Bistable traveling waves in a delayed competitive system

Shuxia Pan; Shengnan Hao

doi:10.3934/dcdsb.2022148

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2023, Volume 28, Issue 3: 1806-1822. Doi: 10.3934/dcdsb.2022148

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Bistable traveling waves in a delayed competitive system

Shuxia Pan^1, , and
Shengnan Hao^2,

1.
School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

^*Corresponding author: Shuxia Pan
^*Corresponding author: Shuxia Pan

Received: January 2022

Revised: June 2022

Early access: August 2022

Published: March 2023

The first author is supported by NSF of China grant 11971213

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Abstract

This paper is concerned with the existence, uniqueness and asymptotic stability of traveling wave solutions in a delayed competitive system with spatial diffusion. When the interspecific competition is strong, the corresponding functional differential system has two stable and two unstable steady states. Firstly, we establish the existence of bistable traveling wave solutions by the theory of monotone semiflows. With the help of squeezing technique based on comparison principle, the asymptotic stability of traveling wave solutions is proved, which further implies the uniqueness of wave speed and wave profile in the sense of phase shift.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 92D25.

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Figure 1. The evolution of (13) with $ r = 0.5 $. From the viewpoint of population dynamics, $ u_2 $ almost invades the habitat of $ u_1 $ at a constant speed $ 0.17 $

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Figure 2. The evolution of (13) with $ r = 2.0 $. From the viewpoint of population dynamics, $ u_1 $ almost invades the habitat of $ u_2 $ at a constant speed $ 0.24 $

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