Persistence in reaction diffusion models with weak allee effect.

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak allele effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation.