Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response

Hongwei Yin; Xiaoyong Xiao; Xiaoqing Wen

doi:10.3934/dcdsb.2018228

Article Contents

2018, Volume 23, Issue 6: 2121-2151. Doi: 10.3934/dcdsb.2018228

This issue Previous Article Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay Next Article On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections

Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response

1.
School of Sciences, Nanchang University Nanchang 330031, China
2.
Numerical Simulation and High-Performance Computing Laboratory, Nanchang University, Nanchang 330031, China

* Corresponding author: Xiaoqing Wen
* Corresponding author: Xiaoqing Wen

Received: June 2016

Revised: March 2017

Early access: July 2018

Published: June 2018

Hongwei Yin is supported by NSF of China (61563033 and 11461044) and NSF of Jiangxi Province (20161BAB201010). Xiaoqing Wen is supported by NSF of China (11563005).

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Abstract

In this paper, a Lévy-diffusion Leslie-Gower predator-prey model with a nonmonotonic functional response is studied. We show the existence, uniqueness and attractiveness of the globally positive solution to this model. Moreover, to its corresponding steady-state model, we obtain the stability of the semi-trivial solutions, the existence and nonexistence of coexistence states by the method of topological degree, the uniqueness and stability of coexistence state, and the multiplicity and stability of coexistence states by Grandall-Rabinowitz bifurcation theorem. In addition, to get these results, we study the property of the Lévy diffusion operator, and give out the comparison principle of the generalized parabolic Lévy-diffusion differential equation, as well as the existence and stability of the solution for the steady-state Logistic equation with Lévy diffusion. Furthermore, we obtain the comparison principle of the steady-state Lévy-diffusion equation. As far as we know, these results are new in the ecological model.

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Mathematics Subject Classification: Primary: 92B05, 92D25; Secondary: 35S05.

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