We examine a generalized KPP equation with a “q-diffusion”, which is a framework that unifies various standard linear diffusion regimes: Fickian diffusion ( \(q = 0\) ), Stratonovich diffusion ( \(q = 1/2\) ), Fokker-Planck diffusion ( \(q = 1\) ), and nonstandard diffusion regimes for general \(q\in \mathbb {R}\) . Using both analytical methods and numerical simulations, we explore how the ability of persistence (measured by some principal eigenvalue) and how the asymptotic spreading speed depend on the parameter q and on the phase shift between the growth rate r(x) and the diffusion coefficient D(x). Our results demonstrate that persistence and spreading properties generally depend on q: for example, appropriate configurations of r(x) and D(x) can be constructed such that q-diffusion either enhances or diminishes the ability of persistence and the spreading speed with respect to the traditional Fickian diffusion. We show that the spatial arrangement of r(x) with respect to D(x) has markedly different effects depending on whether \(q > 0\) , \(q = 0\) , or \(q < 0\) . The case where r is constant is an exception: persistence becomes independent of q, while the spreading speed displays a symmetry around \(q = 1/2\) . This work underscores the importance of carefully selecting diffusion models in ecological and epidemiological contexts, highlighting their potential implications for persistence, spreading, and control strategies.
