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 $N\ge 2$. 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, ${\sigma }_A^2$ and ${\sigma }_B^2$, can even cancel the effects of the population-scaled means, ${\mu }_A$ and ${\mu }_B$. 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 ${\mu }_A-{\sigma }_A^2/2>{\mu }_B-{\sigma }_B^2/2$.