Morse index of steady-states to the SKT model with Dirichlet boundary conditions

Abstract

This paper deals with the stability analysis for steady-states perturbed by the full cross-diffusion limit of the SKT model with Dirichlet boundary conditions. Our previous result showed that positive steady-states consist of the branch of small coexistence type bifurcating from the trivial solution and the branches of segregation type bifurcating from points on the branch of small coexistence type. This paper shows the Morse index of steady-states on the branches and constructs the local unstable manifold around each steady-state of which the dimension is equal to the Morse index.