Random Mating Model Research Articles

The necessity of the poisson distribution for the equivalence of some random mating models

It is shown for a finite, monoecious population that if the two genetically different random mating models, random mating by complete pairing of individuals or random mating subject to monogamy are equivalent to random mating by independent trials, then necessarily in both these models the family size distribution per mating couple is Poisson. Furthermore the mean number of offspring per mating couple is the only unspecified parameter of the family size distribution. In conjunction with the results on random mating by Watterson [ Theor. Popul. Biol. 1, 233–250 (1970)] this now establishes the Poisson family size distribution as necessary and sufficient for equivalence between random mating by complete pairing or subject to monogamy with the multinomial sampling independent trials model.

Random mating and random union of gametes models for finite dioecious populations

This paper compares a random mating model for independent trials with two random union of gametes models in the case when the population is finite and dioecious. The first random union of gametes model is examined as a two-locus dioecious model with unequal recombination values in each sex, which is a generalization of previous work. The second model can be of some biological use under certain circumstances, and it is easier to analyse (when appropriate) than the first. It is examined with and without mutation to obtain both old and new results about the fixation probabilities (in the multilocus case) and the related rates in the autosomal locus case.

A mathematical theory of age structure in sexual populations: random mating and monogamous marriage models

Equations governing the size and age structure of a large population composed of two sexes are developed. It is assumed that the “significant” environment of the population —that which governs its mortality, nuptiality, and natality patterns and rates— remains constant. Two different social arrangements respecting reproduction are considered. It is assumed first that reproduction is by random mating; it is then assumed that reproduction is via strict monogamous marriage. The equations describing these two cases are coupled and nonlinear. In the case of the random mating model, the problem can be reduced to a single nonlinear integral equation. In the case of the monogamous marriage model, a system of partial differential-integral equations results. Applications of these equations to a population in stable growth and in transient growth are discussed.

On the equivalence of random mating and random union of gametes models in finite, monoecious populations

The paper shows that under certain conditions, random mating models for monoecious diploids are equivalent to random union of gametes models, in the case when the population is finite. Two examples are given to illustrate the result, in connection with gametic selection, and for recombination of two loci. It is also shown that under certain conditions three different random mating models can be mathematically equivalent, and likewise three different random union of gametes models can be equivalent.

Linkage and selection: Two locus symmetric viability model

Since Fisher (1930) conjectured on how linkage between two loci could have been reduced, many models have been constructed and analyzed in order to establish more rigorously how natural selection may have produced tightly linked clusters of loci and the evolution of supergenes. The usual approach is to set up a deterministic random mating model with viabilities and recombination. The polymorphic equilibria are determined and conditions on the selection parameters and recombination fraction ascertained such that these are stable. At this equilibrium, a crossover reducing mechanism is then supposed to occur in one of the chromosomes and the progress of this chromosome is examined as a function of the altered recombination fraction and the equilibrium frequencies. Kimura (1956) used the above approach for a special symmetric viability set up. Bodmer and Parsons (1962) treated the same problem with more general, but still symmetric, viabilities. Nei (1967) introduced a model allowing a third locus, independent of the two under investigation to determine the extent of cross-over. Turner (1967) has given a somewhat heuristic treatment of the problem with general viabilities. These authors all found that a population in equilibrium under linkage and selection will be subject to selection for a decrease in recombination fraction. Most of the models analyzed so far can be realized as particular cases of what has become known as the symmetric viability model involving two loci with two alleles at each locus. A detailed discussion of this model and some of its properties was made by Bodmer and Felsenstein (1967). The four chromosomes AB, Ab, aB, and ab occur with frequencies x1 , x2, xs , and x4 , respectively, and the fitnesses of the genotypes are as given in Table I.

Linkage and selection: theoretical analysis of the deterministic two locus random mating model.

Linkage and selection: theoretical analysis of the deterministic two locus random mating model.