Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects

In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction-diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction-diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction-diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction-diffusion system. Finally, the numerical simulation is given to illustrate our results.