ASYMPTOTIC BEHAVIOR FOR THE LOTKA–VOLTERRA EQUATION WITH DISPLACEMENTS AND DIFFUSION

Abstract

In this paper, we consider the Lotka–Volterra equation with displacements and diffusion, that is transport-diffusion system describing the evolution of prey and predator populations with their displacements and their diffusion, in a periodic domain in $\mathbb{R}$. It is shown that the solution to this equation and its logarithm are globally bounded, and that, when the solution converges to the stationary solution in mean value, it converges uniformly with respect to the time variable as well as the space variable. These results are obtained by using $L^2$-estimate of the well-known Lyapunov functional, and, in particular, an estimate of the point-wise growth of the solution by means of the application of the fundamental solution of the heat equation