A Hamilton–Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments

Abstract

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 to determine the dominant terms of the population’s distribution in each case. Our work, which is based on Hamilton–Jacobi equations with constraint, goes further than previous works where such tools were used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. The main elements for the computation of the dominant terms of the population’s distribution are the convergence of the logarithmic transform of the solution to the unique solution of a Hamilton–Jacobi equation and the computation of the correctors. This method allows indeed to go further than the Gaussian approximation commonly used by biologists and makes a connection between the theories of adaptive dynamics and quantitative genetics. Our work being motivated by biological questions, the objective of this paper is to provide the mathematical details which are necessary for our biological results [S. Mirrahimi and S. Gandon, The equilibrium between selection, mutation and migration in spatially heterogeneous environments, in preparation].