Global asymptotic stability of Lotka–Volterra 3-species reaction–diffusion systems with time delays

Abstract

This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion systems and their corresponding ordinary differential systems without diffusion. The time delays may be discrete or continuous, and the boundary conditions for the reaction–diffusion systems are of Neumann type. The goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems. These conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays. The result of global asymptotic stability implies that each of the three model systems coexists, is permanent, and the trivial and all semitrivial solutions are unstable. Our approach to the problem is based on the method of upper and lower solutions for a more general reaction–diffusion system which gives a common framework for the 3-species model problems. Some global stability results for the 2-species competition and prey–predator reaction–diffusion systems are included in the discussion.