# Talk:Fisher's equation

## Fisher Equation is Different from Fisher-Kolgomorov Equation

Fisher equation was published in The Wave of Advance of Advantageous Genes, Annals of Eugenics 7:355-369 (1937). The equation studied in that paper was

$\frac{\partial p}{\partial t}= k \frac{\partial^2 p}{\partial x^2} +mpq$

Fisher conjectured about the existence of a traveling wave. This was proved in 1937 by A.N. Kolmogorov, A.N. Petrovskii, N.S. Piskunov, in A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskovo Gos. Univ. 17 (1937). Kolgomorov et al. studied this equation replacing the term $mpq$ with the logistic term $p(1+p)$.

I have no time to edit this correction now, but will leave the comment in the discussion. If somebody wants to take on this edit, pelase go ahead. Otherwise, I will revisit this article in a few days. 13-FEB-2011. —Preceding unsigned comment added by 75.118.161.179 (talk) 16:46, 13 February 2011 (UTC)

## Replaced

(copied from User talk:Jitse Niesen#Fisher's equation)

I have replaced much of Fisher's equation. The previous contents seems to be taken from EqWorld which has nice references but wrong math. Please check original literature (not websites) in case you consider rv. Thanks. Also, maybe it should be merged with reaction-diffusion equations, which looks a bit sad. Jmath666 08:09, 1 April 2007 (UTC)

Thanks for your work. However, was there anything wrong with the previous version? I can't find a mistake (which admittedly doesn't mean much). Or do you mean that it was rather misleading in that it suggested that there is only one travelling wave solution?
As it happens, I did some work on reaction-diffusion equation yesterday. I agree that it looks sad (I hope to work a bit more on it), but I think it looked even sadder before. I'd be very happy with any help in bringing that page in a better state. It certainly needs a description of travelling waves.
Speaking about original literature, I've never read the book by Fisher that you mention. Does this contain the equation? In my experience, the usual reference is a 1937 paper by Fischer in Ann. Eugenics. -- Jitse Niesen (talk) 09:39, 1 April 2007 (UTC)

Yes, there is more than one traveling wave, moreover the formula made no sense - surely the wave speed does not depend on a. For all I known it might have been the correct formula but context messed up by the author from Eqworld; I'll need to look up the references they cite when I have time. Kolmogorov et al have detailed analysis and it is also outlined in Grindrod. Fisher (1930) was cited by Kolmogorov et al but I do not have it. Do you have a copy of Fisher (1937)? Reaction-diffusion equation looks much better now, very nice context. It needs some meat (=math). If you want to move the 2nd half from Fisher's equation there go ahead. I am reading up on reaction-diffusion equations now from Grindrod in a hurry because I put that on someone's exam. I may end up writing some notes on the phase-plane method in LaTeX for my own use that could be wikified automatically. Jmath666 16:02, 1 April 2007 (UTC)

Here we go. From the abstract of the Ablowitz citation:

The travelling waves for Fisher's equation are shown to be of a simple nature for the special wave speeds $c = \pm 5/\sqrt {(6)}$.... The general solution for this wave speed is found...

So this was not a general formula by any means but a parenthetical info. To put it back we'd need to get that paper and make sure it is quoted right but is it worth the bother? I have asked Mark for a copy out of curiosity anyway. Jmath666 16:29, 1 April 2007 (UTC)

The statement that that the waveform shape is not unique is incorrect. In fact, it follows in a straightforward way from the phase plane analysis that the shape is indeed unique for a given speed. This is no contradiction to the explicit formula that depends on a constant C. In fact, choosing different values for C is equivalent to a shift of the z-coordinate of the form z\mapsto z+const. Such shifts do not affect the shape of the wave. —Preceding unsigned comment added by TECG (talkcontribs) 23:51, 14 January 2010 (UTC)