On the inclusion of self regulating branching processes in the working paradigm of evolutionary and population genetics

Charles J. Mode, Candace K. Sleeman, Towfique Raj

The principal goal of this methodological paper is to suggest to a general audience in the genetics community that the consideration of recent developments of self regulating branching processes may lead to the possibility of including this class of stochastic processes as part of working paradigm of evolutionary and population genetics. This class of branching processes is self regulating in the sense that an evolving population will grow only to a total population size that can be sustained by the environment. From the mathematical point of view the class processes under consideration belongs to a subfield of probability and statistics sometimes referred to as computational applied probability and stochastic processes. Computer intensive methods based on Monte Carlo simulation procedures have been used to empirically work out the predictions of a formulation by assigning numerical values to some point in the parameter space and computing replications of realizations of the process over thousands of generations of evolution. Statistical methods are then used on such samples of simulated data to produce informative summarizations of the data that provide insights into the evolutionary implications of computer experiments. Briefly, it is also possible to embed deterministic non-linear difference equations in the stochastic process by using a statistical procedure to estimate the sample functions of the process, which has interesting methodological implications as to whether stochastic or deterministic formulations may be applied separately or in combination in the study of evolution. It is recognized that the literature on population genetics contains a substantial number of papers in which Monte Carlo simulation methods have been used. But, this extensive literature is beyond the scope of this paper, which is focused on potential applications of self regulating branching processes in evolutionary and population genetics.