Global behavior of a two-species predator-prey chemotaxis model with signal-dependent diffusion and sensitivity

Abstract

In this paper, we consider the global behavior of classical solutions to the following two-species predator-prey chemotaxis model with signal-dependent diffusion and sensitivity$ \left\{\begin{array}{ll} u_{1t} = \nabla\cdot(d_{1}(v)\nabla u_{1})+ \nabla\cdot(\chi_{1}(v)u_{1}\nabla{v})+\mu_{1}u_{1}(1-u_{1}-e_{1}u_{2}), \\ u_{2t} = \nabla\cdot(d_{2}(v)\nabla u_{2})- \nabla\cdot(\chi_{2}(v)u_{2}\nabla{v})+\mu_{2}u_{2}(1+e_{2}u_{1}-u_{2}), \\ v_{t} = d_{3}\Delta{v}+\alpha u_{1}+\beta u_{2}-\gamma v \end{array}\right. $under homogeneous Neumann boundary condition in a bounded smooth domain $ \Omega\subset\mathbb{R}^{2} $, where $ d_{3}, \; \mu_{1}, \; \mu_{2}, \; e_{1}, \; e_{2}, \; \beta, \; \gamma $ are positive constants, $ \alpha\in \mathbb{R} $, and the functions $ d_{i}(v), \chi_{i}(v)(i = 1, 2) $ satisfy the following assumptions:(H1) $ d_{i}(v), \chi_{i}(v)\in C^{2}[0, \infty) $ satisfy $ d_{i}(v), \; \chi_{i}(v)>0 $ for $ v\geq0 $, $ d'_{i}(v)<0 $ and $ \lim\limits_{v\rightarrow \infty}d_{i}(v) = 0 $.(H2) The limits $ \lim\limits_{v\rightarrow \infty}\frac{\chi_{i}(v)}{d_{i}(v)} $ and $ \lim\limits_{v\rightarrow \infty}\frac{d'_{i}(v)}{d_{i}(v)} $ exist.We first apply the Moser iteration method to prove the uniform boundedness and global existence of solutions to the model. Then, by constructing appropriate Lyapunov functions, we establish the following ctiteria on the global asymptotic stability of the equilibria to the model: (i) If $ e_{1}<1 $ and both $ \mu_{1} $ and $ \mu_{2} $ are sufficiently large, the solution $ (u_1, u_2, v) $ converges to a unique positive equilibrium of the model. (ii) If $ e_{1}\geq 1 $ and $ \mu_{2} $ is large enough, the solution $ (u_1, u_2, v) $ converges to the semi-trivial equilibrium $ (0, 1, \frac{\beta}{\gamma}) $. The respective convergence rates are at least exponential if $ e_1\neq1 $, and algebraic if $ e_1 = 1 $.