WIAS Preprint No. 2000, (2014)

The stochastic encounter-mating model



Authors

  • Gün, Onur
  • Yilmaz, Atilla

2010 Mathematics Subject Classification

  • 92D25 60J28 60G55

Keywords

  • population dynamics, pair formation, encounter-mating, assortive mating, random mating, panmixia, homogamy, heterogamy, monogamy, mating preferences, mating pattern, contingency table, multiple hypergeometric distribution, simple point processes, Poisson process, Bernoulli process

Abstract

We propose a new model of permanent monogamous pair formation in zoological populations comprised of kge 2 types of females and males, which unifies and generalizes the encounter-mating models of Gimelfarb (1988). In our model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, which depend on the sex and the type of the animals, we analyze the contingency table Q(t) of permanent pair types at any time tge 0. First, we consider definite mating upon encounter and provide a formula for the distribution of Q(t). In particular, at the terminal time T, the so-called mating pattern Q(T) has a multiple hypergeometric distribution. This implies panmixia which means that female and male types are uncorrelated in the expected mating pattern. Next, when the firing times come from Poisson and Bernoulli point processes, we formulate conditions that characterize panmixia. Moreover, when these conditions are satisfied, the underlying parameters of the model can be changed to yield definite mating upon encounter, and our results for the latter case carry over. Finally, when k=2, we fully characterize heterogamy/panmixia/homogamy, i.e., negative/zero/positive correlation of same type females and males in the expected mating pattern. We thereby rigorously prove, strengthen and generalize Gimelfarb's results.

Appeared in

  • Acta Appl. Math., 148 (2017), pp. 71--102.

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