Fisher–KPP-type models of biological invasion: open source computational tools, key concepts and analysis
Matthew J. Simpson, Scott W. McCue
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:semantics><mml:mrow><mml:mi>c</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math> . To overcome the lack of a well-defined sharp front in solutions of the Fisher–KPP model, we also review two alternative modelling 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mi>c</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt></mml:mrow></mml:mstyle></mml:math> . The second alternative approach is to reformulate the Fisher–KPP model as a moving boundary problem on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:semantics><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>x</mml:mi><mml:mo><</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:semantics></mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>c</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math> and biological recession <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>c</mml:mi><mml:mo><</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:semantics></mml:math> . 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.