Coexistence and stability of a competition—diffusion system in population dynamics

The coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper, where the effect of dispersion (transportation) is taken into consideration. The mathematical problem involves a coupled system of Lotka-Volterra-type reaction-diffusion equations together with some initial and boundary conditions, including the Dirichlet, Neumann and third type. Necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given. For the stability problem, general criteria for the stability and instability of a steady-state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters. Also given are necessary and sufficient conditions for the existence of multiple steady-state solutions and the stability or instability of each of these solutions. Special attention is given to the Neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized. It is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable. The stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients.