Poisson karşılaşma-eşleşme modelinin difüzyon limiti

Encounter-mating models are used to represent mating scenarios in finite zoological populations consisting of males and females, partitioned into different types. Using them, one is able to draw conclusions about the mating process, and ultimately, the resulting pattern of permanent male-female pairs formed between different types, called the mating pattern. Recently, Gün and Yılmaz (Acta Applicandae Mathematicae 148, 1 (2017), 71-102) developed the so-called stochastic encounter-mating (SEM) model to generalize and unify encounter-mating models previously presented in the literature. One particular scenario in the SEM is when the mating clocks of the individuals, within the population being modeled, are Poisson. In this scenario, the process that keeps record of the pair-type formation becomes a continuous-time Markov chain. Gün and Yılmaz show that, despite some inconsistencies (i.e. up to error terms), this process can be incorporated into a framework of density-dependent Markov processes, and derive two separate laws of large numbers, one for the mating process, and another for the mating pattern (Advances in Applied Probability 49, 4 (2017) 1201-1229). In this thesis, we discuss the central limit theorem (CLT) counterparts of these two results. We present two proofs of the CLT for the pair-type process, and provide evidence gathered from MATLAB simulations, to argue the existence of a CLT for the mating pattern.