Convergence to Homogeneous Equilibrium State for Generalized Volterra–Lotka Systems with Diffusion

Saturation terms are added to the classical Volterra–Lotka model and to generalizations of it which maintain its qualitative character. To describe the effect of migration, diffusion terms are introduced. The resulting models possess invariant rectangles, in which the solution remains for all times; in addition, they admit a Lyapunov functional. This enables one to prove the relative compactness of the trajectories, and the convergence to a spatially homogeneous equilibrium solution.