Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source

Ke Lin; Chunlai Mu

doi:10.3934/dcdsb.2017094

Article Contents

2017, Volume 22, Issue 6: 2233-2260. Doi: 10.3934/dcdsb.2017094

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Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source

Ke Lin^, and
Chunlai Mu

College of Mathematics and Statistics, Chongqing University, Chongqing University, Chongqing 401331, China

Author Bio: E-mail address: clmu2005@163.com
Ke Lin, E-mail address: shuxuelk@126.com
Ke Lin, E-mail address: shuxuelk@126.com

Received: January 31, 2016

Revised: May 31, 2017

Published: May 2017

The first author is partially supported by Chongqing graduate student research innovation project (Grant No. CYB15042) and Chongqing Nova program, and the second author is partially supported by NSFC (Grant No. 11371384 and 11571062) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007).

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Abstract

In this paper, we consider a system of three parabolic equations in high-dimensional smoothly bounded domain

$\left\{\begin{array}{llll}u_t=\Delta u-\chi_1\nabla\cdot( u\nabla w)+\mu_1u(1-u-a_1v),\quad &x\in \Omega,\quad t>0,\\v_t=\Delta v-\chi_2\nabla\cdot( v\nabla w)+\mu_2v(1-a_2u-v),\quad &x\in\Omega,\quad t>0,\\w_t=\Delta w- w+u+v,\quad &x\in\Omega,\quad t>0,\\\end{array}\right.$

which describes the mutual competition between two populations on account of the Lotka-Volterra dynamics.

For any cross-diffusivities $\chi_1>0$ and $\chi_2>0$ and the rates $a_1>0$ and $a_2>0$, it is proved that the global classical bounded solutions exist for sufficiently regular initial data when the parameters $\mu_1$ and $\mu_2$ are sufficiently large. In deriving the convergence of solutions to this system, we need to distinguish two cases $a_1, a_2\in[0, 1)$ and $a_1>1$ and $0\leq a_2 < 1$ to prove globally asymptotic stability.

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Mathematics Subject Classification: Primary:35B40, 92C17;Secondary:35B65, 35K45.

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