Coexistence of Three Competing Species with Non-negative Cross-diffusion rate

Abstract

In this paper, we report on our investigation of the existence and non-existence of positive solutions to a 3 x 3 competition interaction system with non-negative cross-diffusion under Dirichlet boundary conditions. First, it is shown that the system with constant diffusions can have a positive solution under suitable assumptions even though each of three 2x2 sub-systems coupled from the three equations of the system does not have a positive solution. Second, we show that the emergence of cross-diffusion in one equation of the system may generate a positive solution in case that the corresponding competition interaction system without cross-diffusion does not have a positive solution.