Population Genetics: Difference Equation Models

The diversity of life is a fundamental empirical fact of nature. Casual observation alone confirms the great variety of species. But diversity also prevails within single species. As in human populations, the individuals of a species vary considerably in their outward traits—size and shape, external markings, fecundity, disease resistance, etc. This is called phenotypic variation, and, since phenotypes are shaped by genes, it reflects an underlying, genetic diversity. The achievements of genetics and molecular biology, as described in Chapter 1, have allowed us to measure and confirm the genotypic variability of populations down to the molecular level. The science of genotypic variation in interbreeding populations is called population genetics. Its goal is understanding how genetic variation changes under the influence of selection, mutation, and the randomness inherent in mating, as one generation succeeds another. Mathematical modeling and analysis play key roles in the subject. Population geneticists combine what is known about the mechanisms of heredity—how DNA carries genetic information, how chromosomes function and give rise to Mendel’s laws, how mutations arise—with hypotheses about mating and selective advantage, to propose mathematical equations for the evolution of genotype frequencies. By comparing the solutions of these equations to field data, they can then test hypotheses and make inferences about genealogy and evolution. This chapter is an introduction to elementary, population genetics models for large populations and simple genotypes. ‘Large’ is a vague term, but it is used here for models formulated by imagining how genotype frequencies evolve in the limit as the population size tends to infinity. The use of this limit is called the “infinite population assumption,” and it is imposed throughout the chapter. Its main consequences is that, in the final models, genotype frequencies become deterministic functions of time, and hence