Urn models, replicator processes, and random genetic drift

To understand the relative importance of natural selection and random genet ic drift in finite but growing populations, the asymptotic behavior of a cl ass of generalized Polya urns is studied using the method of ordinary diffe rential equation (ODE). Of particular interest is the replicator process : tw balls (individuals) are chosen from an urn (the population) at random wi th replacement and balls of the same colors (strategies) are added or remov ed according to probabilities that depend only on the colors of the chosen balls. Under the assumption that the expected number of balls being added a lways exceeds the expected number of balls being removed whenever balls are in the urn, the probability of nonextinction is shown to be positive. On t he event of nonextinction, three results are proven: (i) the number of ball s increases asymptotically at a linear rate, (ii) the distribution chi (n) of strategies at the nth update is a noisy Cauchy Euler approximation to th e 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. U nder a stronger set of assumptions, it is shown that for any attractor of t he 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 prob abilities of nonextinction and convergence to an attractor suggest that ran dom genetic drift is more likely to overcome natural selection in small pop ulations 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.