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Single-crossover dynamics: finite versus infinite populations. (English) Zbl 1139.92018
Summary: Populations evolving under the joint influence of recombination and resampling (traditionally known as genetic drift) are investigated. First, we summarize and adapt a deterministic approach, as valid for infinite populations, which assumes continuous time and single crossover events. The corresponding nonlinear system of differential equations permits a closed solution, both in terms of the type frequencies and via linkage disequilibria of all orders. To include stochastic effects, we then consider the corresponding finite-population model, the Moran model with single crossovers, and examine it both analytically and by means of simulations. Particular emphasis is on the connection with the deterministic solution. If there is only recombination and every pair of recombined offspring replaces their pair of parents (i.e., there is no resampling), then the expected type frequencies in the finite population, of arbitrary size, equal the type frequencies in the infinite population. If resampling is included, the stochastic process converges, in the infinite-population limit, to the deterministic dynamics, which turns out to be a good approximation already for populations of moderate size.

92D15 Problems related to evolution
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60G35 Signal detection and filtering (aspects of stochastic processes)
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