Dynamic transitions for the S-K-T competition system

This paper is concerned with dynamical transition for biological competition system modeled by the S-K-T equations. We study the dynamical behaviour of the S-K-T equations with two different boundary conditions. For the system under non-homogeneous Dirichlet boundary condition, we show that the system undergoes a mixed dynamic transition from the homogeneous state to steady state solutions when the bifurcation parameter cross the critical surface. For the system with Neumann boundary condition, we prove that the system undergoes a mixed dynamic transition, a jump transition and a continuous transition when the bifurcation parameter cross the critical number. Finally, two examples are provided to validate the effectiveness of the theoretical results.