Branching processes in random environment and “bottlenecks” in evolution of populations

A branching process $Z(n)$, $n=0,1\dots$ is considered which evolves in a random environment generated by a sequence of independent identically distributed generating functions $f_0(s),f_1(s),\dots$ . Let $S_0=0$, $S_0=0$, $S_k=\log f'_0(1)+\dots+\log f'_{k-1}(1)$, $k\ge 1$, be the associated random walk and let $\tau (n)$ be the leftmost point of minimum of $\{S_k\}_{k\ge 0}$ on the interval $[0,n]$. Assuming that the random walk satisfies the Spitzer condition $n^{-1}\sum_{k=1}^{n}P\{S_k>0\}\to\rho\in(0,1)$, $n\to\infty$, we show (under the quenched approach) that for each fixed $t\in (0,1]$ and $m=0,\pm 1,\pm 2\dots$ the distribution of $Z(\tau(nt)+m)$ given $Z(n)>0$ converges as $n\to\infty $ to a (random) discrete distribution. Thus, in contrast to fixed points of the form $nt$, where the size of the population is large (even exponentially large, see [V. A. Vatutin and E. E. Dyakonova, Theory Probab. Appl., 49 (2005), pp. 275–308]), the size of the population at (random) points of sequential minima of the associated random walk becomes drastically small and, therefore, the branching process passes through a number of bottlenecks at such moments. As a corollary of our results we find (under the quenched approach) the distribution of the local time of the first excursion of a simple random walk in a random environment, provided this excursion attains a high level.