Traveling wave solutions in a nonlocal reaction-diffusion population model

Bang-Sheng  Han; Zhi-Cheng  Wang

doi:10.3934/cpaa.2016.15.1057

Article Contents

2016, Volume 15, Issue 3: 1057-1076. Doi: 10.3934/cpaa.2016.15.1057

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Traveling wave solutions in a nonlocal reaction-diffusion population model

Bang-Sheng Han^1, and
Zhi-Cheng Wang^1,

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received: October 2015

Revised: December 2015

Published: May 2016

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Abstract

This paper is concerned with a nonlocal reaction-diffusion equation with the form \begin{eqnarray} \frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}+u\left\{ 1+\alpha u-\beta u^{2}-(1+\alpha-\beta)(\phi\ast u) \right\}, \quad (t,x)\in (0,\infty) \times \mathbb{R}, \end{eqnarray} where $\alpha $ and $\beta$ are positive constants, $0<\beta<1+\alpha$. We prove that there exists a number $c^*\geq 2$ such that the equation admits a positive traveling wave solution connecting the zero equilibrium to an unknown positive steady state for each speed $c>c^*$. At the same time, we show that there is no such traveling wave solutions for speed $c<2$. For sufficiently large speed $c>c^*$, we further show that the steady state is the unique positive equilibrium. Using the lower and upper solutions method, we also establish the existence of monotone traveling wave fronts connecting the zero equilibrium and the positive equilibrium. Finally, for a specific kernel function $\phi(x):=\frac{1}{2\sigma}e^{-\frac{|x|}{\sigma}}$ ($\sigma>0$), by numerical simulations we show that the traveling wave solutions may connects the zero equilibrium to a periodic steady state as $\sigma$ is increased. Furthermore, by the stability analysis we explain why and when a periodic steady state can appear.

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Mathematics Subject Classification: 35C07, 35B40, 35K57, 92D25.

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