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 [8] 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-1 log N, if N is the size of geographic space and N-1 the rare mutation rate and α Ie{cyrillic, ukrainian} (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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Persistent URL dx.doi.org/10.1007/978-3-642-23811-6_15
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Dawson, D.A, & Greven, A. (Andreas). (2012). Multiscale Analysis: Fisher-Wright Diffusions with Rare Mutations and Selection, Logistic Branching System. doi:10.1007/978-3-642-23811-6_15