https://github.com/kunzaatko/reactivediffusion
A report on the Fisher-KPP (Kolmogorov-Petrovsky-Piskunov) and the Allee-KPP equations with stading wave solutions. This is a school project for the subject of methods in mathematical bioloby (MBM01)
https://github.com/kunzaatko/reactivediffusion
diffusion-equation university-project
Last synced: 4 months ago
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A report on the Fisher-KPP (Kolmogorov-Petrovsky-Piskunov) and the Allee-KPP equations with stading wave solutions. This is a school project for the subject of methods in mathematical bioloby (MBM01)
- Host: GitHub
- URL: https://github.com/kunzaatko/reactivediffusion
- Owner: kunzaatko
- License: mit
- Created: 2024-06-26T11:32:36.000Z (about 2 years ago)
- Default Branch: trunk
- Last Pushed: 2025-11-05T12:16:33.000Z (8 months ago)
- Last Synced: 2025-11-05T14:20:49.330Z (8 months ago)
- Topics: diffusion-equation, university-project
- Language: TeX
- Homepage:
- Size: 1.33 MB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# 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](https://github.com/kunzaatko/ReactiveDiffusion/releases).
This is the latest version of the [document](https://github.com/kunzaatko/ReactiveDiffusion/releases/latest/download/standing_waves.pdf).
## 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`](https://github.com/tectonic-typesetting/tectonic)
```bash
$ 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.
```bash
$ julia --project fisher_wave.jl
```