Logistic diffusion equations governed by the superposition of operators of mixed fractional order

Abstract

We discuss the existence of stationary solutions for logistic diffusion equations of Fisher–Kolmogoroff–Petrovski–Piskunov type driven by the superposition of fractional operators in a bounded region with “hostile” environmental conditions, modeled by homogeneous external Dirichlet data. We provide a range of results on the existence and nonexistence of solutions tied to the spectral properties of the ambient space, corresponding to either survival or extinction of the population. We also discuss how the possible presence of nonlocal phenomena of concentration and diffusion affect the endurance or disappearance of the population. In particular, we give examples in which both classical and anomalous diffusion lead to the extinction of the species, while the presence of an arbitrarily small concentration pattern enables survival.