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We study fixation in large, but finite, populations with two types, anddynamics governed by birth-death processes. By considering a restricted classof such processes, we derive a continuous approximation for the probability offixation that is valid beyond the weak-selection (WS) limit. From thecontinuous approximations, we then obtain asymptotic approximations forevolutionary dynamics with at most one equilibrium, in the selection-drivenregime, that does not preclude a weak-selection regime. As an application, westudy the fixation pattern when the infinite population limit has an interiorEvolutionary Stable Strategy (ESS): (i) we show that the fixation pattern forthe Hawk and Dove game satisfies what we term the one-half law: if theEvolutionary 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 largepopulations 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 classicalconcepts naturally defined in the discrete case: (i) the definition of an$\textsc{ESS}_N$ strategy; (ii) the definition of a risk-dominant strategy. Wethen present two applications of these restatements: (i) we obtain anasymptotic definition valid in the quasi-neutral regime that recovers both theone-third law under linear fitness and the generalised one-third law for$d$-player games; (ii) we extend the ideas behind the (generalised) one-thirdlaw outside the quasi-neutral regime and, as a generalisation, we introduce theconcept of critical-frequency; (iii) we recover the classification ofrisk-dominant strategies for $d$-player games.

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