Abstract
We study two types of stochastic processes, first a mean-field spatial system of interacting Fisher–Wright diffusions with an inferior and an advantageous type with rare mutation (inferior to advantageous) and selection and second a mean-field spatial system of supercritical branching random walks with an additional death rate, which is quadratic in the local number of particles. The former describes a standard two-type population under selection, mutation and the latter model describes a population under scarce resources causing additional death at high local population intensity. Geographic space is modelled by 1, …, N. The first process starts in an initial state with only the inferior type present or an exchangeable configuration and the second one with a single initial particle. This material is a special case of the theory developed in and describes the results of Section 7 therein. We study the behaviour in two time windows, first between time 0 and T and second after a large time when in the Fisher–Wright model the rare mutants succeed, respectively, in the branching random walk the particle population reaches a positive spatial intensity. It is shown that asymptotically as N → ∞ the second phase for both models sets in after time α− 1logN, if N is the size of geographic space and N − 1 the rare mutation rate and α ∈ (0, ∞) depends on the other parameters. We identify the limit dynamics as N → ∞ in both time windows and for both models as a nonlinear Markov dynamic (McKean–Vlasov dynamic), respectively, a corresponding random entrance law from time − ∞ of this dynamic. Finally, we explain that the two processes are just two sides of the very same coin, a fact arising from a new form of duality, in particular the particle model generates the genealogy of the Fisher–Wright diffusions with selection and mutation. We discuss the extension of this duality in relation to a multitype model with more than two types.
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Dawson, D.A., Greven, A. (2012). Multiscale Analysis: Fisher–Wright Diffusions with Rare Mutations and Selection, Logistic Branching System. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_15
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DOI: https://doi.org/10.1007/978-3-642-23811-6_15
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