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Birth-jump processes and application to forest fire spotting.

T Hillen1, B Greese, J Martin

  • 1a Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology , University of Alberta , Edmonton , Canada.

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|September 5, 2014
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Summary
This summary is machine-generated.

Birth-jump models, which link population growth and spatial spread, can be simplified to reaction-diffusion equations. This finding enhances understanding of phenomena like forest fire spread, showing spotting accelerates invasion speed.

Keywords:
35Q9245K0592B05birth-jump processescritical domain sizediffusion limitintegro-differential equationsminimal wave speedreaction–diffusion equationswildfire spotting

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Area of Science:

  • Mathematical Biology
  • Ecology
  • Population Dynamics

Background:

  • Population models often struggle to decouple growth and spatial spread.
  • Birth-jump models offer a framework for these coupled processes, formulated as nonlinear integro-differential equations.

Purpose of the Study:

  • To present novel derivations of birth-jump models.
  • To approximate birth-jump models with reaction-diffusion equations under specific conditions.
  • To analyze critical domain size and minimal wave speed problems.

Main Methods:

  • Derivation of birth-jump models via random walk and two-compartmental reaction-diffusion approaches.
  • Approximation of integro-differential equations by reaction-diffusion equations using concentrated redistribution kernels.
  • Solving critical domain size and minimal wave speed problems.

Main Results:

  • Two distinct derivations of the birth-jump model equation are provided.
  • Under specific conditions, the birth-jump model simplifies to a reaction-diffusion equation.
  • The proliferation rate influences both diffusion and reaction terms in the approximated model.
  • Solutions for critical domain size and minimal wave speed are obtained.
  • Spotting in forest fires increases the invasion speed of the fire front.

Conclusions:

  • Birth-jump models offer a versatile tool for population dynamics where growth and spread are linked.
  • The approximation by reaction-diffusion equations provides a computationally tractable alternative for certain scenarios.
  • Understanding the impact of spotting is crucial for predicting and managing forest fire dynamics.