Here we’ll consider a classical reaction–diffusion equation, with a logistic nonlinearity. This is often referred to as the Fisher–KPP equation,

\[\pd{u}{t}=\nabla^2 u+ru\left(1-\frac{u}{K}\right),\]

with periodic boundary conditions.

  1. Load the interactive simulation.

  2. Click within the box to visualise a ‘line’ of a population, which will then spread out as a planar wave.

  3. Explore different parameters in this model, namely $D$, $r$, and $K$.

  4. Does the wave speed, $c$, approximately follow the scaling law derived via linearisation of the wavefront (that is, $c \propto \sqrt{rD}$)? One interesting experiment to try is to see what happens if you simultaneously decrease $r$ and increase $D$ (or vice versa). This should have (approximately) the same effective wave speed, but the profile will be different as you have effectively changed the time and space scales in opposite directions.

  5. Does the value of the carrying capacity, $K$, matter for the speed of the wave? Or the profile?

  6. Next change the brush type to a circle and explore how circular waves travel. These are similar to the planar (effectively 1D) waves above, but their speed will be slightly different as the curvature of these wave fronts will influence their speed.

  7. You can also explore this kind of wave in a 1D model. You can press to reset the simulation as you change parameters.

Competitive exclusion

Travelling waves also occur in multispecies models. A model of two competing populations (red and grey squirrels) can be written as,

\[\begin{aligned}\pd{R}{t}&=D\nabla^2 R+R(1-c_{RR}R-c_{RG}G),\\ \pd{G}{t}&=D\nabla^2 G+ G(1-c_{GR}R-c_{GG}G),\end{aligned}\]

where $D$ is a diffusion coefficient, $c_{RG}, c_{GR}$ are interspecific, and $c_{RR},c_{GG}$ are intraspecific competition coefficients. Competitive exclusion can lead to one of the species being driven to extinction by the other. This simulation explores the grey squirrels driving the red to extinction across a map of the United Kingdom.

Epidemic waves

As another example of logistic travelling waves, we can consider the SIS model of infection given by

\[\begin{aligned}\diff{S}{t}&=d I - b S I,\\ \diff{I}{t}&=b S I - d I,\end{aligned}\]

where $S$ is the number of susceptible individuals, $I$ the number of infected individuals, $d$ a recovery rate, and $b$ an infection rate. Since this system is mass conserving (that is, $N=S + I$ must be a constant), we can rewrite this model purely in terms of the proportion of infected individuals $p = I/N$ to get

\[\pd{p}{t}=\nabla^2 p+\beta p(1-p)-\delta p,\]

where we have rescaled the infection and recovery rates and added a diffusion term to model spatial movement of infected individuals. As long as $R_0 = \beta/\delta > 1$, this model will have the same travelling-wave behaviour as the Fisher-KPP equation above, where $p=0$ is an unstable steady state, and $p=(\beta - \delta)/\beta$ is a stable endemic equilibrium. You can see an epidemiological travelling wave across the USA to see how this plays out in time and space.