# American Institute of Mathematical Sciences

October  2017, 37(10): 5433-5454. doi: 10.3934/dcds.2017236

## Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion

 1 School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Hongmei Cheng

Received  June 2016 Revised  June 2017 Published  June 2017

This paper will mainly study the information about the existence and stability of the invasion traveling waves for the nonlocal Leslie-Gower predator-prey model. By using an invariant cone in a bounded domain with initial function being defined on and applying the Schauder's fixed point theorem, we can obtain the existence of traveling waves. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Then we use the weighted energy to prove that the invasion traveling waves are exponentially stable as perturbation in some exponentially as $x\to-\infty$. Finally, by defining the bilateral Laplace transform, we can obtain the nonexistence of the traveling waves.

Citation: Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236
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