Mathematical Models in Biology: Reaction-Diffusion Equations

This repository contains a LaTeX document and associated Julia scripts for generating figures in a report on mathematical models in biology.

The report explores reaction-diffusion partial differential equations, focusing on the Fisher-KPP equation and the Allee effect model, including analytical derivations of travelling wave solutions and phase portraits.

The compiled PDF document is available in releases. This is the latest version of the document.

Equations

The document explores several key equations in reaction-diffusion models:

Basic Diffusion and Reaction-Diffusion

• Diffusion equation: $\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$
• Reaction-diffusion equation: $\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} + F(u)$

KPP Restrictions on Reaction Term

• $F(0) = F(1) = 0$
• $F'(0) = \alpha > 0$
• $F'(u) < \alpha$ for $u \in (0, 1)$

Fisher-KPP Equation

• Reaction term: $F(u) = \alpha u (1 - u)$
• Logistic growth form: $F(u) = \alpha u (1 - u/u_\infty)$
• Dimensionless form: $\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial \chi^2} + u(1-u)$
• Travelling wave ODE: $-c \frac{du}{dz} = \frac{d^2 u}{dz^2} + u(1-u)$

Allee Effect Equation

• Reaction term: $F(u) = \alpha u \left(1 - \frac{u}{u_\infty}\right)\left(\frac{u - u_-}{u_-}\right)$
• Dimensionless form: $\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial \chi^2} + u (1 - u ) \left(\frac{u - u_-}{u_-}\right)$

Figures

Phase portrait of diffusion equation with the Fisher-KPP reactive term

Phase portrait of diffusion equation with the Allee effect reactive term

Compilation

The document can be compiled using tectonic

$ tectonic -X build

after creating the figures.

The figures may be generated with Julia (tested version 1.10) by running the scripts in src/figs/, e.g.

$ julia --project fisher_wave.jl