American Institute of Mathematical Sciences

April  2019, 24(4): 1569-1587. doi: 10.3934/dcdsb.2018220

Global existence and stability in a two-species chemotaxis system

 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

* Corresponding author: S. Guo

Received  December 2017 Revised  March 2018 Published  June 2018

Fund Project: The second author is supported by NSF of China (Grants No. 11671123).

This paper deals with the following two-species chemotaxis system
 $\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), & x\in \Omega ,t>0, & \\ \ \ {{v}_{t}}=\Delta v-v+h(w), & x\in \Omega ,t>0, & \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), & x\in \Omega ,t>0, & \\ \ \ {{z}_{t}}=\Delta z-z+h(u),& x\in \Omega ,t>0, & \\\end{array} \right.$
under homogeneous Neumann boundary conditions in a bounded domain
 $Ω\subset\mathbb{R}^{n}$
with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C1-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium
 $(u_{*}, v_{*}, w_{*}, z_{*})$
may be globally attractive in the weak competition case (i.e.,
 $0 < a_{1}, a_{2} < 1$
), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e.,
 $a_{1}>1>a_{2}>0$
). In the fully strong competition case (i.e.
 $a_{1}, a_{2}>1$
), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous
 $(u_{*}, v_{*}, w_{*}, z_{*})$
. The matter which species ultimately wins out depends crucially on the starting advantage each species has.
Citation: 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
References:

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References:
Solutions of model (3) tend to a positive steady state with parameters (53) and initial condition (54)
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