Inbreeding under a cyclical mating system

General recursion formulae for the coefficient of inbreeding under a cyclical mating system were derived in which one male and one female are selected from each of the n families per generation (population size N = 2 n). Each male is given the family number of his sire in each generation, while his mate comes from another family, varying systematically in different generations. Males of the r-th family in generations 1, 2, 3,..., t' = n-1 within each cycle mate with females from families r+1, r+2, r+3,..., r+t' to produce generations 2, 3, 4,..., t'+1=1, respectively. The change in heterozygosity shows a cyclical pattern of rises and falls, repeating in cycles of n-1 generations. The rate of inbreeding oscillates between <-3% to >6% in different generations within each cycle, irrespective of the population size. The average rate of inbreeding per generation is approximately 1/[4 N-(Log2N+1)], which is the rate for the maximum avoidance of inbreeding. The average inbreeding effective population size is approximately 2 N-2.