Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

Zhaohai Ma; Rong Yuan; Yang Wang; Xin Wu

doi:10.3934/cpaa.2019093

Article Contents

2019, Volume 18, Issue 4: 2069-2092. Doi: 10.3934/cpaa.2019093

This issue Previous Article A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production Next Article On stability properties of the Cubic-Quintic Schródinger equation with

$\delta$

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Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system

1.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2.
School of Mathematical Sciences, Shanxi University, Shanxi 030006, China
3.
School of Sciences, East China Jiao Tong University, Nanchang 330013, China

Received: April 2018

Revised: August 2018

Published: January 2019

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Abstract

In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in $ n $–dimensional space. More precisely, we prove that all planar traveling waves with speed $ c>c^* $ are exponentially stable in $ L^{\infty}(\mathbb{R}^n ) $ in the form of $ t^{-\frac{n}{2\alpha }}{\rm{e}}^{-\varepsilon_{\tau} \sigma t} $ for some constants $ \sigma >0 $ and $ \varepsilon_{\tau} \in (0,1) $, where $ \varepsilon_{\tau} = \varepsilon(\tau) $ is a decreasing function refer to the time delay $ \tau>0 $. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed $ c = c^* $, we show that they are algebraically stable in the form of $ t^{-\frac{n}{2\alpha}} $. The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.

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Mathematics Subject Classification: 35C07, 92D25, 35B35.

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Figure 1. Exact planar traveling wave $ (\phi_1, \phi_2) $ of the system (5.1) with $ b_1 = \frac{1}{2}, b_2 = \frac{19}{2}, r_1 = r_2 = 16, d_1 = 1, d_2 = 7 $

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Figure 2. The left picture denotes the solution $ u_1 $ of system (1.4) with Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution $ u_1(t, x) $ plots at times $ t = 0, 1, 2, 10, 30, 50 $ and behaves as a stable monotone increasing traveling wave (no change of the waves's shape after a large time in the sense of stability) and travels from right to left.

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Figure 3. The left picture denotes the solution $ u_2 $ of system (1.4) with Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution $ u_2(t, x) $ plots at times $ t = 0, 1, 2, 10, 30, 50 $ and behaves as a stable monotone increasing traveling wave (no change of the waves's shape after a large time in the sense of stability) and travels from right to left.

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