  Based on a simple object, an i.i.d. sequence of positive integer-valued
random variables, $\{a_n\}_{n\in \mathbb{Z}}$, we introduce and study two
random structures and their connections. First, a population dynamics, in which
each individual is born at time $n$ and dies at time $n+a_n$. This dynamics is
that of a D/GI/$\infty$ queue, with arrivals at integer times and service times
given by $\{a_n\}_{n\in \mathbb{Z}}$. Second, the directed random graph $T^f$
on $\mathbb{Z}$ generated by the random map $f(n)=n+a_n$. Only assuming
$\mathbb{E}[a_0]<\infty$ and $\mathbb{P}[a_0=1]>0$, we show that, in steady
state, the population dynamics is regenerative, with one individual alive at
each regenerative epochs. We identify a unimodular structure in this dynamics.
More precisely, $T^f$ is a unimodular directed tree, in which $f(n)$ is the
parent of $n$. This tree has a unique bi-infinite path. Moreover, $T^f$ splits
the integers into two categories: ephemeral integers, with a finite number of
descendants of all degrees, and successful integers, with an infinite number.
Each regenerative epoch is a successful individual such that all integers less
than it are its descendants of some order. Ephemeral, successful, and
regenerative integers form stationary and mixing point processes on
$\mathbb{Z}$.
