Date of Award

5-1-2006

Degree Type

Dissertation

University or Center

Clark Atlanta University(CAU)

Degree Name

Ph.D.

Department

Mathematical Sciences

First Advisor

Professor Daniel Bessis

Abstract

This bio-economic model reduces essentially to a set of parametric coupled nonlinear differential equations. The major difficulty stems from the fact that sixteen external parameters appear in various places in the equations. As of today, only numerical methods have been developed to investigate this problem. The practical impossibility to handle such large parametric space obscures the understanding ofthe main features of the solutions of the equations.

A new approach that is analytical and not numerical is proposed. This approach is based on the remark that when one of the sixteen parameters is sent to infmity, the general solution, involving the remaining fifteen parameters can be expressed in terms of elementary functions. This is made possible because, in this limit, besides the time invariance group, an exact invariance scaling group exists. Then the traditional perturbation method of relatively bounded operators can be applied to the exactly solvable previous equations. Furthermore, it seems that the value of the perturbation parameter corresponding to real world fisheries is itself very large. We, therefore, expect excellent results to be achieved with only a first order perturbation.

In order to obtain a global view of the harvesting problem modeled by these types of equations, we have made a dimensional analysis of the sixteen parameters appearing in the equations. We have been able, in all generality, to reduce them to three fundamental dimensionless parameters that carry all the features of the system. This allows making a full analytical study of the fixed points or stationary points of the system. Besides the standard fIXed point, two "catastrophic" new fIXed points have been studied. These are called "catastrophic" because they lead to an exhaustion of the biomass. A global study of the relevance of each stationary fixed point over the range of the dimensionless parameters is presented.

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