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Persistence and extinction in a competition-diffusion system with time delays. (English) Zbl 0817.35043

Summary: A class of time-delay reaction-diffusion systems which arise from the model of two competing ecological species is discussed. We study the global attractivity of the positive steady states in terms of the natural growth rates of the species. Through a monotone iterative scheme, it is proven that when the natural growth rates are in certain unbounded parameter sets, the time-delay reaction-diffusion system has a positive global attractor enclosed by positive steady-state solutions. Asymptotic stability of the trivial and semi-trivial steady states are obtained explicitly. These persistence and extinction results are also demonstrated numerically.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
92D25 Population dynamics (general)
35B40 Asymptotic behavior of solutions to PDEs
35R10 Partial functional-differential equations
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