Study of singularly perturbed models and its applications in ecology and epidemiology.

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.