A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

Liangchen Wang; Chunlai Mu

doi:10.3934/dcdsb.2020114

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December 2020, 25(12): 4585-4601. doi: 10.3934/dcdsb.2020114

A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

Liangchen Wang ^1,, and Chunlai Mu ^2,

1.	Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

^* Corresponding author: Liangchen Wang

Received August 2019 Revised December 2019 Published December 2020 Early access March 2020

This paper deals with the following competitive two-species chemotaxis system with two chemicals

$ \left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u\left( {1 - u - {a_1}w} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta v - v + w,}&{x \in \Omega ,t > 0,}\\{{w_t} = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w\left( {1 - w - {a_2}u} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta z - z + u,}&{x \in \Omega ,t > 0}\end{array}} \right. $

under homogeneous Neumann boundary conditions in a bounded domain

$ \Omega\subset \mathbb{R}^n $

(

$ n\geq1 $

), where the parameters

$ \chi_i>0 $

$ \mu_i>0 $

and

$ a_i>0 $

(

$ i = 1, 2 $

). It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds:

(ⅰ)

$ q_1\leq a_1; $

(ⅱ)

$ q_2\leq a_2 $

;

(ⅲ)

$ q_1>a_1 $

and

$ q_2> a_2 $

as well as

$ (q_1-a_1)(q_2-a_2)<1 $

where

$ q_1: = \frac{\chi_1}{\mu_1} $

and

$ q_2: = \frac{\chi_2}{\mu_2} $

, which partially improves the results of Zhang et al. [53] and Tu et al. [34].

Moreover, it is proved that when

$ a_1, a_2\in(0, 1) $

and

$ \mu_1 $

and

$ \mu_2 $

are sufficiently large, then any global bounded solution exponentially converges to

$ \left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right) $

$ t\rightarrow\infty $

; When

$ a_1>1>a_2>0 $

and

$ \mu_2 $

is sufficiently large, then any global bounded solution exponentially converges to

$ (0, 1, 1, 0) $

$ t\rightarrow\infty $

; When

$ a_1 = 1>a_2>0 $

and

$ \mu_2 $

is sufficiently large, then any global bounded solution algebraically converges to

$ (0, 1, 1, 0) $

$ t\rightarrow\infty $

. This result improves the conditions assumed in [34] for asymptotic behavior.

Keywords: Two-species chemotaxis, boundedness, asymptotic stability, Lotka-Volterra competition.

Mathematics Subject Classification: Primary: 35K35, 35A01, 35B44; Secondary: 35B35, 92C17.

Citation: Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4585-4601. doi: 10.3934/dcdsb.2020114