Fixation in large populations: a continuous view of a discrete problem

Fabio A. C. C. Chalub, Max O. Souza

(Submitted on 27 Aug 2014)

We study fixation in large, but finite populations with two types, and dynamics governed by birth-death processes. By considering a restricted class of such processes, which includes most classical evolutionary processes, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. Indeed, in the derivation three regimes naturally appear: selection-driven, balanced, and quasi-neutral — the latter two require WS, while the former can appear with or without WS. From the continuous approximations, we then obtain asymptotic approximations for evolutions with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we show that the fixation pattern for the Hawk and Dove game satisfies what we term the one-half law: if the Evolutionary Stable Strategy (ESS) is outside a small interval around $\sfrac{1}{2}$, the fixation is of dominance type. We also show that outside of the weak-selection regime the dynamics of large populations can have very little resemblance to the infinite population case. In addition, we also show results for the case of two equilibria. Finally, we present a continuous restatement of the definition of an ESSN strategy, that is valid for large populations. We then present two applications of this restatement: we obtain a definition valid in the quasi-neutral regime that recovers the one-third law under linear fitness and, as a generalisation, we introduce the concept of critical-frequency.