A Quasilinear Predator-Prey Model with Indirect Prey-Taxis

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.