A PROBABILISTIC AND ALGEBRAIC TREATMENT OF REGULAR INBREEDING SYSTEMS
Abstract
The study of regular mating systems was begun by Sewall Wright in 1921. He was interested in, among other things, how much inbreeding a population could tolerate without losing all genetic variability. For example, he showed that a regular system of half-sib mating leads asymptotically to a system in which all individuals are clones of one another, even for an infinite population, while half-first cousin mating does not. We show that regular mating systems, properly defined, can be viewed as graphs with certain natural homogeneity properties. Random walks X(,n) and Y(,n) are introduced on the vertices of these graphs that move from individual to parent. The regularity conditions make the event {X(,n) = Y(,n)} a renewal event. Inbreeding is often studied using the probability of identity by descent (denoted by f) for the two genes at a particular genetic locus in a generic individual. In regular graphs, f = 1 iff P{X(,n) = Y(,n) i.o. (VBAR)X(,0) = Y(,0)} = 1; if f = 1, then all individuals are clones of one another. A certain conjecture about regular mating systems is suggested by a discussion of Jacquard: namely, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where A(,n) is the number of distinct ancestors n generations into the past. Thus A(,n) = 2('n) if there is absolutely no inbreeding. We show this conjecture one way, i.e. if (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) we show f = 1. Unfortunately, the converse is false, as is shown by two very different counterexamples. A special case of regular mating systems comes from graphs of left cancellative semigroups. We study the class of finitely presented semigroups on two generators, with one relation, and an identity, and give conditions on the relation which guarantee these semigroups are cancellative. We verify that for these semigroups the conjecture is true in many cases. Techniques we have employed include embedding (X(,n),Y(,n)) in x or using martingale arguments in specific cases. Finally we show that most regular classical mating systems are graphs of semigroups.
Degree
Ph.D.
Subject Area
Mathematics
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