Establishment of a locally adaptive allele in multidimensional continuous space.

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Local adaptation is widely seen when species adapt to spatially heterogeneous environments. Although many theoretical studies have investigated the dynamics of local adaptation using two-population models, there remains a need to extend the theoretical framework to continuous space settings, reflecting the real habitats of species. In this study, we use a multidimensional continuous space model and mathematically analyze the establishment process of local adaptation, with a specific emphasis on the relative roles of mutation and migration. First, the role of new mutations is evaluated by deriving the establishment probability of a locally adapted mutation using a branching process and a diffusion approximation. Next, the contribution of immigrants from a neighboring region with similar environmental conditions is considered. Theoretical predictions of the local adaptation rate agreed with the results of Wright-Fisher simulations in both mutation-driven and migration-driven cases. Evolutionary dynamics depend on several factors, including the strength of migration and selection, population density, habitat size, and spatial dimensions. These results offer a theoretical framework for assessing whether mutation or migration predominantly drives convergent local adaptation in spatially continuous environments in the presence of patchy regions with similar environmental conditions.