Title:

Spatial population processes

This thesis is a theoretical study of the effect of migration between colonies, each of which is developing according to a simple stochastic birthdeathimmigration process. In Chapters 2 to 7 I investigate the probability structure of the twocolony process. The Kolmogorov forward differential equation for the population size probabilities is developed and from it expressions are derived for the first and secondorder moments. Exact solutions to this forward equation are obtained for three special cases and a recursive solution is developed in a fourth. Three approximate solutions are developed; (i) by modifying the birth mechanism, (ii) by fitting a bivariate negative binomial distribution, and (iii) by placing an upper bound on the total population size. Iterative solutions are then derived by the use of two different techniques. In the first a power series solution is obtained in terms of a common migration rate. In the second sequences of functions are generated which converge to the required solution. The investigation of the twocolony process concludes with a simulation study and an analysis of the probability of extinction. In Chapter 8 I introduce a 'steppingstone' model in which the population is composed of an infinite number of colonies which may be considered to be situated at the integer points of a single coordinate axis. Migration is allowed between nearestneighbours only. Although the Kolmogorov forward differential equation cannot be solved directly, approximate solutions are developed in an analogous manner to those derived for the twocolony process. First and secondorder moments are obtained and an exact stochastic solution is developed for one special case. If the population has a positive rate of growth and is initially concentrated into a relatively small geographic region, we may expect it to diffuse into the surrounding areas and eventually to take over the entire territory. This expanding population may be envisaged as generating a travelling wave and in Chapter 9 I investigate the velocity of propagation and the form of the wave profile. In Chapter 10 I examine nonnearestneighbour migration models and develop expressions for the mean size of each colony at time t for several appropriate migration distributions. To conclude the thesis I present a spatial model in twodimensions and relate it to data on the spatial distribution of flour beetles in a closed container.
