Mean fixation time estimates in constant size populations

Abstract

We consider a population consisting of N particles each of which some type is ascribed to. All particles die at the integer time moments and produce a random amount of particles of the same type as the parent. Moreover, the population retains its size N and the random vectors defining the number of offsprings of each particle have exchangeable distributions. We obtain several upper bounds for the expectation of the variable equal to the number of the generation when all particles in the population become single-type or almost single-type. Here we fix an arbitrary initial configuration of particles according to types.Keywords: Markov chain, exchangeable distribution, evolution of populations, nearest common ancestor, fixation time, simulation modeling § 1. Motivation of the Choice of the Model.The Problems Under ConsiderationMarkov models are used widely in population dynamics and population evolution analysis based on changes of the DNA. One of the popular classes of models of this kind is given by the haploid models of fixed size [1-4] without mutations. These are models with discrete time of the following form. At the beginning, there are some particles. Then each of them dies producing a random amount of offsprings or only some of the particles die and the rest survive. Anyway, the birth and death distribution is such that the total population size does not change. The term "haploid" means that each offspring has at most one parent. The absence of mutations consists in the fact that the particles produce only particles of the same type.The most popular models of the given class were introduced by Wright and Fisher, Moran, Karlin, and McGregor. In the Wright-Fisher model, the numbers of particles of different types in a generation are the parameters of a polynomial distribution of the birth of offsprings. In the Moran model, only one particle dies replacing one or several others with its offsprings. In the Karlin and McGregor model, the process branches and the distribution is redefined by the condition of fixed total number of offsprings.Some generalization of the above models is proposed in [1, 2]: Each ancestor contributes to offspring production additively and the resulting exchangeable distribution of the parent particles. In principle, all this corresponds to the branching condition: each particle generates its own independent branching process but with the strong condition of fixed total number of offsprings. However, application of the traditional methods of the theory of branching processes is rather difficult here.We study the last population model: each of the N particles, living over the time unit, dies and produces a random amount of offsprings of the same type as itself; their distributions are exchangeable and the sum is N .One of the subjects of study in biology is the fixation time of a population, a random time point when the population starting from several groups of heterogeneous particles (which may be all different) first becomes homogeneous, i.e., all particles become single-type [3]. Her...