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Periodic KPP equations: new insights into persistence, spreading, and the role of advection

Nathanaël Boutillon, François Hamel, Lionel Roques

TL;DR

The paper analyzes persistence and spreading for a heterogeneous periodic Fisher–KPP equation with advection, using a principal eigenvalue framework and Freidlin–Gärtner speeds to characterize long-time dynamics. It shows that reversing the advection direction can, in general, affect persistence and spreading, but identifies two structural classes of $(r,b)$ for which these quantities are invariant under $b\mapsto -b$; it also derives homogenization-type limits for small/large environmental periods and presents a bounded-domain counterexample. Beyond these invariances, the work demonstrates nonmonotone relationships between persistence and spreading, establishing that persistence does not generally control spreading and vice versa, though in the no-advection case ($b\equiv0$) spreading is governed by the persistence threshold via $c_\pm[r;0]=\inf_{\lambda>0} k_\lambda[r;0]/\lambda$. Overall, the results clarify when persistence drives propagation in periodic, advective habitats and when advection direction can alter long-time outcomes, with implications for modeling invasion dynamics in heterogeneous environments.

Abstract

We focus on the persistence and spreading properties for a heterogeneous Fisher-KPP equation with advection. After reviewing the different notions of persistence and spreading speeds, we focus on the effect of the direction of the advection term, denoted by $b$. First, we prove that changing $b$ to $-b$ can have an effect on the spreading speeds and the ability of persistence. Next, we provide a class of relationships between the intrinsic growth term $r$ and the advection term $b$ such that changing $b$ to $-b$ does not change the spreading speeds and the ability of persistence. We briefly mention the cases of slowly and rapidly varying environments, and bounded domains. Lastly, we show that in general, the spreading speeds are not controlled by the ability of persistence, and conversely. However, when there is no advection term, the spreading speeds are controlled by the ability of persistence, though the converse still does not hold.

Periodic KPP equations: new insights into persistence, spreading, and the role of advection

TL;DR

The paper analyzes persistence and spreading for a heterogeneous periodic Fisher–KPP equation with advection, using a principal eigenvalue framework and Freidlin–Gärtner speeds to characterize long-time dynamics. It shows that reversing the advection direction can, in general, affect persistence and spreading, but identifies two structural classes of for which these quantities are invariant under ; it also derives homogenization-type limits for small/large environmental periods and presents a bounded-domain counterexample. Beyond these invariances, the work demonstrates nonmonotone relationships between persistence and spreading, establishing that persistence does not generally control spreading and vice versa, though in the no-advection case () spreading is governed by the persistence threshold via . Overall, the results clarify when persistence drives propagation in periodic, advective habitats and when advection direction can alter long-time outcomes, with implications for modeling invasion dynamics in heterogeneous environments.

Abstract

We focus on the persistence and spreading properties for a heterogeneous Fisher-KPP equation with advection. After reviewing the different notions of persistence and spreading speeds, we focus on the effect of the direction of the advection term, denoted by . First, we prove that changing to can have an effect on the spreading speeds and the ability of persistence. Next, we provide a class of relationships between the intrinsic growth term and the advection term such that changing to does not change the spreading speeds and the ability of persistence. We briefly mention the cases of slowly and rapidly varying environments, and bounded domains. Lastly, we show that in general, the spreading speeds are not controlled by the ability of persistence, and conversely. However, when there is no advection term, the spreading speeds are controlled by the ability of persistence, though the converse still does not hold.

Paper Structure

This paper contains 13 sections, 9 theorems, 132 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The influence of inverting the direction of the advection
  4. Relationship between the ability of persistence and the spreading speed
  5. Outline of the paper.
  6. A class of pairs $[r;b]$ such that $k_{\lambda}[r;b]=k_{-\lambda}[r;-b]$ and $c_+[r;b]=c_-[r;-b]$: proof of Theorem \ref{['thm:equality']}
  7. Other results on the influence of the direction of the advection term
  8. General effect of the direction of the advection term: proof of Proposition \ref{['prop:k0neq']}
  9. Small and large periods: proof of Proposition \ref{['ppn:limits_L']}
  10. A counter-example to Theorem \ref{['thm:equality']} in bounded domains: proof of Proposition \ref{['ppn:bounded_domain']}
  11. Relationship between the spreading speed and the ability of persistence: proof of Theorem \ref{['thm:no_general_comparison']}
  12. A condition for local convergence to the positive equilibrium state $p$
  13. Construction of the principal eigenfunctions in item $(ii)$ of Theorem \ref{['thm:equality']}

Key Result

Proposition 2.1

For every $1$-periodic function $b\in\mathcal{C}^{1,\alpha}(\mathbb{R})$ with $b\not\equiv 0$ and $\int_0^1b=0$, there exists a $1$-periodic function $r\in\mathcal{C}^{0,\alpha}(\mathbb{R})$ such that

Figures (2)

  • Figure 1: Three different situations. In each case, the dotted plot is the initial configuration and the full-line plot is taken at a later time.
  • Figure 2: The top panel illustrates the nontrivial relationship between $\phi^+$ and $\phi^-$ when item $(ii)$ of Theorem \ref{['thm:equality']} is satisfied: $b(x)=\cos(2\pi x) + \sin(4 \pi x)$ and $r(x)= 2 + b(x)$. The middle panel shows the solutions $u^+ := u[r; b]$ and $u^- := u[r; -b]$ of \ref{['eq:KPP']} with the same coefficients as in the top panel. The bottom panel shows the solutions $u^+ := u[r; b]$ and $u^- := u[r; -b]$ of \ref{['eq:KPP']} with $b(x) = \cos(2\pi x) + \sin(4 \pi x)$ and $r(x) = 2 + 2 \sin (2 \pi x)$, not satisfying condition $(ii)$ of Theorem \ref{['thm:equality']}.

Theorems & Definitions (22)

  • Proposition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • proof : Proof of Theorem \ref{['thm:equality']}, item $(i)$
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:equality']}, item $(ii)$, for $k_0$
  • ...and 12 more