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Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays. (English) Zbl 1066.92054
Summary: In the Lotka-Volterra competition system with N-competing species if the effects of dispersion and time-delays are both taken into consideration, then the densities of the competing 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 solutions in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary conditions, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to the competing rate constants to ensure the global asymptotic stability of the positive steady-state solutions. This result leads to the permanence of the competing system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solutions. The condition for the global asymptotic stability is independent of diffusion and time-delays, and the conclusions for the reaction-diffusion system are directly applicable to the corresponding ordinary differential system.

MSC:
92D40 Ecology
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
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