A time series analysis of the population genetics of the self-incompatibility polymorphism. 1. Allele frequency distribution of a population with overlapping generations and variation in plant size

A mathematical theory of the population genetics of the self-incompatibility polymorphism is developed for the case where self-incompatibility is controlled by a single, multiallelic gene, S, expressed gametophytically in the pollen. The theory is based on the probability of a particular allele occurring in a new plant and an approximate equation for this is compared, using computer simulation, with the true value, which depends upon the frequencies of the genotypes of the parental generation. This probability is used to express the frequency of a single established allele, from a population with both overlapping generations and variation in plant size, as a time series. The frequency is shown to follow closely the normal distribution which leads to a simple equation for the variance of the distribution. This variance is approximately equal to the average variance of the frequencies of the different alleles in a single generation and the values closely match the computer simulation results of Brooks et al. (1996). The theory allows alternative scenarios to be investigated easily and the effects of varying the parameter values are discussed. The time to steady state can also be calculated.