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.


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


Zbl 1325.60157
Full Text: DOI arXiv Euclid


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