The global attractor of a competitor-competitor-mutualist-reaction-diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction-diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor-mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction-diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state-state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays. (C) 2006 Elsevier Ltd. All rights reserved.