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Matteo Smerlak emailed me notifying me of this paper, which sounds really interesting:

- Matteo Smerlak and Ahmed Youssef, Statistical patterns of Darwinian evolution.

Abstract.In the most general terms, Darwinian evolution is a flow in the space of fitness distributions. In the limit where mutations are infinitely frequent and have infinitely small fitness effects (the "diffusion approximation", Tsimring et al. have showed that this flow admits "fitness wave" solutions: Gaussian-shape fitness distributions moving towards higher fitness values at constant speed. Here we show more generally that evolving fitness distributions are attracted to a one-parameter family of distributions with a fixed parabolic relationship between skewness and kurtosis. Unlike fitness waves, this statistical pattern encompasses both positive and negative (a.k.a. purifying) selection and is not restricted to rapidly adapting populations. Moreover we find that the mean fitness of a population under the selection of pre-existing variation is a power-law function of time, as observed in microbiological evolution experiments but at variance with fitness wave theory. At the conceptual level, our results can be viewed as the resolution of the "dynamic insufficiency" of Fisher's fundamental theorem of natural selection. Our predictions are in good agreement with numerical simulations.

I asked him if he could write a blog article summarizing this paper, and he gladly agreed! I hope he will write it here on the wiki.

If you see this, Matteo, you can announce any progress you make here! Or, if you have questions, you can ask them here.

## Comments

This seems relevant. I'd like to know how the two papers are related.

The distribution of fitness effects among beneficial mutations in Fisher’s geometric model of adaptation, H. Allen Orr. http://www.webpages.uidaho.edu/~joyce/m563_html/papers/Orr2005.pdf

`This seems relevant. I'd like to know how the two papers are related. The distribution of fitness effects among beneficial mutations in Fisher’s geometric model of adaptation, H. Allen Orr. http://www.webpages.uidaho.edu/~joyce/m563_html/papers/Orr2005.pdf`

This work is exciting to me as a biologist, and I'd like to challenge the researchers to further extend their results by relaxing or modifying whatever assumptions result in the one-parameter family with its finite moments. I'd like to see something that looks more like a Lévy stable distribution or some other member of the stable family of PDFs show up when the crank is turned. This outcome would fit my biological intuition far better. There are numerous precedents -- the history of data analysis from biometrician Charlie Winsor to Tukey and beyond, the current extension of phylogenetic signal correlation approaches to methods based on a Lévy random process rather than a Gaussian alternative, and my personal experience as a field biologist and fisheries biometrician. When I see a Gaussian distribution in theoretical or mathematical biology, I often feel like W.D. Hamilton did when he saw the words "group selection". Of course, multilevel selection is very cool now, and theoreticians are very interested in its possible implications for the history of life on earth. High-energy physicists and cosmologists may be wiser about their choices of problems and questions, but evolutionary biology has spent fifty years tilting at the windmill of group selection, enough for entire careers to go far down the wrong road.

Gaussian distributions are ubiquitous tools and highly useful, but biology has gradually learned their limitations along with their undeniably powerful applications to modelling and data analysis. John Tukey, John Chambers, Ron Hardin and Bell Labs generally helped me toward a healthy suspicion of the Central Limit Theorem and the hardened methods of confirmatory data analysis in which I had previously been trained. Three intensely stimulating years at the Labs and many years as a working-stiff fisheries biometrician helped show me what I personally consider the light and the way in evolution, and the research reported in Matteo's post gives me renewed hope that new tools for creative thinking about ecology and evolution are not far off.

`This work is exciting to me as a biologist, and I'd like to challenge the researchers to further extend their results by relaxing or modifying whatever assumptions result in the one-parameter family with its finite moments. I'd like to see something that looks more like a Lévy stable distribution or some other member of the stable family of PDFs show up when the crank is turned. This outcome would fit my biological intuition far better. There are numerous precedents -- the history of data analysis from biometrician Charlie Winsor to Tukey and beyond, the current extension of phylogenetic signal correlation approaches to methods based on a Lévy random process rather than a Gaussian alternative, and my personal experience as a field biologist and fisheries biometrician. When I see a Gaussian distribution in theoretical or mathematical biology, I often feel like W.D. Hamilton did when he saw the words "group selection". Of course, multilevel selection is very cool now, and theoreticians are very interested in its possible implications for the history of life on earth. High-energy physicists and cosmologists may be wiser about their choices of problems and questions, but evolutionary biology has spent fifty years tilting at the windmill of group selection, enough for entire careers to go far down the wrong road. Gaussian distributions are ubiquitous tools and highly useful, but biology has gradually learned their limitations along with their undeniably powerful applications to modelling and data analysis. John Tukey, John Chambers, Ron Hardin and Bell Labs generally helped me toward a healthy suspicion of the Central Limit Theorem and the hardened methods of confirmatory data analysis in which I had previously been trained. Three intensely stimulating years at the Labs and many years as a working-stiff fisheries biometrician helped show me what I personally consider the light and the way in evolution, and the research reported in Matteo's post gives me renewed hope that new tools for creative thinking about ecology and evolution are not far off.`

Nice to hear from you, Robert! And always good to hear from you, Graham.

Matteo will take a while to write his article, since he's busy finishing up other stuff, but I'll point him to you comments.

`Nice to hear from you, Robert! And always good to hear from you, Graham. Matteo will take a while to write his article, since he's busy finishing up other stuff, but I'll point him to you comments.`

Hi all, I'm getting started with the blog post now!

`Hi all, I'm getting started with the blog post now!`

A first version is live at

http://www.azimuthproject.org/azimuth/show/Blog+-+statistical+laws+of+darwinian+evolution

Let me know if you have any comments!

`A first version is live at http://www.azimuthproject.org/azimuth/show/Blog+-+statistical+laws+of+darwinian+evolution Let me know if you have any comments!`

Ever run across Relative Abundance Distribution?

`Ever run across Relative Abundance Distribution?`

WebHubTel, do you mean probability distributions of the relative abundance of species, such as the lognormal? They arise in the context of what has been termed "the species sampling problem", pioneered by Pielou, MacArthur and I.J. Good. Watch out for long, straggling tails and "completeness" estimates that aren't robust to nonnormality and CLT violations. Please tell me more about your interest and current work; I've published (Iong ago) on this topic in an animal behavior context.

`WebHubTel, do you mean probability distributions of the relative abundance of species, such as the lognormal? They arise in the context of what has been termed "the species sampling problem", pioneered by Pielou, MacArthur and I.J. Good. Watch out for long, straggling tails and "completeness" estimates that aren't robust to nonnormality and CLT violations. Please tell me more about your interest and current work; I've published (Iong ago) on this topic in an animal behavior context.`

I had some interest in this briefly, applying max entropy arguments to species diversity. Those give the long tails that you find in abundance data. The problem is that there really isn't enough structure and dynamic range to the data to convince anyone that one distribution is better than another.

See in my book p.434, section entitled "Dispersion, Diversity, and Resilience" https://drive.google.com/open?id=0B8wYusbaTnvMMUZDZHI0Qm5MUDQ

If you have problem with that link, I have others

`I had some interest in this briefly, applying max entropy arguments to species diversity. Those give the long tails that you find in abundance data. The problem is that there really isn't enough structure and dynamic range to the data to convince anyone that one distribution is better than another. See in my book p.434, section entitled "Dispersion, Diversity, and Resilience" https://drive.google.com/open?id=0B8wYusbaTnvMMUZDZHI0Qm5MUDQ If you have problem with that link, I have others`