Traveling wave solutions in a diffusive producer-scrounger model

Junhao Wen; Peixuan Weng

doi:10.3934/dcdsb.2017030

Article Contents

2017, Volume 22, Issue 2: 627-645. Doi: 10.3934/dcdsb.2017030

This issue Previous Article Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise Next Article The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

Traveling wave solutions in a diffusive producer-scrounger model

Junhao Wen and
Peixuan Weng^,

School of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, China

Corresponding author: Peixuan Weng, Tel:0086-20-85213533
Corresponding author: Peixuan Weng, Tel:0086-20-85213533

Received: March 2016

Revised: September 2016

Published: December 2017

Supported by the NSF of China (11171120) and the Natural Science Foundation of Guangdong Province (2016A030313426).

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Abstract

This paper looks into the stability of equilibria, existence and non-existence of traveling wave solutions in a diffusive producer-scrounger model. We find that the existence and non-existence of traveling wave solutions are determined by a minimum wave speed $c_{m}$ and a threshold value $R_{0}$. By constructing a suitable invariant convex set $Γ$ and applying Schauder fixed point theorem, the existence for $c>c_{m}, R_{0}>1$ was established. Besides, a Lyapunov function is constructed subtly to explore the asymptotic behaviors of traveling wave solutions. The non-existences of traveling wave solutions for both $c < c_{m}, R_{0}> 1$ and $R_{0}≤1, c > 0$ were obtained by two-sides Laplace transform and reduction method to absurdity.

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Mathematics Subject Classification: 35K57, 92D25.

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Figure 1. Constant equilibria of model (2)

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