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  • 1
    Electronic Resource
    Electronic Resource
    A probabilistic and algebraic treatment of regular inbreeding systems (1985)
    Springer
    Journal of mathematical biology 22 (1985), S. 175-197 
    Details
    ISSN: 1432-1416
    Keywords: Inbreeding ; Regular Mating Systems ; Markov Chains Martingales ; Renewal Events ; Graph of Finitely Presented Semigroups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Summary A probabilistic and algebraic treatment of regular inbreeding systems is presented. Regular inbreeding systems can be thought of as graphs which have certain natural homogeneity properties. Random walks Xn and Yn are introduced on the nodes of the graphs; the event {Xn = Yn} is a renewal event by the homogeneity property. We show that in such regular inbreeding systems the population becomes genetically uniform if and only if the event {Xn = Yn} is recurrent, which happens if ∑ 1/ An diverges, where An is the number of ancestors n generations into the past. We give two counterexamples to show the converse is false in general, but we verify the converse in the case of the graphs of certain finitely presented semigroups.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275714
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Results for generalized regular inbreeding systems (1988)
    Springer
    Journal of mathematical biology 26 (1988), S. 535-550 
    Details
    ISSN: 1432-1416
    Keywords: Inbreeding ; Regular mating systems ; Markov chains ; Renewal events ; Graphs of finitely presented semigroups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A probabilistic and algebraic treatment of regular inbreeding systems was introduced in Arzberger (1985). In that paper it was shown that (1) regular inbreeding systems can be thought of as graphs of certain semigroups, (2) these graphs reflect a certain natural homogeneity property, (3) a sufficient condition for the population to become genetically uniform is GS 1/A r diverges, where A r is the number of ancestors r generations into the past. In this paper, a specific class of inbreeding systems is studied. For this class, the results of the previous paper are extended to generalized regular inbreeding systems in which overlapping generations in the mating scheme are allowed. A new result about the structure of the set of ancestors of two individuals is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276059
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    Paper (German National Licenses)
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