Boundedness and asymptotic stability in a predator-prey system with density-dependent motilities

Abstract

In this paper, we consider the predator-prey system with density-dependent motilities and pursuit-evasion interaction under homogeneous Neumann boundary conditions. The main obstacle of analysis comes from the term produced by pursuit-evasion interaction. With the $ L^p $-estimate techniques and Moser iteration, we show that the system possesses a global bounded classical solution. Furthermore, with the aid of Lyapunov functional, we establish the asymptotic behavior of solutions to this system under appropriate parameter conditions.