is made of a series of local extinctions and recolonization events, and the rates of extinction and recolonization are equal for all species. What emerges from this scenario is the celebrated Wilson-MacArthur model of island biogeography: the species richness on the island, $S$, satisfies $\dot{S} = -e S + r(Q-S)$, where $e$ is the extinction rate and $r$ is the recolonization rate, both rates will depend on $K$ and on the details of the stochastic process. The Wilson-MacArthur prediction for the average species richness is $\bar{S} = rQ/(r+e)$, the typical size of species richness fluctuations is $\sqrt{S}$, and the statistics of sojourn times (the periods between colonization and extinction, as well as the periods between extinction and colonization) is exponential.

When the only stochastic effect taken into account is demographic noise, as is the case in this paper, the chance of extinction decreases exponentially with $K$ [@kessler2007extinction] and therefore for the typical values of $K$ considered here $e \ll 1$. If this is actually the case, or when $\lambda$ is large so $r \to \infty$, all mainland species are presented on the mainland up to tiny short-term fluctuations.($S \approx Q$). Accordingly, in our model one observes Wilson-MacArthur dynamics on reasonable 