Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiscale analysis: Fisher-Wright diffusions with rare mutations and
  selection, logistic branching system   We study two types of stochastic processes, a mean-field spatial system of
interacting Fisher-Wright diffusions with an inferior and an advantageous type
with rare mutation (inferior to advantageous) and a (mean-field) spatial system
of supercritical branching random walks with an additional deathrate which is
quadratic in the local number of particles. The former describes a standard
two-type population under selection, mutation, the latter models describe a
population under scarce resources causing additional death at high local
population intensity. Geographic space is modelled by $\{1, \cdots, N\}$. 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
\cite{DGsel}.}
  We study the behaviour in two time windows, first between time 0 and $T$ and
secondly 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 the second phase for
both models sets in after time $\alpha^{-1} \log N$, if $N$ is the size of
geographic space and $N^{-1}$ the rare mutation rate and $\alpha \in (0,
\infty)$ depends on the other parameters. We identify the limit dynamics in
both time windows and for both models as a nonlinear Markov dynamic
(McKean-Vlasov dynamic) respectively a corresponding random entrance law from
time $-\infty$ of this dynamic.
  Finally we explain that the two processes are just two sides of the very same
coin, a fact arising from 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 relation to a multitype model with more
than two types.

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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: cs
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: chem-ph