Fisher-KPP equation

Question

Can you help me understand how to derive the Fisher-KPP equation? I.e., how can I figure out where it comes from?

It is easy to find the derivations of the diffusion equation and the heat equation, but I can't find any summary of the way to derive this one...

Will you please help me?

Thanks

Have you read Fisher's own 1937 paper linked to at the Wikipedia article you provided? — horchler, Jul 17 '13 at 21:53
physics SE is more suitable for this question as it only relates the knowledge of physics. — doraemonpaul, Jul 19 '13 at 02:24

score 2 · Accepted Answer · answered Jul 17 '13 at 23:53

I am giving you a rather more intuitive argument here: for equation $$ \frac{\partial u}{\partial t}=r u(1-u)+\color{blue}{\frac{\partial^2 u}{\partial x^2}},\tag{1}$$ if we ignore the blue term, it is just: $$ \frac{\partial u}{\partial t}=ru(1-u),\tag{2} $$ which is a Logistic equation. Above pure Logistic equation models the change of a population having growth rate $r$. With blue term added, it means a diffusion is added spacially, which can be interpreted as: a bump somewhere in the quantity $u$ will propagate in space, in this case, in a nonlinear way like Burgers' equation.

The equilibria and their stability of (1) inherit the ones in (2), which are $u\equiv 1$ (stable) and $u\equiv 0$ (unstable). The traveling wave solution physically describes an traveling wavefront from the region near the stable solution $u=1$ to the region near unstable solution $u=0$.

Also judging by the first few pages of the paper A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem by Kolmogorov-Petrovskii-Piscounov (my library doesn't have this article), the notation is more user-friendly and you might wanna check that out.

Fisher-KPP equation

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