Impact of diffusion mechanisms on persistence and spreading

TL;DR

The paper investigates how a generalized $q$-diffusion operator in a periodic Fisher-KPP framework affects persistence and spreading, revealing that diffusion type can either amplify or dampen invasion depending on the phase relation between growth $r(x)$ and diffusion $D(x)$. It develops a unifying relation $k_q^{\lambda}[r;D]=k_0^{\lambda}[r-h_q;D]$ with $h_q=-\frac{q}{2}D''+\frac{q^2}{4}\frac{(D')^2}{D}$, enabling comparative analysis across $q$; and proves symmetry in the constant-$r$ case about $q=1/2$, with explicit maximal spreading speed at Stratonovich diffusion. The work also provides rigorous proofs, asymptotic limits as $q\to\pm\infty$, and extensive numerical simulations showing tangible phase-shift effects and diffusion-optimization insights for ecological and epidemiological modeling. These results emphasize that diffusion modeling choices materially influence persistence and invasion dynamics, and guide practical diffusion model selection and control strategies in heterogeneous environments.

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.

Figures (4)

• Figure 1: Spreading speed $c^*_q[r;D]$ depending on $q$, with a constant growth term $r\equiv 1.$ Here, $D(x)=0.1+ \cos^2(\pi \, x).$
• Figure 2: Principal eigenvalue $k_q^0$ and spreading speed $c^*_q[r;D]$ depending on the phase shift $\omega$ between $r$ and $D$. Here, $r(x)=\cos^2(\pi \, x)$ and $D(x)=0.1+ \cos^2(\pi \, (x+\omega))$.
• Figure 3: Functions $D$ optimizing the ratio $c_0^*[r,D] / c_1^*[r,D]$ (respectively, $c_1^*[r,D] / c_0^*[r,D]$) with $r$ fixed to $r(x)=\cos^2(\pi \, x)$. The optimization is based on a simulated annealing algorithm. The function $D$ is interpolated from four discrete points using cubic splines.
• Figure 4: The processes $X^1$ and $X^2$ are submitted to a drift bringing them to the center, where $R$ is larger. The process $X^2$ is submitted to a stronger drift than $X^1$, which is what we want to exploit in the proof.

Theorems & Definitions (38)

• Theorem 3.1
• Remark 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Theorem 3.6
• Proposition 3.7
• Proposition 3.8: BouHamRoq25, Theorem 2.2
• Proposition 3.9
• Theorem 3.10