## Multi-type branching processes

This board comes from the office of Peter Keller, a new postdoc working at the University of Edinburgh. He is currently thinking about multi-type branching processes in biology and wanted to draw some pictures on the board to help him figure out what was going on. In a branching process you have a population of individuals who each produce a random number of offspring, and you want to ask questions like “How big will the population be at a certain time in the future?”, or “What is the chance of the population ever becoming extinct?”. In a multi-type branching process, there are different types of individuals. For example, when bacteria reproduce, they may either replicate themselves, or they may produce a mutant form. Mutants can then only replicate into mutants. We can then ask what the proportion of mutants to normal individuals is likely to be after a certain amount of time.

I have no idea what the reference to martingales and larks is. Please leave a comment if you think you know!

Matt JensenA martingale in gambling is a highly-risky strategy of doubling your bets until, eventually, you either win or become bankrupt. In probability theory it has a more generalized and formalized meaning, which I leave to Wikipedia: http://en.wikipedia.org/wiki/Martingale_(probability_theory)

April 1, 2013 at 5:27 pm

Peter KellerA martingale is a stochastic process that resembles a “fair game”, the expectation of the process is constant over time.

Regarding the larks and martingales, it is of course a modified citation from Shakespeare’s Romeo and Juliet. Like Juliet tried to convince Romeo to stay a little longer, Martingales enrich the theory and allow in an easy way to calculate things that are otherwise difficult…

May 16, 2013 at 12:48 pm