On a link between a species survival time in an evolution model and the Bessel distributions

Abstract

We consider a stochastic model for species evolution. A new species is born at rate $\lambda$ and a species dies at rate $\mu$. A random number, sampled from a given distribution $F$, is associated with each new species and assumed as its fitness, at the time of birth. Every time there is a death event, the species that is killed is the one with the smallest fitness. We consider the (random) survival time of a species with a given fitness $f$. We show that the survival time distribution depends crucially on whether $f<f_{c}$, $f=f_{c}$ or $f>f_{c}$ where $f_{c}$ is a critical fitness that is computed explicitly.