A stochastic model for the evolution of species with random fitness

Abstract

We generalize the evolution model introduced by Guiol, Machado and Schinazi (2010). In our model at odd times a random number X of species is created. Each species is endowed with a random fitness with arbitrary distribution on [0, 1]. At even times a random number Y of species is removed, killing the species with lower fitness. We show that there is a critical fitness fc below which the number of species hits zero i.o. and above of which this number goes to infinity. We prove uniform convergence for the distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.

An evolution model with uncountably many alleles

“…In the first case the process is usually a birth-death process and one can focus either on the equilibrium when time grows, or on the trajectories. For instance the asymptotic distribution of fitnesses is studied by [15,3,4] in an evolution scheme where a random number of least fit individuals die at each generation, while [6,18] study the effect of random/deterministic events on this distribution; [2] and [7] study the convergence of the evolutionary process as the population size goes to infinity. The population is assumed to have a constant size in classical models such as the Wright-Fisher and the Moran models, but even with this assumptions there are still many theoretical challenges and applications (see for instance [10,21,22]).…”

An evolution model with event-based extinction

“…Several extensions and generalizations of the GMS model were subsequently introduced and studied, as in Ben-Ari et al [5], Michael and Volkov [6], Bertacchi et al [7] and Grejo et al [8]. Further, Guiol et al [9] proposed a variation of the model, where the evolution is given in continuous time.…”