Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size

TL;DR

This work investigates spreading in a one-dimensional Fisher–KPP equation with a moving, bounded central patch of higher growth rate in a three-zone heterogeneous environment. The authors develop a moving-frame eigenproblem and construct a generalized principal eigenpair that governs sharp barriers for front propagation. They derive a complete piecewise formula for the spreading speed when the patch has constant size and speed, and analyze slow/fast patches and slowly oscillating patches, revealing phenomena such as front locking, nonlocal pulling, and ill-defined spreading speeds. The results connect to multi-species invasion theory and climate-change-type models, and they outline extensions to more general heterogeneities and optimization of growth-rate landscapes.

Abstract

This paper is concerned with spreading properties of space-time heterogeneous Fisher--KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or intermediate growth, a central patch with fast growth and a right half-line with slow or intermediate growth. The central patch moves at various speeds. The behavior of the front changes drastically depending on the speed of the central patch. Among other things, intriguing phenomena such as nonlocal pulling and locking may occur, which would make the behavior of the front further complicated. The problem we discuss here is closely related to questions in biomathematical modelling. By considering several special cases, we illustrate the remarkable diversity of possible behaviors. In particular, when the central patch has constant size and constant speed, we provide a complete set of explicit formulas for the spreading speed.

Figures (5)

• Figure 1: Illustration of the heterogeneous intrinsic growth rate under assumption \ref{['ass:piecewise_constant_r']}.
• Figure 2: Illustration of the heterogeneous intrinsic growth rate under the assumptions of Theorem \ref{['thm:one_interface']}.
• Figure 3: The spreading speed as a function of the environmental speed under the assumptions of Theorem \ref{['thm:one_interface']}.
• Figure 4: The spreading speed $c^\star$ of Theorem \ref{['thm:constant_case']} as a function of the speed of the environmental heterogeneity $c_A$. Note that we fix values of $\lambda_1$ instead of fixing values of $L$. This is rigorously equivalent, cf. Proposition \ref{['prop:lambda1_L0']}. Note also that the figures where $L\leq\underline{L}$, i.e., $\lambda_1=-\max(r_1,r_3)$, are independent of $L$ and $\lambda_1$.
• Figure 5: The spreading speed $c^\star$ of Theorem \ref{['thm:constant_case']} as a function of the generalized principal eigenvalue $\lambda_1$ in the case $c_A>2\sqrt{\max(r_1,r_3)}$ (parameter values: $r_1=1$, $r_2=16$, $r_3=4$, $c_A=7$). Due to the monotonic dependence of $\lambda_1$ on $r_2$ or $L$ (the larger the patch, the larger $-\lambda_1$; cf. Propositions \ref{['prop:lambda1_L0']} and \ref{['prop:lambda1_r2']}), this graph is a proxy for $c^\star$ as a function of $L$ or $r_2$. It can be interpreted as follows: when the environmental speed $c_A$ is larger than $2\sqrt{\max(r_1,r_3)}$, a small patch has no effect, an intermediate patch induces nonlocal pulling, a large patch induces locking. Similar figures in the cases $2\sqrt{r_3}\leq c_A\leq 2\sqrt{r_1}$ and $c_A<2\sqrt{r_3}$ could be produced and would show that the spreading speed in these cases is never impacted by the patch.

Theorems & Definitions (26)

• Theorem 1.1: A single transition Lam_Yu_2021
• Theorem 2.1: Patch of constant size, constant speed
• Corollary 2.2: Slow patch
• Corollary 2.3: Fast patch
• Remark 2.1
• Proposition 3.1: Berestycki_Ros
• Lemma 3.2
• proof : Step 1
• proof : Step 2
• Lemma 3.3