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Self-sustaining traveling fronts for a model related to bushfires

Serena Dipierro, Enrico Valdinoci, Glen Wheeler, Valentina-Mira Wheeler

TL;DR

The paper analyzes a nonlocal bushfire propagation model $\partial_t u = c\Delta u + \int (u(y,t)-\Theta)_+\,K(x,y)\,dy$, deriving sharp criteria for invasion versus extinction, and establishing the existence of unbounded traveling fronts for kernels that are either mild or short-range. It proves there are no vertically translating burning fronts, but traveling waves exist for prescribed speeds, with detailed monotonicity and stability properties: they are not generally evolutionarily stable yet are stable to localized perturbations. The work also develops numerical schemes and provides rigorous error bounds, and it rules out self-similar burning fronts, highlighting the distinct dynamics induced by nonlocal interactions. Together, these results offer precise, quantitative insights into fire-front propagation, extinction regimes, and the robustness of traveling fronts under localized disturbances, with potential applications to predictive fire management and modeling. $ $

Abstract

This article investigates a mathematical model for bushfire propagation, focusing on the existence and properties of translating solutions. We obtain quantitative bounds on the environmental diffusion coefficient and ignition kernels, identifying conditions under which fires either propagate across the entire region or naturally extinguish. Our analysis also reveals that vertically translating solutions do not exist, whereas traveling wave solutions with a front moving at any prescribed velocity always exist for kernels that are either of mild intensity or short range. These traveling waves exhibit unbounded profiles. Although evolutionary unstable, these traveling waves demonstrate stability under perturbations localized in a small region.

Self-sustaining traveling fronts for a model related to bushfires

TL;DR

The paper analyzes a nonlocal bushfire propagation model , deriving sharp criteria for invasion versus extinction, and establishing the existence of unbounded traveling fronts for kernels that are either mild or short-range. It proves there are no vertically translating burning fronts, but traveling waves exist for prescribed speeds, with detailed monotonicity and stability properties: they are not generally evolutionarily stable yet are stable to localized perturbations. The work also develops numerical schemes and provides rigorous error bounds, and it rules out self-similar burning fronts, highlighting the distinct dynamics induced by nonlocal interactions. Together, these results offer precise, quantitative insights into fire-front propagation, extinction regimes, and the robustness of traveling fronts under localized disturbances, with potential applications to predictive fire management and modeling.

Abstract

This article investigates a mathematical model for bushfire propagation, focusing on the existence and properties of translating solutions. We obtain quantitative bounds on the environmental diffusion coefficient and ignition kernels, identifying conditions under which fires either propagate across the entire region or naturally extinguish. Our analysis also reveals that vertically translating solutions do not exist, whereas traveling wave solutions with a front moving at any prescribed velocity always exist for kernels that are either of mild intensity or short range. These traveling waves exhibit unbounded profiles. Although evolutionary unstable, these traveling waves demonstrate stability under perturbations localized in a small region.
Paper Structure (32 sections, 21 theorems, 228 equations, 6 figures)

This paper contains 32 sections, 21 theorems, 228 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Mathematical framework and consistency checks
  4. Fire invasion or extinction
  5. Numerical schemes and approximations
  6. Traveling fire waves
  7. Monotonicity issues
  8. Stability of traveling waves
  9. Self-similar fire fronts
  10. Organization of the paper
  11. Proof of Lemma \ref{['COMPAPI']}
  12. Proof of Lemma \ref{['NECE']}
  13. Proof of Theorem \ref{['SOPTH']}
  14. Proof of Theorem \ref{['SOPTH:2']}
  15. Proof of Theorem \ref{['DBDARYEFF']}
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $u$ and $v$ be such that in $\Omega\times(0,+\infty)$, with $u(x,0)\leqslant v(x,0)$ for all $x\in\Omega$ and $u(x,t)\leqslant v(x,t)$ for all $x\in\partial\Omega$ and $t\in[0,+\infty)$. Then, $u(x,t)\leqslant v(x,t)$ for all $x\in\Omega$ and $t\in[0,+\infty)$.

Figures (6)

  • Figure 1: Delineation maps of actual bushfire events. Left: EMSR747 Wildfire in Central Macedonia region, Greece, August 17, 2024. Right: EMSR253 Forest fire in Piemonte, Italy, October 27, 2017. Images from the Copernicus Emergency Management Service, https://rapidmapping.emergency.copernicus.eu/EMSR747, https://emergency.copernicus.eu/mapping/list-of-components/EMSR253.
  • Figure 2: Plot of the function $u(x,t)= 1-e^{\frac{x}{1-t}}$ for $t\in\{-0.9, -0.5, 0, 0.5, 0.9\}$.
  • Figure 3: Plot of the traveling wave given by Theorem \ref{['TRAVE']} for $\omega=3$ (blue). The idealized wave is $v_3$ (red).
  • Figure 4: Plot of the traveling wave given by Theorem \ref{['TRAVE']} for $\omega=2$ (blue). The idealized wave is $v_2$ (red).
  • Figure 5: Plot of the traveling wave given by Theorem \ref{['TRAVE']} for $\omega=\sqrt3$ (blue). The idealized wave is $v_{\sqrt3}$ (red). This figure shows the wave on the same domain as in Figures \ref{['fig1']} and \ref{['fig2']}.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Lemma 2.1: Comparison Principle
  • Lemma 2.2: Necessity of the initial ignition
  • Theorem 2.3: Description of a fire invading the whole region
  • Theorem 2.4: Description of a fire being extinguished by environmental thermal diffusion
  • Theorem 2.5: Boundary ignition
  • Theorem 2.6: Error bounds in the presence of simplified convolutions
  • Theorem 2.7: Absence of vertically translating solutions
  • Theorem 2.8: Existence of traveling waves for kernels with short-range interactions
  • Theorem 2.9: Existence of traveling waves for kernels with mild interactions
  • Theorem 2.10: Absence of bounded traveling waves
  • ...and 15 more