×

A stochastic model for the evolution of species with random fitness. (English) Zbl 1409.60109

Summary: We generalize the evolution model introduced by H. Guiol et al. [Markov Process. Relat. Fields 17, No. 2, 253–258 (2011; Zbl 1325.60157)]. 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.

MSC:

60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
92D15 Problems related to evolution

Citations:

Zbl 1325.60157
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

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. · Zbl 1266.60125 · doi:10.1016/j.spa.2012.09.009
[3] I. Ben-Ari, A. Matzavinos, A. Roitershtein, On a species survival model. Electron. Commun. Probab. 16 (2011), 226-233. · Zbl 1225.60124
[4] 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. · Zbl 1336.60145
[5] H. Guiol, F. P. Machado, R. B. Schinazi, A stochastic model of evolution. Markov Process. Related Fields 17 (2011), n. 2, 253-258. · Zbl 1325.60157
[6] 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. · Zbl 1401.60146
[7] R. Meester, D. Znameski, Limit behavior of the Bak-Sneppen evolution model. Ann. Probab. 31 (2003), 1986-2002. · Zbl 1044.60095
[8] S. Michael, S. Volkov, On the generalization of the GMS evolutionary model. Markov Process. Related Fields 18 (2012), n. 2, 311-322. · Zbl 1287.60116
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.