We introduce the following model for the evolution of a population. At every discrete time n ≥ 0 exactly one individual is introduced in the population and is assigned a death probability cn sampled from C, a fixed probability distribution. We think of cn as a genetic marker of this individual. At every time n ≥ 1 every individual in the population dies or not independently of each other with its corresponding death probability cn. We show that the population size goes to infinity if and only if E(1/C) = ∞. This is in sharp contrast with the model with constant c and with the model in random environment (same random cn for all individuals at time n). Both of these models are always positive recurrent. Thus, it is really the randomness of individual c’s that makes the population survive! 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.
Fontes et al. (Thu,) studied this question.