This seminar will be presented in hybrid mode. The speaker will deliver his talk in person.
Title: When k tribes go to war
Abstract: A population of k murderous tribes exist. At rate C(i,j), an individual from tribe i meets with an individual from tribe j (for j = 1, . . . , k), killing them and reducing the j-th tribe’s by one individual. This includes infighting within tribes. Eventually, there is one surviving individual. One may think of this stochastic process as a multi-type version of Kingman’s coalescent death chain. In this talk, we investigate what happens as the total population across all tribes tends to infinity. In essence, if we describe n(t) = (n1(t), . . . ., nk(t)) as the process describing the number of individualsin each tribe at time t > 0, we discuss what it means for this process to “come down from infinity”. In doing so, we uncover a remarkable connection with the so-called replicator equations, describing the dynamical system of an evolutionary game theory. For this reason, we call our stochastic process the “replicator coalescent” It turns out that, from the many “infinities” the process can come down from, there is a “bottleneck” that the process must always pass through which corresponds to an evolutionary stable state in the replicator equations. Prior to this bottle neck, which occurs at arbitrarily large population size, through a time change, we see that the replicator coalescent behaves essentially like the solution to the replicator equations. After the bottleneck, stochastic effects become more pronounced and the process behaves more like a Markov chain.
The talk will be followed by refreshments in the Huxley Common Room at 4pm.