On a quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction

Abstract

In this paper, we study the quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction

$$\begin{aligned} \begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot \left( D_{1}(u)\nabla u\right) -\chi \nabla \cdot \left( S_{1}(u)\nabla z\right) +u\left( \alpha v-a_{1} -b_{1}u\right) ,&x \in \varOmega , t>0, \\&v_t=\nabla \cdot \left( D_{2}(v)\nabla v\right) +\xi \nabla \cdot \left( S_{2}(v)\nabla {w}\right) +v\left( a_{2} -b_{2} v-u\right) ,&x \in \varOmega , t>0, \\&{w_t}=\Delta w+\beta {u}-\gamma {w},&x \in \varOmega , t>0,\\&{z_t}=\Delta z+\delta {v}-\rho z,&x \in \varOmega , t>0,\\ \end{aligned} \right. \end{aligned} \end{aligned}$$

under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\varOmega \subset \mathbb {R}^{n}(n\ge 1)\), where \( \chi , \xi , \alpha , \beta , \gamma , \delta , \rho , a_{1},a_{2},\) \(b_{1},b_{2}\) are positive parameters, the functions \(D_{i} \in C^{2}([0,\infty ))\) and \(S_{i}\in C^{2}([0,\infty ))\) with \(S_{i}(0)=0(i=1,2)\). Firstly, under certain suitable conditions, we prove that the system admits a unique globally bounded classical solution when \(n\le 4\). Moreover, we investigate the asymptotic stability and precise convergence rates of globally bounded solutions by constructing appropriate Lyapunov functionals. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.