Mathematical Properties of the Agent-Based Model of Extinction–Recolonization for Population Genetics

The individual-based model describes the dynamics of genetic diversity of a population scattered on a spatial continuum in the case of a finite number of individuals. During extinction events in a certain area, a portion of the population dies, after which new individuals with the genotype of the parent are born during a recolonization event. In this paper we examine the model, as well as its modification, and derive properties related to population parameters. The study demonstrates that the lifespan of individuals follows an exponential distribution, allele probabilities remain constant over time, and the average heterozygosity, constrained by the number of individuals during extinction and recolonization, equals a similar quantity in the Moran model. The joint distribution of alleles is generalized for populations continuously scattered in space. Joint allele distribution and heterozygosity are computed through simulations.