Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays.

*(English)*Zbl 1066.92054Summary: 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 |

##### Keywords:

Lotka-Volterra competition; Reaction-diffusion system; Ordinary differential equations system; Time-delays; Global asymptotic stability; Permanence
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\textit{C. V. Pao}, Nonlinear Anal., Real World Appl. 5, No. 1, 91--104 (2004; Zbl 1066.92054)

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