Impact of diffusion mechanisms on persistence and spreading

TL;DR

This paper investigates how different diffusion mechanisms, unified under a generalized $q$-diffusion model, influence persistence and spreading speeds in ecological and epidemiological systems through analytical and numerical methods.

Contribution

It introduces a comprehensive framework for $q$-diffusion, analyzing its impact on persistence and spreading, revealing how spatial arrangements and the parameter $q$ alter these dynamics.

Findings

Persistence and spreading depend on $q$ and spatial configuration.

For constant $r$, persistence is independent of $q$; spreading speed is symmetric around $q=1/2.

Different $q$ values can either enhance or diminish persistence and spreading.

Abstract

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.