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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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 $f_c$ 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1543028983

Digital Object Identifier
doi:10.1214/18-ECP190

Subjects
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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

  • [1] P. Bak, K. Sneppen, Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 74 (1993), 4083–4086.
  • [2] I. Ben-Ari, An empirical process interpretation of a model of species survival. Stochastic Process. Appl. 123 (2013), n. 2, 475–489.
  • [3] I. Ben-Ari, A. Matzavinos, A. Roitershtein, On a species survival model. Electron. Commun. Probab. 16 (2011), 226–233.
  • [4] W. Feller, An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley & Sons Inc., New York-London-Sydney, 1968, xviii+509 pp.
  • [5] W. Feller, An introduction to probability theory and its applications. Vol. II, Second edition, John Wiley & Sons Inc., New York-London-Sydney, 1971, xxiv+669 pp.
  • [6] C. Grejo, F.P. Machado, A. Roldán-Correa, The fitness of the strongest individual in the subcritical GMS model. Electron. Commun. Probab. 21 (2016), n. 12, 5 pp.
  • [7] H. Guiol, F. P. Machado, R. B. Schinazi, A stochastic model of evolution. Markov Process. Related Fields 17 (2011), n. 2, 253–258.
  • [8] H. Guiol, F. P. Machado, R. B. Schinazi, On a link between a species survival time in an evolution model and the Bessel distributions. Braz. J. Probab. Stat. 27 (2013), n. 2, 201–209.
  • [9] R. Meester, D. Znameski, Limit behavior of the Bak-Sneppen evolution model. Ann. Probab. 31 (2003), 1986–2002.
  • [10] S. Michael, S. Volkov, On the generalization of the GMS evolutionary model. Markov Process. Related Fields 18 (2012), n. 2, 311–322.
  • [11] D. Williams, Probability with martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1991. xvi+251 pp.