Many models of continuous trait evolution assume that traits evolve according to Brownian motion. What is the biological or physical basis for this choice?

I realize there are models that do not assume Brownian motion, but what I am interested in asking is why the null model is so often chosen to be Brownian motion.

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    $\begingroup$ What else would you choose? $\endgroup$ – Bryan Krause Sep 23 at 20:00
  • $\begingroup$ Please define "Brownian motion". $\endgroup$ – Rodrigo de Azevedo Sep 25 at 8:17
  • $\begingroup$ Brownian motion models the random motion of particles suspended in a medium. In biology it is often used to model situations where change in some system occurs via a random walk. $\endgroup$ – jmah Sep 25 at 19:07
  • $\begingroup$ I asked because some people use "Brownian motion" as a synonym for Wiener process. I conclude you are not one of those people. $\endgroup$ – Rodrigo de Azevedo Sep 25 at 19:23
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    $\begingroup$ Lévy flight. $\endgroup$ – Rodrigo de Azevedo Sep 25 at 22:00

I think that the simple answer to this question is that the present comparative methodology was largely established by Felsenstein 1985, American Naturalist. For mathematical convenience, he suggested Brownian motion as a null hypothesis, because "...the variance of the distribution of change of a branch is proportional to the length of time of the branch...", and then "...it is easy to see that the differences between pairs of tips... must be independent." Also: "...after one unit of time, the contrast [between a pair of tips] has expectation 0 and [easily defined] variance..."

He explicitly discusses whether Brownian motion is a reasonable model in the section "What if we lack an acceptable statistical model of character change?"

I would suggest reading that paper in more detail if you are interested in the details of Brownian motion applied to phylogenies.

A recent historical perspective on this influential paper can be found here.

For a more extensive bibliography/more details you can see here.

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  • $\begingroup$ thank you very much for the papers! Yes, this is exactly what I was looking for. Felsenstein's a giant - I should have guessed he had something to do with it... $\endgroup$ – jmah Sep 23 at 20:30
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    $\begingroup$ @jmah i feel like my main contribution to this website is as a felsenstein stan $\endgroup$ – Maximilian Press Sep 24 at 18:20

Just came across a chapter by Felsenstein on phylogenetic inference with quantitative characters. In it, he states this biological justification for using Brownian motion:

A quantitative trait that has genetic variation controlled by a single locus will change as the gene frequencies at the locus undergo genetic drift... Brownian motion is a reasonable approximation to change of a quantitative character by genetic drift, provided that ... [additive genetic variance] ... remains approximately constant.

Felsenstein also mentions that Cavalli-Sforza and Edwards (1967) consider that varying selection at a locus can be approximated by Brownian motion.

So I suppose that's another slant on why Brownian motion is used.

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