Some two-dimensional markov processes

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

Total variation approximation for quasi-equilibrium distributions

Quasi-stationary distributions, as discussed by Darroch & Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.Comment: 16 page

Lifetime and reproduction of a marked individual in a two-species competition process

The interest is in a stochastic model for the competition of two species, which was first introduced by Reuter [18] and Iglehart [11], and then analyzed by Ridler-Rowe [19]. The model is related to the two-species autonomous competitive model (Zeeman [24]), where individuals compete either directly or indirectly for a limited food supply and, consequently, birth and death rates depend on the population size of one or both of the species. The aim is to complement the treatment of the model we started in [8,9] by focusing here on probabilistic descriptors that are inherently linked to an individual: its residual lifetime and the number of direct descendants. We present an approximating model based on the maximum size distribution, and we discuss on various models defined in terms of the underlying killing and reproductive strategies. Numerical examples are presented to show the effects of the killing and reproductive strategies on the behavior of an individual, and how the impact of these strategies on the descriptors vanishes in highly competitive ecosystems

On the number of births and deaths during an extinction cycle, and the survival of a certain individual in a competition process

Competition processes, as discussed by Iglehart (1964) [26] and Reuter (1961) [25], have been frequently used in biology to describe the dynamics of population models involving some kind of interaction among various species. Our interest is in the stochastic model of a competition process analyzed by Ridler-Rowe (1978) [23], which is related to an ecosystem of two species. The ecosystem is closed in the sense that no immigration or emigration is supposed to take place. Individuals compete either directly or indirectly for common resources and, consequently, births and deaths depend on the population sizes of one or both of the species. In this paper, we focus on the number of births and deaths during an extinction cycle. Specifically, we discuss an approximation method inspired from the use of the maximum size distribution, which is equally applicable to the survival of a certain individual. We analyze three models defined in terms of the way individuals within each species are selected to die. Our results are illustrated with reference to simulated data