Uniqueness and stability of a predator-prey model with C-M functional response

In this paper, we are concerned with positive solutions of a predator-prey model with Crowley-Martin functional response under Dirichlet boundary conditions. First, we establish the existence of positive solutions when (a,c) is close to (λ1,λ1). Next, the stability properties of nonnegative solutions are discussed by spectral analysis. In addition, we obtain a complete understanding of the uniqueness and non-uniqueness of positive solutions by the Lyapunov-Schmidt procedure and the perturbation technique. At last, some numerical simulations are presented to supplement the analytic results in one dimension.