A Quasilinear Predator-Prey Model with Indirect Prey-Taxis

Abstract

This paper deals with a quasilinear predator-prey model with indirect prey-taxis

$$\begin{aligned} \left\{ \begin{aligned}{}&u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla w)+rug(v)-uh(u),&(x,t)\in \Omega \times (0,\infty ), \\&w_t=d_{w}\Delta w- \mu w+\alpha v,&(x,t)\in \Omega \times (0,\infty ),\\&v_t=d_{v}\Delta v+f(v)-ug(v),&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset \mathbb {R}^{n}$, $n\ge 1$, where $d_{w},d_{v},\alpha ,\mu ,r>0$ and the functions $g,h,f \in C^{2}([0,\infty ))$. The nonlinear diffusivity D and chemosensitivity S are supposed to satisfy

$$\begin{aligned} D(s)\ge a(s+1)^{-\gamma } \;\;\;and \;\;\;0\le S(s)\le bs(s+1)^{\beta -1} \;\;\text {for all}\;\; s\ge 0, \end{aligned}$$

with $a,b>0$ and $\gamma ,\beta \in \mathbb {R}$. Suppose that $\gamma +\beta <1+\frac{1}{n}$ and $\gamma <\frac{2}{n}$, it is proved that the problem has a unique global classical solution, which is uniformly bounded in time. In addition, we derive the asymptotic behavior of globally bounded solution in this system according to the different predation conditions.