An Extension of a Galton-Watson Process to a Two-Sex Density Dependent Model.

Abstract

A two-sex, density dependent, genetic model is formulated as an extension of a multitype Galton-Watson process to a. non-linear case. The formulation presented and partially analyzed in this paper is new in at least three ways to the field of branching processes. First, a new parameterization of the mating system is presented based on, among other things, the Farlie-Morgenstern class of bivariate distribution functions. Second, the viabilities of genotypes are formulated as a function of total population size, which allows for accommodating density dependence into the model to take into account the carrying capacity of an environment. Third, a set on nonlinear difference equations is embedded in the stochastic process by operating with conditional expectation of the random functions of the process at each point in time, given the past, to obtain functions which are viewed as estimates of the sample functions of the process. These equations resemble a model that would arise if one worked within deterministic paradigm. Computer intensive methods are used to partially analyze the model by computing a sample of Monte Carlo realizations of the process, summarizing them statistically, and comparing the results to the predictions based on the embedded deterministic model. The results of two computer experiments are presented, which suggest that numerical solutions of the deterministic model are not always good measures of central tendency for the sample functions of the process, particularly when the deterministic model exhibits chaotic behavior or when the carrying capacity of the environment forces the numbers of some genotypes to be small. The results of these experiments suggest that density dependent formulations in which population size is limited by the carrying capacity of the environment may lead to alternative working paradigms for the study of genetic drift, a subject introduced into evolutionary genetics by two founding fathers Fisher and Wright.