In this work, we characterize the solution of a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection and migration between two habitats. Assuming that the effects of the mutations are small but nonzero, we show that the population’s phenotypical distribution has at most two peaks and we give explicit conditions under which the population will be monomorphic (unimodal distribution) or dimorphic (bimodal distribution). More importantly, we provide a general method,based on Hamilton-Jacobi equations with constraint, to determine the dominant terms of the population’s distribution in each case. This method allows indeed to go further than the Gaussian approximation commonly used by biologists and and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics in theoretical evolutionary biology.
This is a joint work with Sylvain Gandon.