# American Institute of Mathematical Sciences

## 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
##### References:

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##### References:
 [1] Li Ma, Shangjiang Guo. Bifurcation and stability of a two-species diffusive Lotka-Volterra model. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1205-1232. doi: 10.3934/cpaa.2020056 [2] Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 [3] Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137 [4] Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477 [5] Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015 [6] Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 [7] Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132 [8] Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178 [9] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [10] Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220 [11] Youshan Tao, Michael Winkler. Boundedness vs.blow-up in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3165-3183. doi: 10.3934/dcdsb.2015.20.3165 [12] Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156 [13] Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure & Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001 [14] Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771 [15] Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131 [16] Tahir Bachar Issa, Rachidi Bolaji Salako. Asymptotic dynamics in a two-species chemotaxis model with non-local terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3839-3874. doi: 10.3934/dcdsb.2017193 [17] Ting-Hui Yang, Weinian Zhang, Kaijen Cheng. Global dynamics of three species omnivory models with Lotka-Volterra interaction. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2867-2881. doi: 10.3934/dcdsb.2016077 [18] Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 [19] Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650 [20] Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953

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