Traveling waves for nonlocal Lotka-Volterra competition systems

Bang-Sheng Han; Zhi-Cheng Wang; Zengji Du

doi:10.3934/dcdsb.2020011

Article Contents

2020, Volume 25, Issue 5: 1959-1983. Doi: 10.3934/dcdsb.2020011

This issue Previous Article Asset price volatility and price extrema Next Article On the threshold dynamics of the stochastic SIRS epidemic model using adequate stopping times

Traveling waves for nonlocal Lotka-Volterra competition systems

†.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, China
§.
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan, 611756, China
‡.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, China

^* Corresponding author: Zhi-Cheng Wang
^* Corresponding author: Zhi-Cheng Wang

Received: February 2019

Revised: May 2019

Early access: December 2019

Published: May 2020

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Abstract

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium $ (0, 0) $ to some unknown positive steady state for wave speed $ c>c^* = \max\left\{2, 2\sqrt{dr}\right\} $ and there is no such traveling wave solutions for $ c<c^* $, where $ d $ and $ r $ respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.

Keywords:

Lotka-Volterra diffusion-competition system,
nonlocal,
traveling wave solution,
existence,
numerical simulation.

Mathematics Subject Classification: 35C07, 35K40, 35K57, 35R10.

Citation:

Full Text(HTML)

Figure 1. The time and space evolution of $ u(x,t) $ in nonlocal equation (36) with kernel $ \phi_{\sigma}(x) = \frac{3a}{2\sigma}e^{-\frac{a}{\sigma}|x|}-\frac{1}{\sigma}e^{-\frac{|x|}{\sigma}} $. Our computational domain is $ x\in [0,85], t \in [0,30] $. The corresponding parameter values are: $ L_0 = 15,\ a = 0.7,\ a_1 = 0.4,\ a_2 = 0.5,\ d = 0.1,\ r = 2 $ and $ \sigma $ follows by 0.3, 0.6, 1.2, 1.5

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Figure 2. The time and space evolution of $ v(x,t) $ in nonlocal equation (36) with kernel $ \phi_{\sigma}(x) = \frac{3a}{2\sigma}e^{-\frac{a}{\sigma}|x|}-\frac{1}{\sigma}e^{-\frac{|x|}{\sigma}} $. Our computational domain is $ x\in [0,85],k ∈ [0,30] $. The corresponding parameter values are: $ L_0 = 15,\ a = 0.7,\ a_1 = 0.4,\ a_2 = 0.5,\ d = 0.1,\ r = 2 $ and $ \sigma $ follows by 0.3, 0.6, 1.2, 1.5

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