Population models at stochastic times

In this article, we consider time-changed models of population evolution $mathcal{X}^f(t)=mathcal{X}(H^f(t))$, where $mathcal{X}$ is a counting process and $H^f$ is a subordinator with Laplace exponent $f$. In the case $mathcal{X}$ is a pure birth process, we study the form of the distribution, the intertimes between successive jumps and the condition of explosion (also in the case of killed subordinators). We also investigate the case where $mathcal{X}$ represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size $n_0$. Finally, the subordinated linear birth-death process is considered. A special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.