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

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 many of the evolutionary processes usually discussed in the literature, 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 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): (1) we show that the fixation pattern for the Hawk and Dove game satisfies what we term the one-half law: if the ESS is outside a small interval around 1/2, the fixation is of dominance type; (2) 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, and show that even when there is weak-selection the long-term dynamics can be dramatically different from the one predicted by the replicator dynamics. Finally, we present continuous restatements valid for large populations of two classical concepts naturally defined in the discrete case: (1) the definition of an ESSN strategy; (2) the definition of a risk-dominant strategy. We then present three applications of these restatements: (1) 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; (2) 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; (3) we recover the classification of risk-dominant strategies for d-player games.