Electronic Communications in Probability

A stochastic model for the evolution of species with random fitness

Daniela Bertacchi, Jüri Lember, and Fabio Zucca

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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 $f_c$ 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 fitness distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 88, 13 pp.

Received: 19 April 2018
Accepted: 7 November 2018
First available in Project Euclid: 24 November 2018

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Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J15

generalized GMS model birth and death process survival fitness queuing process limit distribution

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Bertacchi, Daniela; Lember, Jüri; Zucca, Fabio. A stochastic model for the evolution of species with random fitness. Electron. Commun. Probab. 23 (2018), paper no. 88, 13 pp. doi:10.1214/18-ECP190. https://projecteuclid.org/euclid.ecp/1543028983

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