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

 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  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
 $\left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right)$
as
 $t\rightarrow\infty$
; When
 $a_1>1>a_2>0$
and
 $\mu_2$
is sufficiently large, then any global bounded solution exponentially converges to
 $(0, 1, 1, 0)$
as
 $t\rightarrow\infty$
; When
 $a_1 = 1>a_2>0$
and
 $\mu_2$
is sufficiently large, then any global bounded solution algebraically converges to
 $(0, 1, 1, 0)$
as
 $t\rightarrow\infty$
. This result improves the conditions assumed in [34] for asymptotic behavior.
Citation: Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020114
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