# Conditions under Which the Mean Fertility Is Maximized When a Population Is at a Stable Equilibrium

###### Scott, M.E.; Scott, D.M.; Pollak, E.

#### Genetics 118(4): 713-720

#### 1988

ISSN/ISBN: 0016-6731 PMID: 17246422 Document Number: 244399

It is assumed that there is a population with two alleles at one locus, random mating of adults and selection only involving differential fertilities. By making use of the Kuhn-Tucker theory of optimization under constraints, conditions are derived under which stable equilibrium frequencies x, y and z of the three genotypes are the same as those that maximize the mean fertility of the population. We derive all sets of frequencies of this type for the Hadeler-Liberman symmetric fertility model and all such sets for which at least one genotype is missing for the general model. If the population has frequencies that are initially near those at which there is both a stable equilibrium and maximization of the mean fertility, then the mean fertility .vphi.t at time t is nondecreasing with t as t .fwdarw. .infin. It is found that it is possible for the stable equilibrium maximum points (x, y, z) to be one or two points on a ridge on which the mean fertility .vphi. is maximized or the entire set of points on the ridge. Furthermore, .vphi. may be smaller on this ridge than at another stable equilibrium point at which .vphi. is not even locally maximized.