  Considering a nonlocal version of the Fisher-KPP equation, we explore the
impact of heterogeneous diffusion on pattern formation within a bounded region
(refuge). Under homogeneous diffusion, it is established that, nonlocality can
lead to spontaneous pattern formation under certain parameter conditions,
otherwise, when the homogeneous state is stable, spatial perturbations such as
the existence of a refuge (high quality region within a hostile environment)
can induce patterns. To understand how density-dependent diffusivity influences
these forms of pattern formation, we examine diffusivity reactions to both
rarefaction or overcrowding. Additionally, for comparison, we investigate the
scenario where diffusivity levels vary spatially, inside and outside the
refuge. We find that state-dependent diffusivity affects the shape and
stability of patterns, potentially triggering either explosive growth or
fragmentation of the population distribution, depending on how diffusion
responses to density changes.
