Social system and reproduction of helpers in a cooperatively breeding cichlid fish (Julidochromis ornatus) in Lake Tanganyika: field observations and parentage analyses
Algorithm for the computerized calculation of probability of paternity likelihood was investigated. The probability is calculated by Essen-Möller's formula as W = X/(X+Y) = 1/(1 + Y/X). The X value is also given as X = Hl, m, n/Kl, m, where kl, m and Hl, m, n are the probabilities of mother-child and mother-child-father combinations, respectively. In this study, four functions as F(PQ) = [1-(P not equal to Q)] x p x q, Z(RS) = (1- (R = S)), K(PQ,RS) = 1/2([(R = P) + (R = Q)].s + [(S = P) + (S = Q)].r).F (PQ)/Z(RS) and H(PQ, RS, TU) = 1/4([(R = P) + (R = Q)] [(S = T) + (S = U)] + [(S = P) + (S = Q)] [(R = T) + (R = U)]) x F(PQ).F(TU)/Z(RS) were created, where PQ, RS and TU were the genotypes of mother, child and the alleged father, P, Q, R, S, T and U were their alleles, and p, q, r, s, t and u were the allele frequencies. The equality or inequality in parenthesis was the relation operator which gave -1 or 0 when the expression was true of false, respectively. Then, three formulae as Y = n sigma k = l F([TU]k), Kl, m = 1 sigma i = l m sigma j = l K ([PQ]i, [RS]j) and Hl, m, n = l sigma i = l m sigma j = l n sigma k = l H([PQ]i, [RS]j, [TU]k) were obtained, where [PQ]i, [RS]j and [TU]k were one of the mother's, one of the child's and one of the putative father's genotypes considered from their phenotypes, respectively. Using these formulae, the probability of paternity likelihood could be calculated in every case. These formulae were programmed in BASIC language using a personal computer. Algorithm for the calculation of the probability in the deceased party was also investigated.