Asymptotic behavior of solutions for a Lotka-Volterra mutualism reaction-diffusion system with time delays

Authors:
Yuan-Ming Wang
Affiliations:
Department of Mathematics, East China Normal University, Shanghai 200241, People's Republic of China and Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Sci ...
Venue:
Computers & Mathematics with Applications
Year:
2009

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Abstract

This paper is to investigate the asymptotic behavior of solutions for a time-delayed Lotka-Volterra N-species mutualism reaction-diffusion system with homogeneous Neumann boundary condition. It is shown, under a simple condition on the reaction rates, that the system has a unique bounded time-dependent solution and a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the constant positive steady-state solution as time tends to infinity. This convergence result implies that the trivial steady-state solution and all forms of semitrivial steady-state solutions are unstable, and moreover, the system has no nonconstant positive steady-state solution. A condition ensuring the convergence of the time-dependent solution to one of nonnegative semitrivial steady-state solutions is also given.