The effect of fecundity derivatives on the condition of evolutionary branching in spatial models
Introduction
Evolutionary branching is a process in which the trait of an evolving monomorphic population first approaches a so-called singular trait, but then disruptive selection causes the population to become dimorphic, i.e., to contain two different resident traits, and these two traits evolve away from each other (Metz et al., 1992, Metz et al., 1996, Geritz et al., 1997, Geritz et al., 1998). When mutations are so frequent that there is no clear separation between ecological and evolutionary time-scales, evolutionary branching means that a unimodal trait distribution first concentrates around the singular strategy, and then the distribution becomes bimodal.
Invasion fitness (Metz et al., 1992) is the long-term exponential growth rate of a rare mutant in an environment set by the resident. At singular strategies the first-order derivative of the invasion fitness vanishes. The condition for evolutionary branching is usually given by calculating the second-order derivatives of invasion fitness at a singular strategy. There is, however, another approach to study the branching condition. Instead of considering a mutant–resident system, we can study the dynamics of a continuous trait distribution and identify evolutionary branching as the increase of the variance of the distribution (Sasaki and Dieckmann, 2011, Wakano and Iwasa, 2012). In a case of a well-mixed population, the branching condition derived by calculating invasion fitness and that by calculating variance dynamics have been shown to be identical when the trait distribution is approximated by the Gaussian distribution. In case of a spatially structured population, comparing these approaches requires more detailed calculations.
The metapopulation reproduction ratio (metapopulation fitness) is a fitness proxy that measures the growth of a mutant population between dispersal generations in an environment set by resident. (Metz and Gyllenberg, 2001, Parvinen and Metz, 2008). By investigating the metapopulation fitness, the branching conditions have been studied for several different metapopulation models (Parvinen, 2002, Parvinen, 2006, Nurmi and Parvinen, 2008, Nurmi and Parvinen, 2011). On the other hand, the trait distribution approach can also be extended to spatially structured populations and an analytic expression for the branching condition has been derived by Wakano and Lehmann (2014) for a specific model. In structured populations, the trait distribution cannot be described by a single Gaussian distribution (as in a well-mixed case) because different demes (local patches) can have different trait distributions and because individuals in the same deme tend to have similar trait values. In other words, the individual trait value is no longer an independent random variable sampled from the same distribution and we need to take into account the positive correlation of trait values within a deme. This correlation can be expressed in terms of relatedness and as a result the branching condition is given by a combination of fitness derivatives and relatedness coefficients. The analytically derived condition by Wakano and Lehmann (2014) agreed with their simulations.
In this article we investigate Wright's island model, which is a discrete-time metapopulation model in which the number of adults in each deme is fixed through generations. The relative fecundity of each adult depends on its own inheritable trait and the traits of other adults in the same deme. The individuals to become adults in the next generation are randomly chosen among philopatric and dispersed offspring.
Assuming locally a fixed number of adults is not very realistic, and also not strictly speaking even necessary, because evolution in metapopulation models with more realistic local population dynamics has been successfully analysed using the metapopulation fitness (see references above). However, this simplifying assumption allows one to obtain general analytic expressions for the selection gradient and conditions for convergence stability and evolutionary stability. Ajar (2003) obtained such expressions by calculating the metapopulation fitness, while Wakano and Lehmann (2014) used the trait distribution approach. Both studies express their main results in such terms of relatedness coefficients, which might discourage researchers to apply these results to practical questions if they are not very familiar with inclusive fitness theory.
The first goal of this study is to explicitly show the selection gradient and conditions for convergence stability and evolutionary stability in terms of derivatives of the fecundity function and original spatial parameters (deme size, dispersal rate and the probability to survive dispersal). The use of our expressions is straightforward, and they are valid for any fecundity function. In this form it will be clearly observed that singular strategies in the spatial model are the same as in the well-mixed case (Taylor, 1992a, Taylor and Irwin, 2000), also called a cancellation result. Also the condition for convergence stability remains unchanged, whereas the condition for evolutionary stability is affected by the spatial structure.
The second goal is to study whether spatial structure promotes or inhibits evolutionary branching. For the direction of evolution in spatial models (e.g., evolution of cooperation), tremendous amount of papers have been published. Compared to them, the effect of spatial structure on evolutionary branching has been far less studied. Wakano and Lehmann (2014) have shown that when fecundity is determined by repeated snowdrift games (Doebeli et al., 2004) between individuals within the deme, a smaller dispersal rate inhibits branching. This was confirmed by their individual-based simulations but their analysis is only a numerical calculation of the general formula of the condition for evolutionary stability. Thus, it is not clear whether spatial structure always inhibits branching for any kind of local interactions or there exist some kind of interactions that trigger branching only when spatial structure is introduced. We aim to answer this question by investigating the explicit expression determining evolutionary stability.
This paper is organized as follows. In Section 2 we describe the model and formulate the metapopulation reproduction number. The general explicit expression for the selection gradient and the second order derivatives are presented in Section 3. Especially, in the condition of evolutionary stability the coefficients of each fecundity derivative (=derivative of the fecundity function) have fixed signs. In Section 4 we prove general results suggesting that the spatial structure of Wright's island model often, but not always, inhibits evolutionary branching. As a counterexample we present an artificial fecundity function for which branching occurs only for intermediate values of the dispersal rate. In Section 5 we apply our results to situations in which fecundity is determined by any pairwise game (not just the snowdrift game), or by a public-goods game.
Section snippets
Island model and fecundity function
We consider an extended version of Wright's island model (Wright, 1931). We assume that there are infinitely many habitat patches (demes). In the beginning of the season each patch contains adult individuals. These adults may differ in their strategies s, which affect their fecundity that represents the number of juveniles that they produce. Throughout the manuscript, γ is considered to be very large (actually ). More precisely, the relative fecundity for an adult with strategy s1,
First-order results
Because of the symmetry property of , there are essentially only two different first-order derivatives of F. One is the first-order derivative with respect to the strategy of self, which is defined asThe other is the first-order derivative with respect to the strategy of anybody else in the patch, defined asbecause the right-hand side of that equality is independent of the choice of k. Note
Spatial structure inhibits branching in a wide class of fecundity functions
Using a continuous snowdrift game (Doebeli et al., 2004) as an example, Wakano and Lehmann (2014) have shown that a branching point (evolutionarily attracting singular strategy, which is not uninvadable) in a well-mixed model changes to be evolutionarily stable (uninvadable) as the migration rate decreases below a threshold value. We can generalize this result in the form of the following theorem. . Theorem 4 Evolutionary branching is not possible for a sufficiently small value of d (that is, small m or
Pairwise games
Assume that individuals in the deme play pairwise games among each other and that the total payoff from these games determines the fecundity of each individual. We can either assume that a certain number of games is played, and the game participants are randomly chosen, or that all possible combinations of games take place. In some games the role of individuals matters. In such a situation, let denote the payoff of the individual using strategy in role i matched with
Discussion
We have studied evolution by natural selection in Wright's island model in which there is an infinite number of patches (demes) of constant, finite size. In each season adults produce offspring, and the fecundity of each adult depends on its own strategy as well as the strategies of other individuals in the focal patch. A proportion of juveniles disperses to other patches. Since adults do not survive until the next season, the fixed number of offspring to become adults are randomly chosen among
Acknowledgements
KP wishes to thank Akira Sasaki for the invitation to the JSMB/SMB 2014 meeting in Osaka, Japan, during which the collaboration, that eventually lead to this work, was initiated. We want to thank anonymous reviewers for insightful comments. Support from JSPS KAKENHI Grant number 25118006 to HO, and 25870800 and 16K05283 to JYW are gratefully acknowledged.
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