Protection zone in a modified Lotka-Volterra model

Xiao  He; Sining Zheng

doi:10.3934/dcdsb.2015.20.2027

Article Contents

2015, Volume 20, Issue 7: 2027-2038. Doi: 10.3934/dcdsb.2015.20.2027

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Protection zone in a modified Lotka-Volterra model

Xiao He^1, and
Sining Zheng^2,

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024
2.
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024

Received: December 31, 2013

Revised: April 30, 2015

Published: August 2015

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Abstract

This paper studies the dynamic behavior of solutions to a modified Lotka-Volterra reaction-diffusion system with homogeneous Neumann boundary conditions, for which a protection zone should be created to prevent the extinction of the prey only if the prey's growth rate is small. We find a critical size of the protection zone, determined by the ratio of the predation rate and the refuge ability, to ensure the existence, uniqueness and global asymptotic stability of positive steady states for general predator's growth rate $\mu>0$. Bellow the critical size the dynamics of the model would be similar to the case without protection zones. The known uniqueness results for the protection problems with other functional responses, e.g., Holling II model, Leslie model, Beddington-DeAngelis model, were all required that the predator's growth rate $\mu>0$ is large enough. Such a large $\mu$ assumption is not needed for the uniqueness and asymptotic results to the modified Lotka-Volterra reaction-diffusion system considered in this paper.

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Mathematics Subject Classification: Primary: 35J47, 35K57, 92D40.

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