On the~Time of Supplanting All Particles by Particles of One Type in a~Fixed Size Population

S. A. Klokov ; V. A. Topchii

Matematičeskie trudy, Tome 8 (2005) no. 2, pp. 168-183

Résumé

We consider a population of $N$ particles of each of which some type is ascribed to. At the integer time moments each particle splits into two particles of the same type as their parent, and then $N$ particles are instantly equiprobably excluded from the population of $2N$ particles. Let $\tau$ be a random variable denoting the number of generation when all particles become of the same type for the first time. We obtain upper bounds for the expectation of $\tau$. In particular, if all particles have different types originally then $\tau$ coincides, in terminology of branching processes, with the distance (in time) to the nearest common ancestor of the population with infinite long history. In simple cases, simulation results and approximate numerical solutions of systems of equations show that the resultant bound is about half as much again.