  We introduce the following model for the evolution of a population. At every
discrete time $j\geq 0$ exactly one individual is introduced in the population
and is assigned a death probability $c_j$ sampled from $C$, a fixed probability
distribution. We think of $c_j$ as a genetic marker of this individual. At
every time $n\geq 1$ every individual in the population dies or not
independently of each other with its corresponding death probability $c_j$. We
show that the population size goes to infinity if and only if $E(1/C)=\infty$.
This is in sharp contrast with the model with constant $c$ and with the model
in random environment (same random $c_n$ for all individuals at time $n$). Both
of these models are always positive recurrent. Thus, genetic diversity does
seem to help survival! We also study the point process associated with our
model. We show that the limit point process has an accumulation point near 0
for the $c'$s. For certain $C$ distributions, including the uniform, the limit
process properly rescaled is also shown to converge to a non-homogeneous
Poisson process.
