# American Institute of Mathematical Sciences

August  2017, 22(6): 2233-2260. doi: 10.3934/dcdsb.2017094

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

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

Received  February 2016 Revised  June 2017 Published  March 2017

Fund Project: 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)

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.
Citation: Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094
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##### References:
 [1] Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789 [2] Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299 [3] Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034 [4] Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324 [5] Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150 [6] Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125 [7] Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503 [8] 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 [9] Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018 [10] Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180 [11] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 [12] Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268 [13] Rachidi B. Salako, Wenxian Shen. Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 293-319. doi: 10.3934/dcdss.2020017 [14] Rachidi B. Salako. Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5945-5973. doi: 10.3934/dcds.2019260 [15] Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 203-209. doi: 10.3934/dcdss.2020011 [16] Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Decay in chemotaxis systems with a logistic term. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 257-268. doi: 10.3934/dcdss.2020014 [17] Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198 [18] Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019 [19] Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147 [20] 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

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