Dynamics of food-chain models with density-dependent diffusion and ratio-dependent reaction function

This paper is concerned with a 3-species and a 2-species food-chain reaction diffusion systems in a bounded domain where the diffusion coefficients may be density dependent and the reaction functions are ratio-dependent. These equations are quasilinear where the diffusion coefficients may be degenerate on the boundary of the domain. Three basic types of Dirichlet, Neumann and Robin boundary conditions are considered, and in each case some very simple conditions are obtained to ensure the dynamical behavior of the time-dependent solution in relation to some positive solutions or quasi-solutions of the steady-state problem, including the existence of these solutions. This dynamical behavior leads to the coexistence and global attractor of the food-chain systems. In the case of Neumann boundary condition sufficient conditions are given to ensure that the steady-state problem has a unique positive constant solution which is a global attractor of the time-dependent system.