  A reaction-diffusion Leslie-Gower predator-prey model, incorporating the fear
effect and prey refuge, with Beddington-DeAngelis functional response, is
introduced. A qualitative analysis of the solutions of the model and the
stability analysis of the coexistence equilibrium, are performed. Sufficient
conditions guaranteeing the occurrence of Turing instability have been
determined either in the case of self-diffusion or in the case of
cross-diffusion. Different types of Turing patterns, representing a spatial
redistribution of population in the environment, emerge for different values of
the model parameters.
