Interplay between scales in the nonlocal FKPP equation

We consider a generalization of the FKPP equation for the evolution of the spatial density of a single-species population where all the terms are nonlocal. That is, the spatial extension of each process (growth, competition and diffusion) is ruled by an influence function, with a characteristic shape and range of action. Our purpose is to investigate the interference between these different components in pattern formation. We show that, while competition is the leading process behind patterns, the other two can act either constructively or destructively. For instance, diffusion that is commonly known to smooth out the concentration field can actually favor pattern formation depending on the shape and range of the dispersal kernel. The results are supported by analytical calculations accompanied by numerical simulations.