Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue

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We consider the system of reaction-diffusion equations proposed in [8] as a population dynamics model. The first equation stands for the population density and models the ecological effects, namely dispersion and growth with a Allee effect (bistable nonlinearity). The second one stands for the Allee threshold, seen as a trait mean, and accounts for evolutionary effects. Precisely, the Allee threshold is submitted to three main effects: dispersion (mirroring ecology), asymmetrical gene flow and selection. The strength of the latter depends on the population density and is thus coupling ecology and evolution. Our main result is to mathematically prove evolutionary rescue: any small initial population, that would become extinct in the sole ecological context, will persist and spread thanks to evolutionary factors.