arXiv:1502.05508v1 [q-bio.PE] 19 Feb 2015
The neutral evolution of iteroparity M´elissa V´erin1,2 , Fr´ed´eric Menu1 and Etienne Rajon1,∗ 1
CNRS, UMR5558, Laboratoire de Biom´etrie et Biologie Evolutive,
Universit´e de Lyon, F-69000, Lyon, Universit´e Lyon 1, F-69622, Villeurbanne, France 2
Section of Population Genetics, Center of Life and Food Sciences Weihenstephan Technische Universit¨ at M¨ unchen, Freising, Germany ∗
Corresponding author:
[email protected]
Abstract Iteroparous individuals reproduce several times instead of just once, as semelparous do. Many species are iteroparous at various degrees (i.e. they differ by the mean number of reproductive events), despite a well-known advantage to semelparity: semelparous genotypes grow more rapidly in stable environments where resources are in excess. Existing theory might still explain the evolutionary success of iteroparity, either by a direct advantage (it can provide higher lifetime fecundity) or by its buffering effect in varying environments. Here we confront this Malthusian view of the evolution of iteroparity by considering that resources are limiting, such that a population can only grow in size up to a point where it stabilizes. Using an adaptive dynamics approach, we show that semelparity looses its selective advantage in this context, and that all reproduction strategies (i.e. semelparous and iteroparous at any degree) are strictly neutral when they bear equal lifetime fecundity. Then we develop a neutral model for the evolution of these strategies, showing that the most probable outcome is the highest degree of iteroparity. In light of these results, the observation that many species are semelparous becomes puzzling. We argue that some ecological contexts may favor semelparity, either directly (e.g. when survival until the next reproductive season is unlikely) or by yielding population dynamics that scarcely limit growth. Evolutionary demographics – Ecology – Density dependence – Genetic drift – Aging
Strategies of parental investment are distributed along a negative relationship (i.e. a trade-off) between the number of litters produced during an adult’s lifetime (‘longevity’ hereafter) and the size of each litter [1–4]. In a stable, non-limiting environment, the shape of the trade-off is sufficient to determine which strategy grows the most rapidly and outcompetes the others (fig. 1) [4]. A convex trade-off means that litter size decreases rapidly with longevity, such that semelparous individuals, who reproduce only once, produce a larger number of offspring in their adult life on average. Such a trade-off would appropriately describe the case of a species where adults have a fixed amount of resources to allocate to either reproduction or survival, such that a high adult survival would be at the cost of a decreased lifetime fecundity. Unsurprisingly, given this trade-off, semelparity outcompetes any other reproduction strategy.
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Changing the form of the trade-off to concave can give a direct selective advantage to iteroparous individuals (fig. 1), whenever the number of adults transmitted to the next generation (i.e. the sum of the numbers of adults surviving and of offspring successfully reaching adulthood) overcomes that of semelparous ones [1, 2, 5]. This kind of trade-off may encompass many situations where adults can feed between reproduction events, so the resources necessary for reproduction can be partly acquired during an adult’s lifetime. Nevertheless, iteroparity is expected to only provide a slight selective advantage over semelparity in a stable, non-limiting environment: in the extreme situation where iteroparous individuals produce litters of the same size as semelparous ones, iteroparity only adds one adult (through survival) to the next generation [5]. mean number of reproduction events 1.33
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Figure 1: Three different shape of the trade-off between litter size and adult survival, and their impact on the asymptotic growth rate. The mean number of reproduction events (represented as a supplementary x-axis above the figure) is a function of adult survival between reproduction events (see text). When the trade-off is convex, semelparous individuals (i.e. those with adult survival 0) have the highest lifetime fecundity and the highest growth rate. When the trade-off is concave, iteroparous individuals (with adult survival above 0) have higher lifetime fecundities, and the strategy with the highest growth rate depends on the precise shape of the trade-off (it equals 0.39 in the example shown). The linear trade-off ensures that lifetime fecundity is constant along the trade-off, and theory predicts that semelparity provides the highest growth rate (as with a convex trade-off), assuming that the population is not regulated. This advantage to iteroparity vanishes when the overall number of individuals produced by any strategy is kept constant by a linear trade-off (fig. 1) – that is, when a fixed number of offspring may be produced on a single (semelparity) or several litters (iteroparity). In this situation, semelparous individuals have the maximum growth rate [4], as they produce all their offspring at once, which will reach adulthood (and the ability to reproduce) sooner than the average offspring of any iteroparous adult. This apparent advantage to semelparity, as well as the limited advantage to iteroparity, has led researchers to ask why iteroparity is so widespread [6]. One reason might lie in the fact that the growth rate of an iteroparous genotype is relatively immune to random environmental events affecting juvenile life history traits, compared to a semelparous genotype [2, 7, 8]. Offspring from semelparous parents develop concurrently, such that randomly occurring environmental events may affect all of them and, in the long run, yield dramatically variable growth 2
rates. Offspring from iteroparous parents, however, are spread among several litters that may encounter different environmental conditions. Iteroparity thus acts as a bet-hedging strategy [9] that increases the geometric mean of the growth rates by decreasing their temporal variance. Current theory thus predicts that semelparity should be predominant in stable and in slightly variable environments, while iteroparity should evolve when environmental variance is above some threshold [8]. Quite surprisingly, the evolution of iteroparity has scarcely been studied in a stable, density-dependent environment – but see study [4], which we discuss later. In this situation, the advantage to semelparity in a stable environment associated with its quicker growth might become irrelevant. Using the linear tradeoff described above and thus eliminating any direct advantage to a specific strategy, we show that any strategy along this trade-off is indeed neutral relative to others – in other words, no reproduction strategy is expected to increase or decrease in frequency in response to selection. This result is independent of the density-dependency function used, as long as it yields a stable population at equilibrium. Next we investigate the neutral evolutionary dynamics of these reproduction strategies. Quite ironically, the most probable genotype in the dynamical equilibrium hence created has the highest level of iteroparity (i.e. maximum longevity and minimum litter size). Iteroparity should thus be expected as a result of evolution in the context of a stable environment that realistically includes population regulation, whenever no specific advantage to a given reproduction strategy is assumed.
Dynamics of a regulated population We first consider a monomorphic ‘resident’ population constituted by NJr juveniles and NAr adults, whose dynamics is described by the discrete system [1]. ( NJr (t + 1) = NAr (t) × F × (1 − itr ) × d (1) NAr (t + 1) = NAr (t) × itr + sJ × NJr (t) The reproduction strategy is described by the continuous variable itr , the survival of adults from one reproductive event to the next. Adults with itr = 0 are semelparous: they reproduce once and die. Iteroparity arises as soon as itr is above 0, and the mean number of reproductive events increases with itr (this number equals 1/(1 − itr )). We assume that the number of offspring produced by an adult in its lifetime is independent of its reproduction strategy, to avoid any direct advantage to a particular strategy – this corresponds to the linear trade-off described above. This is done by setting the fecundity per reproduction event to F (1−itr ). Each egg produced survives with the density-dependent probability d, which decreases as the overall number of eggs produced, NAr (t)F (1 − itr ), increases. We do not assign a specific density-dependence function to d at this point. Instead, we assume that this function eventually yields an equilibrium where NAr (t + 1) = NAr (t) = NA and NJr (t + 1) = NJr (t) = NJ . This property is shared by many classical density-dependence functions, given a parameterization that avoids cyclic or chaotic dynamics (see SI texts 1-2 and figs. S1-S4). In consequence, the population dynamics simplifies to: ( NJ = NA × F × (1 − itr ) × d (2) NA = NA × itr + sJ × NJ By solving the system, we show that at equilibrium the product of the lifetime potential fecundity F and of the density-dependent egg survival d equals 1/sJ . This property also emerges when d is formulated explicitly (SI text 1 and figs. S1-S4). This implies that all strategies reach a stable equilibrium where they have the same lifetime density-dependent fecundity.
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All reproduction strategies are neutral We model the evolutionary dynamics of the reproduction strategy by considering the fate of a single mutant with strategy itm appearing in a monomorphic population (with strategy itr ) whose dynamics is described by the system [2]. We use here the adaptive dynamics framework, which commonly assumes that mutations are rare enough for the resident population to reach an equilibrium before a mutant appears [10]. As we have seen, at this equilibrium the resident’s lifetime fecundity F × d equals 1/sJ . Another classical assumption of adaptive dynamics is that the resident population is large enough for the mutant to be negligibly rare at the beginning of the invasion. The density-dependent parameter d thus only depends on NAr (t)F (1 − itr ) and the mutant’s lifetime fecundity F × d equals that of the resident, 1/sJ – we relax the two assumptions above later (see Iteroparity also evolves in polymorphic populations and supplement), with no visible impact on our results. The dynamics of the mutant can thus be modeled with the following system: ( NJm (t + 1) = NAm (t) × (1 − itm )/sJ (3) NAm (t + 1) = NAm (t) × itm + sJ × NJm (t) Initially, the mutant is a juvenile so NJm (0) = 1 and NAm (0) = 0. The mutant population may grow from this point, and its ability to do so is given by the Lyapunov exponent, or invasion fitness, denoted log λ(itm , itr ) [11–15]: N (T ) 1 log , (4) log λ(itm , itr ) = lim T →∞ T N (0) where N (t) = NJm (t) + NAm (t). Because F × d is equal for all residents, the invasion dynamics of the mutant is independent of the resident’s reproduction strategy itr (see system [3]). Moreover, the asymptotic growth rate of the mutant can be calculated as the first eigenvalue of the matrix: itm sJ M= , (5) (1 − itm )/sJ 0 which equals 1 for all itm (see appendix). Therefore, the invasion fitness equals 0 for all mutants and no reproduction strategy should increase (or decrease) in frequency in response to selection. Changes in frequency might nonetheless occur due to neutral processes like genetic drift: we study their impact on the evolutionary dynamics of reproduction strategies in the following section.
The neutral evolution of iteroparity Here we model the production of offspring, their mutation and fixation, using a Markov chain. As before, the number of offspring produced by an adult with reproduction strategy iti at any time step equals (1 − iti )/sJ . With probability µ, an offspring carries a mutation that changes its reproduction strategy. Here again, µ is small enough that a mutant can appear and fix before any other mutation occurs. Because reproduction strategies are neutral, a mutant fixes with probability ǫ regardless of its strategy itm . Ignoring the transient dynamics of mutant fixation, a population will always be monomorphic. We assume that mutations have small effects on the reproduction strategy, such that a population with strategy iti can only mutate to the lower iti−1 or the higher iti+1 . The probability of transition to either of these states equals: µ 1 − iti × × ǫ, (6) pi,i−1 = pi,i+1 = sj 2 and the probability of remaining in state i equals: pi,i = 1 − pi,i+1 − pi,i−1 4
(7)
For the first and last of the N states in the Markov chain, only the probabilities of transition p1,2 and pN,N −1 can be calculated with equation [6], and the probabilities of remaining in those states are p1,1 = 1 − p1,2 and pN,N = 1 − pN,N −1 , respectively. pi−1,i i-1
pi,i+1 i+1
i pi,i−1
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Figure 2: Schematic representation of the model for the neutral evolution of iteroparity. The states i − 1, i and i+1 correspond to populations with different reproduction strategies, with iti−1 < iti < iti+1 – i.e. the degree of iteroparity increases from left to right. The values of pi,i−1 , pi−1,i , pi+1,i and pi,i+1 can be calculated using eq. [6]; higher probabilities are represented by thicker arrows. pi,i−1 equals pi,i+1 according to eq. [6], is lower than pi−1,i (because 1 − iti−1 > 1 − iti ) and is higher than pi+1,i (because 1 − iti+1 < 1 − iti ). The probabilities of remaining in a given state are not shown, but can be calculated from eq. [7]. The evolutionary dynamics of the reproduction strategy can thus be described by the Markov chain represented in Fig. 2. Using equations [6] and [7], one can create a Markov chain of any length by dividing the range of possible reproduction strategies into a given number of adjacent values of it. We used the set {0, 0.01, ..., 0.99} and calculated the equilibrium state of the Markov chain using the markovchain package of the R software [16, 17]. Strikingly, at equilibrium, the most likely reproduction strategy has the highest level of iteroparity (it = 0.99; Fig. 3). Under this model, semelparity is expected to be observed less than 0.2% of the time on an extremely long time series where many neutral mutations have fixed, or in less than 2 populations, in a sample of 1000 populations that would have split long ago. Medium to high degrees of iteroparity – i.e. it > 0.5, that is more than 2 reproductive events on average – are expected about 86% of the times. High degrees of iteroparity are more likely to evolve because a population with reproduction strategy iti−1 produces more offspring per reproduction event, and thus more mutants, than a population with a slightly higher level of iteroparity iti+1 . Therefore, a population that switches from state i to state i + 1 is less likely to switch back to i than a population switching from i to i − 1 (see fig. 2). This yields a small bias towards increasing the degree of iteroparity at each step, which does not stop until the highest degree of iteroparity has been reached. To formulate this idea in biological terms, a population with a higher degree of iteroparity generates fewer mutants per timestep – most adults at t + 1 are adults at t that survived – than a semelparous population that is renewed every time. Therefore, while exploring every possible reproduction strategies neutrally, a population remains iteroparous longer – more so if the degree of iteroparity is high – and is more likely to be observed in that state.
Iteroparity also evolves in polymorphic populations The models described in the two sections above rely on the assumption that mutations are rare, so that the population is only transiently polymorphic. Moreover, in the neutral Markov chain, mutants can only reach neighboring reproduction strategies, even though the degree of iteroparity might vary continuously. These assumptions can be relaxed in individual based simulations where mutations occur with probability µ and change the offspring strategy from its parent’s by an amount sampled in a continuous distribution (see Simulation procedure in the Material and methods section). The model otherwise matches the discrete life cycle described above: any juvenile survives with probability sJ , in which case it becomes an adult. An adult with genotype i survives with probability iti at each time step and produces F (1 − iti ) eggs on average.
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Figure 3: High degrees of iteroparity are the most likely evolutionary outcome for a regulated population living in stable environment. In this context, only neutral processes can impact the evolution of the reproduction strategy. The black bars (left y -axis) represent the equilibrium probability that a given population will have a given it, under the neutral model described in Fig. 2. The red line (right y -axis) represents the probability that a population be at a given it or below. P The i NEi eggs produced by all genotypes are inPcompetition for survival. The function d needs to − i NEi /K throughout this paper, such that survival be defined explicitly Phere: we use the exponential e is close to 1 when i NEi is small and decreases as this number increases. It is worth noting that within this simulation framework, the study of adaptive dynamics and of neutral evolution are no longer decoupled. Neutrality may arise from the density dependent process – if this result remains valid under the assumptions of the current model – in which case a high degree of iteroparity may evolve neutrally, as expected from the Markov chain above. We ran 100 replicate simulations with µ = 0.001 and F = 5. The simulations were initiated with a single genotype with it1 = 0.5 (fig. 4, t = 0). Initially, the replicate populations diverge and exhibit a large range of reproduction strategies (fig. 4, t = 105 ), with a slight tendency towards increasing it. At higher simulation times (fig. 4, t = 4 × 105 and t = 2 × 106 ), the mean reproduction strategy is, in most replicate simulations, very close to the maximum possible it. Highly iteroparous genotypes evolve under a wide range of parameter values, including even higher mutation rates, lower fecundity and different initial it (see SI text 2 and fig. S5). We also studied another form of density-dependency, and obtain very similar results (SI text 2 and fig. S6). These results confirm that high levels of iteroparity are generally expected to evolve in a stable, density-dependent environment.
Discussion Considering a linear trade-off between litter size and adult survival, such that any strategy along the trade-off has the same overall fecundity, we found that those strategies are effectively neutral in a densitydependent environment yielding stable population dynamics. This result contrasts with the usual prediction – made by neglecting population regulation but considering an otherwise similar context – that semelparity has a strong selective advantage.
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Figure 4: Simulations confirm that high it should evolve in polymorphic populations living in a stable densitydependent environment. The simulation procedure is described in the material and Methods section. The distribution of the mean iteroparity it among 100 replicate population is represented at different simulation times. Parameter values: F = 5, µ = 0.001. Two previous studies have obtained similar results without clearly emphasizing them. In his densitydependent model, Takada [4] found that a genotype with adult survival it has a fitness equal to f /(1 − it), with f the litter size. In order to prevent any direct selective advantage to a given reproduction strategy, we assumed a constant lifetime fecundity F across strategies, by setting f = F (1 − it). Under this assumption, Takada’s model predicts that all reproduction strategies should have equal fitness F – i.e. they should be neutral, as we predict. Bulmer [8] predicted the result of a competition between an annual (semelparous) and a perenial plant (iteroparous) in a stochastically varying and density-dependent environment. Studying a case where the trade-off is linear – in our framework, Bulmer’s perenial would have an it equal to 0.5 and an annual fecundity divided by 2 – and where environmental variations have small effects, he predicted that either strategy may fix, with probabilities proportional to their initial frequencies. This result is very similar to our expectation that any strategy should be neutral in a constant, density-dependent environment, and it indicates that this result might hold even when small environmental fluctuations occur. Neither Bulmer nor Takada studied the neutral evolutionary dynamics of the reproduction strategies, which we found should result in a majority of replicate populations adopting a highly iteroparous strategy. The reason is simple: semelparous individuals produce more offspring (and thus more mutants) during a given time period than iteroparous, so a semelparous population or subpopulation is more likely to be replaced by a genotype with a different strategy. Iteroparity is therefore more likely to evolve because it reduces population turnover and thus the occurrence of mutants with different reproduction strategies. Of course, selection for a specific reproduction strategy may overcome these neutral dynamics, and the range of conditions where neutral processes dominate remains to be explored. Large environmental fluctuations provide a well-known selection pressure for iteroparity, as iteroparity can act as a bet-hedging strategy [8, 9, 18]. Such fluctuations should thus make semelparity even less likely to evolve; they might also select for an intermediate iteroparity strategy and decrease the probability that the highest level of iteroparity evolves. 7
So why would semelparity ever evolve? Or, to be more precise, why is it more frequent – we are not aware of a precise estimate, but semelparous species are indeed frequently observed in nature – than the 0.2% that our neutral model predicts? Interestingly, Mittledorf and Martins recently showed that a variable environment might select for a rapid population turnover – a property of semelparity – because it increases the number of mutations occurring during a given time period, which may facilitate adaptation towards a scarcely changing phenotypic optimum [19]. This apparent theoretical conflict – a changing environment can favor both semelparity and iteroparity – needs to be resolved. The frequency of environmental changes and the number of possible environmental optima should be key variables, as bet-hedging strategies (such as iteroparity) are most relevant when the environment changes often between few optima. Aside from environmental variation, the only other selection pressure known to act on reproduction strategies arises from the form of the trade-off between litter size and adult survival, with some trade-offs – convex ones – increasing the overall fecundity of semelparous strategies and thus providing a direct advantage to them. This likely represents a stronger selection pressure for semelparity than infrequent environmental changes, suggesting that the form of this trade-off probably has a prominent role on the evolution of reproduction strategies. As we described in the introduction, the feeding strategy may affect the form of the trade-off between litter size and adult survival, and thereby control the evolution of the reproduction strategy. For instance, an early acquisition of resources leads adults to deal with a limited amount of resources, such that they need to sacrifice future reproduction for survival. This constraint thus yields a convex trade-off and presumably favors semelparity over iteroparity. In the opposite situation where adults can acquire resources between reproduction events, iteroparity may provide an increased lifetime fecundity and should thus be favored. There is little doubt that the feeding strategy can change and evolve, and this evolution probably occurs conjointly with that of the position along the trade-off – i.e. the reproduction strategy. The evolution of semelparity and iteroparity cannot be clearly understood, in our opinion, before a theory incorporates this joint evolutionary process – as well as the underlying eco-physiological processes [20]. This theory should allow neutral and adaptive processes to act, and the present study suggests that neutrality should play a key role whenever the ecological context does not favor the evolution of a markedly convex or concave trade-off between adult survival and fecundity.
Material and methods Simulation procedure We simulated a population of a variable number N of genotypes with different reproduction strategies iti . Each genotype j is represented by NJi juveniles and NAi adults. At a given timestep, adults with genotype i produce a number of eggs NEi sampled from a Poisson distribution with mean NAi × F × (1 − iti ). PN Eggs survive with the density dependent probability e− i=1 NEi /1000 ; for each genotype, the number of surviving eggs is sampled from a binomial with this probability and NEi trials. Note that we also use a different density-dependent function in the supplement. The surviving eggs constitute the juveniles at the beginning of the next timestep. Each offspring of genotype i can mutate with probability µ. When this event occurs, the number of genotypes N is incremented by 1 and the new genotype N has NAN = 0, NJN = 1, and itN = iti + ǫ. ǫ is sampled from a normal distribution with mean 0 and standard deviation 0.05; mutants with itN below 0 or above 0.99 take the value 0 and 0.99, respectively. The number of juveniles with genotype i at the beginning of the next timestep is decremented by 1. After reproduction, the number of surviving adults of genotype j is sampled from a binomial with NAj trials and probability itj . These will constitute the adults with genotype i at the next timestep,
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together with the offspring produced at the previous timestep that survive, whose number is sampled from a binomial with NJi trials and probability sJ . At each timestep, a genotype is suppressed if NAi = NJi = 0. In each replicate simulation, we simulated the process described above during 2 × 106 timesteps, and recorded NAi , NJi and iti for each of the i genotypes present in the population. We ran 100 replicate simulations for each parameter set explored; the distribution of the mean values of it in the 100 replicates is represented in fig. 4 at different timesteps. The program was written in the R programming language and is available as a supplement.
Appendix: mutant’s asymptotic growth rate The mutant’s asymptotic growth rate is calculated as the largest eigenvalue of M (defined in eq. [5]), which can be obtained by solving |M − λI| = 0. This yields the characteristic polynomial: (itm − λ) × (−λ) − (1 − itm )/sJ × sJ = 0,
(8)
(λ − 1)(λ + 1 − itm ) = 0
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which simplifies to: Hence the polynomial has roots λ1 = 1 and λ2 = itm − 1. λ1 > λ2 for 0 ≤ itm < 1 so the asymptotic growth rate of the mutant equals 1.
Acknowledgments The authors thank Marine Ginoux and Anne Nguyen for stimulating discussions at the beginning of this project, and Jean-Fran¸cois Lemaitre for helpful suggestions. This work has been supported by the French National Research Agency (grant reference “ANR-08-MIE-007”), by the Centre National de la Recherche Scientifique (CNRS, UMR 5558), and by the Universit´e Claude Bernard Lyon 1.
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[10] Geritz S, Kisdi E, Mesz´ena G, Metz J (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12:35–57. [11] Metz J, Nisbet R, Geritz S (1992) How should we define ’fitness’ for general ecological scenarios? Trends Ecol. Evol. 7:198–202. [12] van Dooren TJM, Metz JAJ (1998) Delayed maturation in temporally structured populations with non-equilibrium dynamics. J. Evol. Biol. 11:41–62. [13] Rand D, Wilson H, McGlade J (1994) Dynamics and evolution: Evolutionarily stable attractors, invasion exponents and phenotype dynamics. Phil. Trans. R. Soc. B 343:261–283. [14] Ferriere R, Gatto M (1995) Lyapunov exponents and the mathematics of invasion in oscillatory or chaotic populations. Theor. Popul. Biol. 48:126–171. [15] Roff DA (2008) Defining fitness in evolutionary models. Journal of genetics 87:339–348. [16] Spedicato GA (2014) markovchain: discrete time Markov chains made easy. [17] R Core Team (2013) R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, Austria). [18] Orzack SH, Tuljapurkar S (1989) Population dynamics in variable environments. vii. the demography and evolution of iteroparity. American Naturalist pp 901–923. [19] Mitteldorf J, Martins ACR (2014) Programmed life span in the context of evolvability. Am Nat 184:289–302. [20] Bonnet X (2011) The evolution of semelparity. Reproductive biology and phylogeny of snakes. Reproductive Biology and Phylogeny Series. Science Publishers Inc 17:645–672.
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Supplementary information SI text 1:
Density-dependent population dynamics
Here we study the population dynamics described by the system [1] in the main paper, replacing d by two density-dependency functions. Density-dependency function 1: d = e−
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P Here the population is monomorphic, so the number of eggs produced at a given timestep, i NEi equals NA (t) × F × (1 − it). We simulated the system with sJ = 0.9 and F = 5, starting with NA = 10 and NJ = 10. As shown in fig. S1, this function yields a stable dynamic equilibrium when F < 8, with juvenile survival sJ = 0.9 and adult survival it = 0 (i.e semelparous). Lower sJ yield unstable population dynamics at higher values of F (e.g. at F = 15 for sJ = 0.5). Increasing it has a similar impact, as shown in fig. S1 (bottom panel) for it = 0.3. We ran simulations in the stable regime (F = 5, sJ = 0.9) for 100 generations with it = 0.1, 0.5 and 0.9 (fig. S2). The population equilibrium is indeed stable and quickly reached. At the equilibrium, the density-dependent lifetime fecundity F × d equals 1/sJ , as expected from the more general theoretical approach in the main paper. Density-dependency function 2: d =
1
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P This function yields a sigmoidal relationship between dP and the number of eggs produced, i NEi – as before, the population is monomorphic at this point so i NEi = NA (t) × F × (1 − it). Two parameters control the shape of the function: K is the number of eggs at which d = 0.5, and beta controls the slope of the function at this point (it is steeper at higher β). As before, the population dynamics become cyclic or chaotic as F increases; For the parameters used in fig. S3 (top panel, β = 0.001, K = 400, sJ = 0.9, it = 0), this occurs when F > 8. The shape of the function plays an obvious role, as shown in the bottom panel of fig. S3, where we see that the population dynamics are unstable when β is over 0.0033. We also ran simulations in the stable regime with this function (F = 5, sJ = 0.9, β = 0.001) for 100 generations with it = 0.1, 0.5 and 0.9 (fig. S4). At the equilibrium, the density-dependent lifetime fecundity F × d equals 1/sJ , as with the first function above and as we show should generally be expected in regulated populations with stable population dynamics (see main text).
SI text 2:
Evolutionary dynamics
Density-dependency function 1: d = e−
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We simulated the evolution of it in 100 replicate populations, as described in the Material and methods section of the main paper. We used different parameter sets, showing that neither the mutations rate µ, the potential fecundity F nor the initial value of it in the population change the evolutionary outcome (fig. S5): in every case studied, a very high it evolves, presumably as a result of neutral evolutionary dynamics (see paper). Note that the simulations were run for 2 × 106 generations when µ = 10−3 and for 10−5 generations when µ = 10−2 . Density-dependency function 2: d =
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the main text and in the previous section. Changing d does not impact the results: in most replicate simulations, a very high level of iteroparity evolves, independently of the mutation rate µ, potential fecundity F and initial level of iteroparity it.
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Potential lifetime fecundity, F
5
10 Potential lifetime fecundity, F
Figure S1: Bifurcation diagram for the density-dependency function 1. For each value of F , the total population size (NA + NJ ) is represented for 100 timesteps after an initial simulation period of 100 steps. More than one value of N (t) mean that the equilibrium is unstable (cyclic or chaotic), which occurs here when F > 8 (top panel) or F > 13 (bottom).
12
F×d
5 0
200
1 sJ
400
2.5
600
800
NA NJ F×d
0
number of individuals
it = 0.1
0
20
40
60
80
100
60
80
100
60
80
100
time
5 2.5 1 sJ
F×d
800 600 400 200
0
0
number of individuals
it = 0.5
0
20
40 time
2.5 1 sJ 0 0
20
40 time
Figure S2: Population dynamics simulated using the density-dependency function 1.
13
F×d
2500 1500 500 0
number of individuals
5
it = 0.9
800 1000 600 400 0
200
N(t)
5
10
15
20
0
500
N(t)
1000
1500
Potential lifetime fecundity, F
0.001
0.003
0.005
0.007
0.009
Slope of the density−dependency function, beta
Figure S3: Bifurcation diagram for the density-dependency function 2. For each value of F (top panel) or β (bottom), the total population size (NA + NJ ) is represented for 100 timesteps after an initial simulation period of 100 steps. More than one value of N (t) mean that the equilibrium is unstable (cyclic or chaotic), which occurs here when F > 8 (top panel) or β > 0.0033 (bottom).
14
F×d
5 0
200
1 sJ
400
2.5
600
800
NA NJ F×d
0
number of individuals
it = 0.1
0
20
40
60
80
100
60
80
100
60
80
100
time
5 2.5 1 sJ
F×d
800 600 400 200
0
0
number of individuals
it = 0.5
0
20
40 time
2.5 1 sJ 0 0
20
40 time
Figure S4: Population dynamics simulated using the density-dependency function 2.
15
F×d
2500 1500 500 0
number of individuals
5
it = 0.9
0.8 0.6 0.0
0.2
0.4
it 0.0
0.2
0.4
it
0.6
0.8
1.0
µ = 10−2
1.0
µ = 10−3
itinit = 0
itinit = 0.5 F=2
itinit = 0
itinit = 0.5
itinit = 0
F=5
itinit = 0.5 F=2
itinit = 0
itinit = 0.5 F=5
Figure S5: Distribution of the mean value of it evolving in 100 replicate simulations (for each parameter set) where the density dependent egg survival is modeled by function 1, with K = 1000. We used two high mutation rates µ = 10−3 (left panel) or µ = 10−2 (right), with two values for the potential fecundity F (2 or 5). We initiated the simulations with two initial values of it: 0 and 0.5.
16
0.8 0.6 0.0
0.2
0.4
it 0.0
0.2
0.4
it
0.6
0.8
1.0
µ = 10−2
1.0
µ = 10−3
itinit = 0
itinit = 0.5 F=2
itinit = 0
itinit = 0.5
itinit = 0
F=5
itinit = 0.5 F=2
itinit = 0
itinit = 0.5 F=5
Figure S6: Distribution of the mean value of it evolving in 100 replicate simulations (for each parameter set) where the density dependent egg survival is modeled by function 2, with K = 400 and β = 0.001. We used two high mutation rates µ = 10−3 (left panel) or µ = 10−2 (right), with two values for the potential fecundity F (2 or 5). We initiated the simulations with two initial values of it: 0 and 0.5.
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