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Urn models, replicator processes, and random genetic drift. (English) Zbl 0993.92026

Summary: To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equations (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and balls of the same colors (strategies) are added or removed according to probabilities that depend only on the colors of the chosen balls. Under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn, the probability of nonextinction is shown to be positive.
On the event of nonextinction, three results are proven: (i) the number of balls increases asymptotically at a linear rate, (ii) the distribution \(x(n)\) of strategies at the n th update is a ‘noisy’ Cauchy–Euler approximation to the mean limit ODE of the process, and (iii) the limit set of \(x(n)\) is almost surely a connected internally chain recurrent set for the mean limit ODE. Under a stronger set of assumptions, it is shown that for any attractor of the mean limit ODE there is a positive probability that the limit set for \(x(n)\) lies in this attractor.
Theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes, and is less likely to overcome natural selection in large populations with the potential for rapid growth.

MSC:

92D15 Problems related to evolution
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F99 Limit theorems in probability theory
60C05 Combinatorial probability
92D10 Genetics and epigenetics
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