John Harte of U. C. Berkeley spoke about the maximum entropy method as a method of predicting patterns in ecology. Annette Ostling of the University of Michigan spoke about some competing theories, such as the ‘neutral model’ of biodiversity—a theory that sounds much too simple to be right, yet fits the data surprisingly well!

We managed to get a video of Ostling’s talk, but not Harte’s. Luckily, you can see the slides of both. You can also see a summary of Harte’s book *Maximum Entropy and Ecology*:

• John Baez, Maximum entropy and ecology, *Azimuth*, 21 February 2013.

Here are his talk slides and abstract:

• John Harte, Maximum entropy as a foundation for theory building in ecology.

Abstract.Constrained maximization of information entropy (MaxEnt) yields least-biased probability distributions. In statistical physics, this powerful inference method yields classical statistical mechanics/thermodynamics under the constraints imposed by conservation laws. I apply MaxEnt to macroecology, the study of the distribution, abundance, and energetics of species in ecosystems. With constraints derived from ratios of ecological state variables, I show that MaxEnt yields realistic abundance distributions, species-area relationships, spatial aggregation patterns, and body-size distributions over a wide range of taxonomic groups, habitats and spatial scales. I conclude with a brief summary of some of the major opportunities at the frontier of MaxEnt-based macroecological theory.

Here is a video of Ostling’s talk, as well as her slides and some papers she recommended:

• Annette Ostling, The neutral theory of biodiversity and other competitors to maximum entropy.

Abstract:I am a bit of the odd man out in that I will not talk that much about information and entropy, but instead about neutral theory and niche theory in ecology. My interest in coming to this workshop is in part out of an interest in what greater insights we can get into neutral models and stochastic population dynamics in general using entropy and information theory.I will present the niche and neutral theories of the maintenance of diversity of competing species in ecology, and explain the dynamics included in neutral models in ecology. I will also briefly explain how one can derive a species abundance distribution from neutral models. I will present the view that neutral models have the potential to serve as more process-based null models than previously used in ecology for detecting the signature of niches and habitat filtering. However, tests of neutral theory in ecology have not as of yet been as useful as tests of neutral theory in evolutionary biology, because they leave open the possibility that pattern is influenced by “demographic complexity” rather than niches. I will mention briefly some of the work I’ve been doing to try to construct better tests of neutral theory.

Finally I’ll mention some connections that have been made so far between predictions of entropy theory and predictions of neutral theory in ecology and evolution.

These papers present interesting relations between ecology and statistical mechanics. Check out the nice ‘analogy chart’ in the second one!

• M. G. Bowler, Species abundance distributions, statistical mechanics and the priors of MaxEnt, *Theoretical Population Biology* **92** (2014), 69–77.

Abstract.The methods of Maximum Entropy have been deployed for some years to address the problem of species abundance distributions. In this approach, it is important to identify the correct weighting factors, or priors, to be applied before maximising the entropy function subject to constraints. The forms of such priors depend not only on the exact problem but can also depend on the way it is set up; priors are determined by the underlying dynamics of the complex system under consideration. The problem is one of statistical mechanics and it is the properties of the system that yield the correct MaxEnt priors, appropriate to the way the problem is framed. Here I calculate, in several different ways, the species abundance distribution resulting when individuals in a community are born and die independently. In

the usual formulation the prior distribution for the number of species over the number of individuals is 1/n; the problem can be reformulated in terms of the distribution of individuals over species classes, with a uniform prior. Results are obtained using master equations for the dynamics and separately through the combinatoric methods of elementary statistical mechanics; the MaxEnt priors then emerge a posteriori. The first object is to establish the log series species abundance distribution as the outcome of per capita guild dynamics. The second is to clarify the true nature and origin of priors in the language of MaxEnt. Finally, I consider how it may come about that the distribution is similar to log series in the event that filled niches dominate species abundance. For the general ecologist, there are two messages. First, that species abundance distributions are determined largely by population sorting through fractional processes (resulting in the 1/n factor) and secondly that useful information is likely to be found only in departures from the log series. For the MaxEnt practitioner, the message is that the prior with respect to which the entropy is to be maximised is determined by the nature of the problem and the way in which it is formulated.

• Guy Sella and Aaron E. Hirsh, The application of statistical physics to evolutionary biology, *Proc. Nat. Acad. Sci.* **102** (2005), 9541–9546.

A number of fundamental mathematical models of the evolutionary process exhibit dynamics that can be difficult to understand analytically. Here we show that a precise mathematical analogy can be drawn between certain evolutionary and thermodynamic systems, allowing application of the powerful machinery of statistical physics to analysis of a family of evolutionary models. Analytical results that follow directly from this approach include the steady-state distribution of fixed genotypes and the load in finite populations. The analogy with statistical physics also reveals that, contrary to a basic tenet of the nearly neutral theory of molecular evolution, the frequencies of adaptive and deleterious substitutions at steady state are equal. Finally, just as the free energy function quantitatively characterizes the balance between energy and entropy, a free fitness function provides an analytical expression for the balance between natural selection and stochastic drift.

This musing just popped up in my brain: What about the global ecosystem, i.e. Gaia. Currently Lovelock’s Gaia hypothesis seems dead, i.e. Gaia is just a lucky thing. But it looks the art of Daisyworlds is not yet dead, i.e. is just starting off in a general mathematics sense (e.g. this 2013 paper http://eprints.soton.ac.uk/354179/ by James G Dyke http://www.jamesgdyke.info on the emergence of ecosystem homeostasis). Maybe the MaxEnt principle can help explain Gaia? (Some weird theory of “most probably lucky” without appealing to an anthropic principle…)