On the classification of regular systems of inbreeding

Regular systems of inbreeding are defined as those with discrete, nonoverlapping generations and with the same number of individuals and mating pattern in every generation. Given the number of individuals in a generation, there are many possible regular mating systems. A notion of when two such mating systems are equivalent is introduced, and several necessary conditions are given for such an equivalence. The use of these conditions is illustrated for N = 2, 3, 4 and 5 individuals, and a complete enumeration has been found for these cases: the numbers of inequivalent mating systems are 1, 5, 57 and 858, respectively. The maximal eigenvalue of the matrix q that specifies the recursion relations satisfied by the probabilities of identity have also been found for these cases. For N = 3 and 4 (and 2 trivially), circular mating gives the slowest rate of approach to genetic uniformity of those systems that do evolve to uniformity, but for N = 5 there are two other mating systems that have a slower rate of convergence, and for N = 6 partial results show that there are many such examples.