ISSN:
1432-1416
Keywords:
Inbreeding
;
Regular mating
;
Cubic ancestral growth
;
Random walks
;
Graph of finitely presented semigroups
Source:
Springer Online Journal Archives 1860-2000
Topics:
Biology
,
Mathematics
Notes:
Abstract A specific regular inbreeding system of quadruple half-second cousin mating is considered. A regular inbreeding system can be thought of as a graph which satisfies certain natural homogeneity properties. Random walks X n and Y n are introduced on the nodes of the graph; the event {X n=Yn} is a renewal event by the homogeneity properties. In Arzberger (1985) it is shown that 1) graphs associated with left cancellative semigroups are regular, and 2) for regular systems, the population becomes genetically uniform if and only if the event {X n=Yn} is recurrent. In Arzberger (1986) the system of quadruple half-second cousin mating is associated with a cancellative semigroup, thus the system is regular. In this paper we show that 1) An is asymptotically of the form cn 3, where A n is the number of ancestors n generations into the past, 2) {X n=Yn} is not recurrent (this is shown by associating (X n, Y n) with a random walk in Z 3, 3) P[X 3n =Y 3n ] is asymptotically of the form cn −3/2. Thus, in this example, genetic heterogeneity is maintained, with a cubic rate of growth for An, not by an exponential growth rate, as in all previous examples of regular inbreeding systems in which genetic heterogeneity is maintained.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00276058
