Abstract
Models of evolution by natural selection often make the simplifying assumption that populations are infinitely large. In this infinite population limit, rare mutations that are selected against always go extinct, whereas in finite populations they can persist and even reach fixation. Nevertheless, for mutations of arbitrarily small phenotypic effect, it is widely believed that in sufficiently large populations, if selection opposes the invasion of rare mutants, then it also opposes their fixation. Here, we identify circumstances under which infinite-population models do or do not accurately predict evolutionary outcomes in large, finite populations. We show that there is no population size above which considering only invasion generally suffices: for any finite population size, there are situations in which selection opposes the invasion of mutations of arbitrarily small effect, but favours their fixation. This is not an unlikely limiting case; it can occur when fitness is a smooth function of the evolving trait, and when the selection process is biologically sensible. Nevertheless, there are circumstances under which opposition of invasion does imply opposition of fixation: in fact, for the n-player snowdrift game (a common model of cooperation) we identify sufficient conditions under which selection against rare mutants of small effect precludes their fixation—in sufficiently large populations—for any selection process. We also find conditions under which—no matter how large the population—the trait that fixes depends on the selection process, which is important because any particular selection process is only an approximation of reality.
Highlights
Adaptive dynamics is a widely used and extremely successful framework for investigating the evolution of continuous traits by natural selection (Brännström et al 2013)
We ask more precisely: for any given finite population size, if selection opposes the invasion of mutants playing a strategy sufficiently similar to the residents’, does it necessarily oppose their fixation? In Sects. 2 and 4, we show that the answer to this question is “no”: for any population size N, no matter how large, it is possible to construct well-behaved payoff functions such that there is a singular strategy at which selection opposes the invasion but favours the fixation of mutations of arbitrarily small effect
We identify a simple condition on n-player snowdrift games under which, if selection opposes the invasion of sufficiently similar mutants, it generically opposes their fixation in sufficiently large populations, regardless of the selection process
Summary
Adaptive dynamics is a widely used and extremely successful framework for investigating the evolution of continuous traits by natural selection (Brännström et al 2013). 2 and 4, we show that the answer to this question is “no”: for any population size N , no matter how large, it is possible to construct well-behaved payoff functions (and a selection process) such that there is a singular strategy at which selection opposes the invasion but favours the fixation of mutations of arbitrarily small effect Such a singular strategy is evolutionarily stable according to adaptive dynamics, but is not an ESSN. We identify a simple condition on n-player snowdrift games under which, if selection opposes the invasion of sufficiently similar mutants, it generically opposes their fixation in sufficiently large populations, regardless of the selection process When this condition holds, it provides a justification for analyzing evolutionary stability in finite populations using infinite-population models, and in particular, for ignoring the selection process in this context. Our analyses are presented in the context of a standard model for the evolution of cooperation, the n-player snowdrift game
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