# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020315

## Asymptotic stability in a chemotaxis-competition system with indirect signal production

 College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

* Corresponding author: Pan Zheng

Received  May 2019 Revised  October 2019 Published  August 2020

Fund Project: This work is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), Natural Science Foundation of Chongqing (Grant No: cstc2019jcyj-msxmX0082) and China-South Africa Young Scientist Exchange Programme

This paper deals with a fully parabolic inter-species chemotaxis-competition system with indirect signal production
 $\begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} = d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), &(x,t)\in \Omega\times (0,\infty), \\ & w_{t} = d_{w}\Delta w-\lambda w+\alpha v, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*}$
under zero Neumann boundary conditions in a smooth bounded domain
 $\Omega\subset \mathbb{R}^{N}$
(
 $N\geq 1$
), where
 $d_{u}>0, d_{v}>0$
and
 $d_{w}>0$
are the diffusion coefficients,
 $\chi\in \mathbb{R}$
is the chemotactic coefficient,
 $\mu_{1}>0$
and
 $\mu_{2}>0$
are the population growth rates,
 $a_{1}>0, a_{2}>0$
denote the strength coefficients of competition, and
 $\lambda$
and
 $\alpha$
describe the rates of signal degradation and production, respectively. Global boundedness of solutions to the above system with
 $\chi>0$
was established by Tello and Wrzosek in [J. Math. Anal. Appl. 459 (2018) 1233-1250]. The main purpose of the paper is further to give the long-time asymptotic behavior of global bounded solutions, which could not be derived in the previous work.
Citation: Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020315
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