On a two-species competitive predator-prey system with density-dependent diffusion.

This paper deals with a two-species competitive predator-prey system with density-dependent diffusion, i.e., \begin{eqnarray*}\label{1a} \left{ \begin{split}{}&u_t=\Delta (d_{1}(w)u)+\gamma_{1}uF_{1}(w)-uh_{1}(u)-\beta_{1}uv,&(x,t)\in \Omega\times (0,\infty),\&v_t=\Delta (d_{2}(w)v)+\gamma_{2}vF_{2}(w)-vh_{2}(v)-\beta_{2}uv,&(x,t)\in \Omega\times (0,\infty),\&w_t=D\Delta w-uF_{1}(w)-vF_{2}(w)+f(w),&(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, with the nonnegative initial data $\left( {u_{0}, v_{0}, w_{0}} \right) \in (W^{1,p}(\Omega))^{3}$ with $p>2$, where the parameters $D,\gamma_{1},\gamma_{2},\beta_{1},\beta_{2}>0$, $d_{1}(w)$ and $d_{2}(w)$ are density-dependent diffusion functions, $F_{1}(w)$ and $F_{2}(w)$ are commonly called the functional response functions accounting for the intake rate of predators as the functions of prey density, $h_{1}(u)$ and $h_{2}(v)$ represent the mortality rates of predators, and $f(w)$ stands for the growth function of the prey. First, we rigorously prove the global boundedness of classical solutions for the above general model provided that the parameters satisfy some suitable conditions by means of $L^{p}$-estimate techniques. Moreover, in some particular cases, we establish the asymptotic stabilization and precise convergence rates of globally bounded solutions under different conditions on the parameters by constructing some appropriate Lyapunov functionals. Our results not only extend the previous ones, but also involve some new conclusions.