Dynamics of Lotka–Volterra competition reaction–diffusion systems with degenerate diffusion

Abstract

This paper is dealt with a class of Lotka–Volterra competition reaction–diffusion system and a competitor–competitor–mutualist system with density-dependent diffusion in a bounded domain under the three basic types of Dirichlet, Neumann and Robin boundary conditions. The governing equations for the competition system consist of an arbitrary number of degenerate quasilinear parabolic equations while the competitor–competitor–mutualist system involves three degenerate equations. The goal of the paper is to show: the existence of positive steady-state solutions or quasi-solutions, the existence and uniqueness of a classical global time-dependent solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions or quasi-solutions. The above goals are achieved under a very simple condition on the reaction rates of the reaction functions and these results yield a global attractor and the coexistence of the competing species. In the case of Neumann boundary condition the system has a unique constant positive steady-state solution which is a global attractor of all the competing species. The above conclusions lead to some interesting distinct dynamic behavior between degenerate quasilinear reaction–diffusion systems and the corresponding semilinear reaction–diffusion systems in which some or all of the competing species may be in extinction.