| Summary: | A thesis submitted to the Faculty of Science,
University of the Witwatersrand, Johannesburg,
in fulfilment of the requirements for the degree
of Doctor of Philosophy
February 1976 === This thesis treats systems of ordinary differential equations that
ar*? extracted from ch-_ Kolmogorov forward equations of a class of Markov
processes, known generally as birth and death processes. In particular
we extract and analyze systems of equations which describe the dynamic
behaviour of the second-order moments of the probability distribution
of population governed by birth and death processes. We show that
these systems form an important class of stochastic population models
and conclude that they are superior to those stochastic models derived
by adding a noise term to a deterministic population model. We also
show that these systems are readily used in population control studies,
in which the cost of uncertainty in the population mean size is taken
into account.
The first chapter formulates the univariate linear birth and
death process in its most general form. T i«- prvbo'. i: ity distribution
for the constant parameter case is obtained exactly, which allows one
to state, as special cases, results on the simple birth and death,
Poisson, Pascal, Polya, Palm and Arley processes. Control of a popu=
lation, modelled by the linear birth and death process, is considered
next. Particular attention is paid to system performance indecee
which take into account the cost associated with non-zero variance
and the cost of improving initial estimates of the size of the popula”
tion under control.
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