Dynamics of the time to the most recent common ancestor in a large branching population
If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting stochastic process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright-Fisher dynamics. We investigate cases in which the population varies in size and evolves according to a class of models that includes suitably conditioned $(1+\beta)$-stable continuous state branching processes (in particular, it includes the conditioned Feller continuous state branching process). We also consider the discrete time Markov chain that tracks the MRCA age just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent, and compute stationary distributions when they exist. We also introduce a new family of Markov processes that stand in a relation with respect to the $(1+\beta)$-stable continuous state branching process that is similar to the one between the Bessel-squared diffusions and the Feller continuous state branching process.