There are some tutorials starting today here:
• Tutorials on discrete mathematics and probability in networks and population biology, Institute of Mathematical Sciences, National University of Singapore, 2-6 May 2011. Organized by Andrew Barbour, Malwina Luczak, Gesine Reinert and Rongfeng Sun.
Back in 1954, Armitage and Doll noticed that log-log plots of cancer incidence as a function of age are close to linear, except for breast cancer, which slows down in older women. They suggested an explanation: a chain of independent random events have to occur before cancer can start. A simple model based on a Markov process gives a simple formula for how many events it must take—see the first batch of slides for details. This work was the first of a series of ever more sophisticated multi-stage models of carcinogenesis.
One of the first models Durrett explained was the Moran process: a stochastic model of a finite population of constant size in which things of two types, say and are competing for dominance. I believe this model can be described by a stochastic Petri net with two states, and , and two transitions:
Since I like stochastic Petri nets, I’d love to add this to my collection.
Suppose a population of animals roams around randomly and whenever two meet, they engage in some sort of conflict… or more generally, any sort of ‘game’. Suppose each can choose from some set of strategies. Suppose that if one chooses strategy and the other chooses strategy , the expected ‘payoff’ to the one is , while for the other it’s .
More generally, the animals might choose their strategies probabilistically. If the first chooses the ith strategy with probability and the second chooses it with probability then the expected payoff to the first player is
where the angle brackets are the usual inner product in I’m saying this in an overly fancy way, and making it look like quantum mechanics, in the hope that some bright kid out there will get some new ideas. But it’s not rocket science; the angle bracket is just a notation for this sum:
Let me tell you what it means for a probabilistic strategy to be ‘evolutionarily stable’. Suppose we have a ‘resident’ population of animals with strategy and we add a few ‘invaders’ with some other strategy, say . Say the fraction of animals who are invaders is some small number , while the fraction of residents is .
If a resident plays the game against a randomly chosen animal, its expected payoff will be
Indeed, it’s just as if the resident was playing the game against an animal with probabilistic strategy ! On the other hand, if an invader plays the game against a randomly chosen animal, its expected payoff will be
The strategy is evolutionarily stable if the residents do better:
for all probability distributions and sufficiently small .
Canning showed us how to manipulate this condition in various ways and prove lots of nice theorems. His slides will appear online later, and then I’ll include a link to them. Naturally, I’m hoping we’ll see that a dynamical model, where animals with greater payoff get to reproduce more, has the evolutionary stable strategies as stable equilibria. And I’m hoping that some model of this sort can be described using a stochastic Petri net—though I’m not sure I see how.
On another note, I was happy to see Persi Diaconis and meet his wife Susan Holmes. Both will be speaking later in the week. Holmes is a statistician who specializes in “large, messy datasets” from biology. Lately she’s been studying ant networks! Using sophisticated image analysis to track individual ants over long periods of time, she and her coauthors have built up networks showing who meets who in ant ant colony. They’ve found, for example, that some harvester ants interact with many more of their fellows than the average ant. However, this seems to be due to their location rather than any innate proclivity. They’re the ants who hang out near the entrance of the nest!
That’s my impression from a short conversation, anyway. I should read her brand-new paper:
• Noa Pinter-Wollman, Roy Wollman, Adam Guetz, Susan Holmes and Deborah M. Gordon, The effect of individual variation on the structure and function of interaction networks in harvester ants, Journal of the Royal Society Interface, 13 April 2011.
She said this is a good book to read:
• Deborah M. Gordon, Ant Encounters: Interaction Networks and Colony Behavior, Princeton U. Press, Princeton New Jersey, 2010.
There are also lots of papers available at Gordon’s website.