# American Institute of Mathematical Sciences

May  2018, 38(5): 2591-2607. doi: 10.3934/dcds.2018109

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

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

* Corresponding author: Mingxin Wang

Received  July 2017 Revised  January 2018 Published  March 2018

Fund Project: This work was supported by NSFC Grants 11771110 and 11371113.

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.

Citation: Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
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