A reaction-diffusion system arising in population genetics

This paper is concerned with a nonweakly coupled system of parabolic equations that models the reaction-diffusion process of a population of continuously reproducing diploids with two alleles at one locus of genes. We assume that the fitnesses of genotypes are density dependent and the boundary condition is Dirichlet type with the boundary data given in the form such that the system has a unique uniform steady-state solution. The existence of nonuniform steady-state solutions is examined in relation with the stability of uniform steady-state solutions. In several cases we show that a nonuniform steady-state solution exists if the uniform one is unstable. We also investigate the stability and attractivity of the uniform steady-state solution in various cases. Some results of global convergence of solutions to a uniform steady-state solution are also given.