Reaction-diffusion models of biological invasion: Open source computational tools, key concepts and analysis

Author:

Matthew J Simpson, Scott W McCue

Keyword:

Nonlinear Sciences, Pattern Formation and Solitons, Pattern Formation and Solitons (nlin.PS), Populations and Evolution (q-bio.PE)

journal:

date:

2024-03-04 00:00:00

Abstract

This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction-diffusion models used to study biological invasion. Starting with the classic Fisher-Kolomogorov (Fisher-KPP) model, we illustrate how computational methods can be used to explore time-dependent partial differential equation (PDE) solutions in parallel with phase plane and regular perturbation techniques to explore invading travelling wave solutions moving with dimensionless speed $c \ge 2$. To overcome the lack of a well-defined sharp front in solutions of the Fisher-KPP model, we also review two alternative modeling approaches. The first is the Porous-Fisher model where the linear diffusion term is replaced with a degenerate nonlinear diffusion term. Using phase plane and regular perturbation methods, we explore the distinction between sharp- and smooth-fronted invading travelling waves that move with dimensionless speed $c \ge 1/\sqrt{2}$. The second alternative approach is to reformulate the Fisher-KPP model as a moving boundary problem on $0 < x < L(t)$, leading to the Fisher-Stefan model with sharp-fronted travelling wave solutions arising from a PDE model with a linear diffusion term. Time-dependent PDE solutions and phase plane methods show that travelling wave solutions of the Fisher-Stefan model can describe both biological invasion $(c > 0)$ and biological recession $(c < 0)$. Open source Julia code to replicate all computational results in this review is available on GitHub; we encourage researchers to use this code directly or to adapt the code as required for more complicated models.

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