Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities

This work deals with a general cross-diffusion system modeling the dynamics behavior of two predators and one prey with signal-dependent diffusion and sensitivity subject to homogeneous Neumann boundary conditions. Firstly, in light of some Lp-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p-$$\end{document}estimate techniques, we rigorously prove the global existence and uniform boundedness of positive classical solutions in any dimensions with suitable conditions on motility functions and the coefficients of logistic source. Moreover, by constructing some appropriate Lyapunov functionals, we further establish the asymptotic behavior of solutions to a specific model with Lotka-Volterra type functional responses and density-dependent death rates for two predators as well as logistic type for the prey. Our results not only generalize the previously known one, but also present some new conclusions.