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Invading and Receding Sharp-Fronted Travelling Waves.

Maud El-Hachem1, Scott W McCue1, Matthew J Simpson2

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|February 21, 2021
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Summary
This summary is machine-generated.

The Fisher-Stefan model, a new generalization of the Fisher-KPP equation, can now model biological recession, not just invasion. This mathematical advancement allows estimation of unknown parameters from experimental invasion speed data.

Keywords:
InvasionMoving boundary problemPartial differential equationReaction–diffusionStefan problem

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Area of Science:

  • Mathematical Biology
  • Theoretical Ecology
  • Biophysics

Background:

  • Biological invasion is often modeled using the Fisher-KPP equation.
  • The Fisher-KPP model cannot simulate population decrease or recession.
  • A limitation exists in modeling receding populations with classical models.

Purpose of the Study:

  • To study the Fisher-Stefan model, a generalization of the Fisher-KPP equation.
  • To analyze the model's ability to simulate biological recession.
  • To establish a relationship between model parameters and traveling wave speed.

Main Methods:

  • Reformulating the Fisher-KPP model as a moving boundary problem.
  • Numerical simulation and phase plane analysis.
  • Perturbation analysis to construct approximate solutions.

Main Results:

  • Approximate solutions for slowly invading, slowly receding, and rapidly receding waves were constructed.
  • A relationship between the model parameter [Formula: see text] and the traveling wave speed c was determined.
  • Reinterpretation of the Fisher-KPP phase plane revealed overlooked features.

Conclusions:

  • The Fisher-Stefan model successfully models both invasion and recession.
  • The derived relationship allows estimation of the parameter [Formula: see text] from experimental data.
  • The study offers practical applications and mathematical insights into population dynamics.