Some two-dimensional markov processes

Abstract

This thesis is primarily concerned with the mathematical analysis of some Markov processes which take place on a two-dimensional lattice of points in the first two chapters, mathematical models of two biological phenomena are considered, namely the competition for survival between two species, and the effect of an epidemic on a population. These models are obtained by a known method which permits certain random variations in the population sizes. For the model of the competition process, it is found that one of the species almost certainly becomes extinct, and the likelihood of the extinction of a given species is investigated. Also, the expectation of the the time-at which extinction occurs is bounded, irrespective of the initial state, and an estimate is made of the total number of births and deaths that occur before this time. For the epidemic model, it is found that the epidemic almost certainly dies out, and the expectation of the time at which this event first occurs is estimated when the initial population is large. Various questions on the eventual state of the population are also considered. In the third chapter, a class of recurrent two-dimensional random walks in discrete time is considered. A limiting law is found for the probability distribution of first passage times which is identical to the limiting law in the analogous situation for Brownien motion. The method is also applied to certain continuous time random walks and to certain random walks in three dimensions. The last problem considered is the distribution of points at which e simple unsymmetric discrete time random walk makes its first passage through the boundaries of the half and quarter planes. The limiting distribution is found to be a form of either normal distribution or stable distribution of order half.