Influence of density-dependent diffusion on pattern formation in a refuge

We investigate a nonlocal generalization of the Fisher-KPP equation, which incorporates logistic growth and diffusion, for a single species population in a viable patch (refuge). In this framework, diffusion plays an homogenizing role, while nonlocal interactions can destabilize the spatially uniform state, leading to the emergence of spontaneous patterns. Notably, even when the uniform state is stable, spatial perturbations, such as the presence of a refuge, can still induce patterns. These phenomena are well known for environments with constant diffusivity. Our goal is to investigate how the formation of winkles in the population distribution is affected when the diffusivity is density-dependent. Then, we explore scenarios in which diffusivity is sensitive to either rarefaction or overcrowding. We find that state-dependent diffusivity affects the shape and stability of the patterns, potentially leading to either explosive growth or fragmentation of the population distribution, depending on how diffusion reacts to changes in density.