Study of singularly perturbed models and its applications in ecology and epidemiology.
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Date
2017
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Abstract
In recent years the demand for a more accurate description of real life processes and advances in experimental
techniques have resulted in construction of very complex mathematical models, consisting of tens, hundreds,
if not thousands, of highly coupled di erential equations. The sheer size and complexity of such models
often preclude any robust, theoretical or numerical, analysis of them. Fortunately, often such models describe
phenomena occurring on vastly di erent time or size scales. We focused on complex processes with two time/size
scales described by systems of ordinary di erential equations. In such a case, there is a small parameter that
multiplies one or more derivatives. Using the Tikhonov Theorem, we have been able to understand the asymptotic
behaviour of the solution to some complex epidemiological models. Furthermore, we present analysis based on
the Butuzov theorem, which, for the purpose of the discussed models, was generalized to two dimensional
non-autonomous problems. We applied the developed theory on an ecological model with interactions given by
the mass action law.
Description
Doctoral degree. University of KwaZulu-Natal, Durban.