Stable Coexistence States in the Volterra–Lotka Competition Model with Diffusion

We study the existence, uniqueness, and stability of coexistence states in the Lotka–Volterra model with diffusion for two competing species. We assume that the parameters describing the interaction and self-limitation of the species are constant, and consider two types of growth and boundary conditions: periodic growth rates with Neumann boundary conditions, and constant growth rates with Dirichlet boundary conditions. In the case of periodic growth rates we find sufficient conditions for the existence of an attracting periodic coexistence state, thus extending results obtained by Gopalsamy in the model without diffusion. In the cases of constant growth rates and homogeneous Dirichlet boundary conditions we obtain sufficient conditions for the existence, uniqueness, and stability of time independent coexistence states; when the diffusion and growth rates are equal for the two species we give necessary and sufficient conditions for the existence of such states. The main tools for our investigations are existence and comparison theorems based on the maximum principle, upper and lower solutions, and the variational characterization of eigenvalues.