Stochastic viability in an island model with partial dispersal: Approximation by a diffusion process in the limit of a large number of islands

In this paper, we investigate a finite population undergoing evolution through an island model with partial dispersal and without mutation, where generations are discrete and non-overlapping. The population is structured into D demes, each containing N individuals of two possible types, A and B, whose viability coefficients, s_A and s_B, respectively, vary randomly from one generation to the next. We assume that the means, variances and covariance of the viability coefficients are inversely proportional to the number of demes D, while higher-order moments are negligible in comparison to 1/D. We use a discrete-time Markov chain with two timescales to model the evolutionary process, and we demonstrate that as the number of demes D approaches infinity, the accelerated Markov chain converges to a diffusion process for any deme size N≥2. This diffusion process allows us to evaluate the fixation probability of type A following its introduction as a single mutant in a population that was fixed for type B. We explore the impact of increasing the variability in the viability coefficients on this fixation probability. At least when N is large enough, it is shown that increasing this variability for type B or decreasing it for type A leads to an increase in the fixation probability of a single A. The effect of the population-scaled variances, σ_A² and σ_B², can even cancel the effects of the population-scaled means, μ_A and μ_B. We also show that the fixation probability of a single A increases as the deme-scaled migration rate increases. Moreover, this probability is higher for type A than for type B if the population-scaled geometric mean viability coefficient is higher for type A than for type B, which means that μ_A-σ_A²/2>μ_B-σ_B²/2.