Fluctuation in Population Dynamics

A dynamics for a large population of randomly mating individuals is formulated as a Markov process on a large product space which involves a pairwise interaction between components. Each individual in the population possesses a character (the state of individual). An individual undergoes Markovian change of its character during its life time. Two individuals may make random mating after which they die, leaving two children with new characters. Each individual is a unique member of a lineage at each time. We are interested in the number of lineages which, comprise individuals possessing the character x_i at time t_i for i = 1,…,m, where x_l,…,x_m and t_l,…,t_m are given in advance. If the initial distribution is i.i.d. and the size n of the population is very large, the proportion of this number to n is approximately computed by means of a joint distribution of a nonlinear Markov process (for a single individual in a infinite population). We shall study the law of the error in this approximation and establish a limit theorem about it.