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Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects. (English) Zbl 1201.35030
Summary: 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.
MSC:
35B25Singular perturbations (PDE)
92D25Population dynamics (general)
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