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Forward and backward processes in bisexual models with fixed population sizes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

M. Möhle
M. Möhle*
Affiliation:
Johannes Gutenberg-Universität Mainz
*
∗ Postal address: Johannes Gutenberg-Universität Mainz, Fachbereich 17 Mathematik, Saarstraße 21, 55099 Mainz, Germany.
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Abstract

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random.

First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N.

Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.

Keywords

EXCHANGEABLE PROCESSESGALTON–WATSON PROCESSORNSTEIN–UHLENBECK PROCESSWRIGHT-FISHER MODEL

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 60J85: Applications of branching processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 309 - 332
Copyright
Copyright © Applied Probability Trust 1994 

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References

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