# 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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