This thesis is concerned initially with a stochastic logistic, (SL), analogue of VERHULST's equation to describe the growth of an isolated population in a limited environment. An essential feature of the SL process is that extinction will occur with probability 1, and so to begin with the analysis is concerned with the distribution of extinction times in terms of the basic parameters, which are the birth and death rates and the upper limit, N, to the integer state variable, j. The analysis then continues with an investigation into the form of the distribution of j at time t, and for small t an approximate analytic result for the related PGF is obtained. For larger t a LAPLACE Transform solution, inverted numerically, is found to be an effective method of evaluating the moments of this distributon, and this may be extended to very large t , if necessary, by considering the process conditional on non - extinction. A diffusion approximation to the SL process is then derived, and this not only extends the numerical methodology developed so far to the large N situation, but also provides a simple overall view of the SL process itself. The model is then modified so as to incorporate age and/or time dependent birth and death parameters, and theoretically it is shown that extinction will occur with probability 1 for a large class of such processes. The way in which the distribution of extinction times, in particular, and the overall behavior of the process, in general, depend on the additional parameters is then investigated by simulation, and in this respect it found useful to consider also the corresponding deterministic processes for which numerical solutions can be found. Finally the model is developed futher so as to describe the sheep populations of the islands of St. Kilda.
|Date of Award||1980|