Asymptotic Behaviors of a Parabolic System Related to a Planktonic Prey and Predator Model

Abstract

There is considerable current interest in plankton population distribution and movement in the sea. We consider two models with two different types of plankton between which there exists a predator-prey relation. Denoting by $U(t,x) = (u_1 (t,x),u_2 (t,x))$ the densities of the plankton populations at time t and position $x \in \mathbb{R}^r $, we assume that $U(t,x)$ satisfies a certain semilinear parabolic system. This system is treated as a mixed initial-boundary value problem with zero Neumann boundary conditions. Results concern the asymptotic behavior of the solution for large time.