Coexistence states for a Lotka-Volterra competitive model with aggressive movement strategy

This paper is concerned with a model of reaction-diffusion-advection equations arising in ecology. The equations model a situation in which the foreign species and the native species are both assumed to be competing for space in the same ecological niche, and the foreign species is just random diffusion while the native species is "braver" - a combination of random diffusion and a directed movement up the gradient of the foreign species. Our aim is to give the necessary and sufficient conditions for the coexistence of the foreign species and native species. For this, we first establish an a priori estimate result. Then we apply eigenvalue theory and homogenization principle to derive the necessary conditions for the coexistence. Finally, the sufficient conditions for the coexistence are given by using a global bifurcation method.