The Stochastic Encounter-Mating Model
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We propose a new model of permanent monogamous pair formation in zoological populations with multiple types of females and males. According to this 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, we analyze the contingency table of permanent pair types in three cases: (i) definite mating upon encounter; (ii) Poisson firing times; and (iii) Bernoulli firing times. In the first case, the contingency table has a multiple hypergeometric distribution which implies panmixia. The other two cases generalize the encounter-mating models of Gimelfarb (Am. Nat. 131(6):865–884, 1988) who gives conditions that he conjectures to be sufficient for panmixia. We formulate adaptations of his conditions and prove that they not only characterize panmixia but also allow us to reduce the model to the first case by changing its underlying parameters. Finally, when there are only two types of females and males, we provide a full characterization of panmixia, homogamy and heterogamy.
KeywordsPopulation dynamics Pair formation Encounter-mating Assortative mating Random mating Panmixia Homogamy Heterogamy Monogamy Mating preferences Mating pattern Contingency table Multiple hypergeometric distribution Simple point process Poisson process Bernoulli process
Mathematics Subject Classification (2000)92D25 60J28 60G55
We are indebted to A. Courtiol for introducing us to Gimelfarb’s work on encounter-mating and for suggesting interesting problems. We also thank F. Rezakhanlou and P. Diaconis for valuable comments and discussions. O. Gün gratefully acknowledges support by DFG SPP Priority Programme 1590 “Probabilistic Structures in Evolution”. A. Yilmaz is supported in part by European Union FP7 Marie Curie Career Integration Grant no. 322078.
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