Dynamics for the diffusive Leslie-Gower model with double free boundaries

Mingxin Wang; Qianying Zhang

doi:10.3934/dcds.2018109

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2018, Volume 38, Issue 5: 2591-2607. Doi: 10.3934/dcds.2018109

This issue Previous Article Traveling waves for a microscopic model of traffic flow Next Article On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components

Dynamics for the diffusive Leslie-Gower model with double free boundaries

Mingxin Wang^, and
Qianying Zhang

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Mingxin Wang
* Corresponding author: Mingxin Wang

Received: June 30, 2017

Revised: December 31, 2017

Published: June 2018

This work was supported by NSFC Grants 11771110 and 11371113

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In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to∞$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u, v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.

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Mathematics Subject Classification: Primary: 35K51, 35R35, 35A02; Secondary: 35B40, 92B05.

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