocation strategy gives a higher fitness than the old strategy, retain the new strategy and repeat 4. Otherwise, return to the old strategy and repeat 4.
We use an initial Δ of 0.1, and halve this value after every 100 iterations without improvement in fitness, down to a minimum of 1 × 10–4. Five thousand iterations are ample for convergence on a good solution (most changes in fitness occur within the first 1000 iterations), and repeat simulations always converge on the same solution (to several decimal places) for a given set of parameters. Where analytical solutions are possible (for deterministic season length), the algorithm converges on the analytical solution.
Appendix 2. Algorithm for Cases 2A & 2B (deterministic or stochastic season length, density dependence)
1 Find a combination of the optimal reproductive allocation strategy and associated optimal hatching fraction as described in Appendix 1. We set the initial abundance of resting stages to a low value (1 × 10–4, measured just before the start of the favourable season). Although the optimal strategy depends on the abundance of resting stages, the algorithm converges on an equilibrium population size and set of strategies that does not depend on the initial abundance chosen here.
2 Find the equilibrium abundance. Numerically project the dynamics of this combination for many seasons using eqns 1–3 (with density-dependent reproduction as described by eqn 4), until the population is close to equilibrium. We use 5000 seasons (see step 4, below).
3 Attempt an invasion. Search for combinations of reproductive allocation strategy and hatching fraction that have a positive growth rate when rare and the combination found in step 1 is at its equilibrium abundance. We use the algorithm described in Appendix 1 to perform the search, and assume that the density-dependent effects of active stage abundance on reproductive output are determined only by the abundance of the common combination.
4 Find the new equilibrium. If there is a combination with positive growth rate when rare, set the abundance of this combination to a low value (1 × 10–4 resting stages, measured just before the start of the favourable season, which is typically a relative abundance of <2 × 10–5 for the parameter values we explore). Then iterate the joint dynamics of the two combinations until close to an equilibrium (5000 seasons: the median (over all invasion attempts and simulations) of the largest proportional change in abundance of any strategy in the last season was 0.0002, and the maximum was 0.009). We set the abundance of a strategy to zero when it fell to 1 × 10–9 resting stages.
5 Look for invaders of the new equilibrium. Repeat steps 2–4 for each new invasion attempt. We use 50 invasion attempts in all simulations. The growth rate when rare of the best invader usually declines as the number of successful invasions increases. In some cases, we find no more successful invaders after a few invasions, while in other cases all 50 invasions are successful, for reasons we discuss in the text (cases 2A and 2B).
Repeat runs with the same parameters give consistent results. Using 20 000 rather than 5000 seasons for approach to equilibrium does not affect the results.