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. © 2010 Elsevier Inc.
Wang, C. you, Wang, S., Yang, F. ping, & Li, L. rui. (2010). Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects. Applied Mathematical Modelling, 34(12), 4278–4288. https://doi.org/10.1016/j.apm.2010.05.003