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

Abstract

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, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. From the continuous approximations, we then obtain asymptotic approximations for evolutionary dynamics with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we study the fixation pattern when the infinite population limit has an interior Evolutionary Stable Strategy (ESS): (i) 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; (ii) we also show that, outside of the weak-selection regime, the long-term dynamics of large populations can have very little resemblance to the infinite population case; in addition, we also present results for the case of two equilibria. Finally, we present continuous restatements valid for large populations of two classical concepts naturally defined in the discrete case: (i) the definition of an $\textsc{ESS}_N$ strategy; (ii) the definition of a risk-dominant strategy. We then present two applications of these restatements: (i) we obtain an asymptotic definition valid in the quasi-neutral regime that recovers both the one-third law under linear fitness and the generalised one-third law for $d$-player games; (ii) we extend the ideas behind the (generalised) one-third law outside the quasi-neutral regime and, as a generalisation, we introduce the concept of critical-frequency; (iii) we recover the classification of risk-dominant strategies for $d$-player games.

References in corpus (14)

• Stochastic Dynamics of Invasion and Fixation
• Coevolutionary Dynamics: From Finite to Infinite Populations
• Evolutionary games in the multiverse
• Fixation of strategies for an evolutionary game in finite populations
• A microscopic probabilistic description of a locally regulated population and macroscopic approximations
• Quasi-stationary distributions and population processes
• Universality of weak selection
• Coevolutionary dynamics in large, but finite populations
• Stochasticity and evolutionary stability
• Fixation times in evolutionary games under weak selection
• Evolutionary Multiplayer Games
• Stochastic differential equations for evolutionary dynamics with demographic noise and mutations
• The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations
• Large Fluctuations and Fixation in Evolutionary Games

Cited by in corpus (5)

• On the stochastic evolution of finite populations
• Fixation probabilities for the Moran process in evolutionary games with two strategies: graph shapes and large population asymptotics
• Fitness potentials and qualitative properties of the Wright-Fisher dynamics
• Population dynamics and games of variable size
• Insuperable strategies in two-player and reducible multi-player games