The Fisher equation is partial differential equation of the form:

$$\frac{\partial u(x,t)}{\partial t}=D \nabla^2 u + r u(1-u)$$

where

*u(x,t)*represents the density of cancer cells at time t and location x, and the parameters of the model are D, the diffusivity of cancer cells (i.e. how fast they migrate) and

*r*, the rate of cell division.

The Fisher equations exhibits travelling wave solution, i.e. a fixed front profile that is being translated in space as time progresses. These solutions are typically characterised by their velocity

*c*and slope

*s*. It has been shown that the wave speed is given by

$$c=2\sqrt{Dr}.$$

A partial proof of this can be found in Mathematical Biology by James Murray. In the same book it is claimed that the slope

$$s=1/4c,$$

which implies that faster waves are less steep. This statement is accompanied by the below figure.

Substituting the expression for

*c*one is lead to believe that

$$s=\frac{1}{8\sqrt{D r}}.$$

This expression is never mentioned in Murray's book, but I would claim that the presentation is quite misleading, because if one looks closer at the analysis that leads up to the statement

*s=1/4c*, then one finds that the analysis is done on a non-dimensional version of the Fisher equation, which has wavespeed

*c = 2*. This implies that the statement

*s=1/4c*simply means

*s = 1/8*.

If one instead carries out the exact same analysis on the dimensional Fisher equation one finds that

$$s=1/8\sqrt{r/D}.$$

I think this fact has been missed by many mathematical biologist and I hope this blog post can shed some light on this misunderstanding.

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