On the generalization of the GMS evolutionary model

We study a generalization of the evolution model proposed by Guiol, Machado and Schinazi (2010). In our model, at each moment of time a random number of species is either born or removed from the system; the species to be removed are those with the lower fitnesses, fitnesses being some numbers in $[0,1]$. We show that under some conditions, a set of species approaches (in some sense) a sample from a uniform distribution on $[f,1]$ for some $f\in [0,1)$, and that the total number of species forms a recurrent process in most other cases.