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.