Date of Award

Spring 2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Denis L. Blackmore

Second Advisor

Roy Goodman

Third Advisor

F. C. Hoppensteadt

Fourth Advisor

Horacio G. Rotstein

Fifth Advisor

Kevin Russell

Abstract

There are many population models in the literature for both continuous and discrete systems. This investigation begins with a general discrete model that subsumes almost all of the discrete population models currently in use. Some results related to the existence of fixed points are proved. Before launching into a mathematical analysis of the primary discrete dynamical model investigated in this dissertation, the basic elements of the model - pioneer and climax species - are described and discussed from an ecological as well as a dynamical systems perspective. An attempt is made to explain why the chosen hierarchical form of the model to explain why the chosen hierarchical form of the model can be expected to follow the real-world evolution of pioneer-climax species. Following the discussion of the discrete dynamical model from the applications viewpoint, an extensive dynamical systems investigation is conducted using analytical and simulation tools. Fixed and periodic points are found and their stability is investigated. Sufficient conditions for the existence and stability of n -cycles are proved and illustrated for several values of n. For eample, the existence of a stable, attracting 3-cycle is proved for a certain range of parameters for an all-pioneer model. It is also observed that the hierarchical system has a predisposition to period-doubling behavior.

Bifurcations of the hierarchical model are studied in considerable depth. It is proved, for example, that the model cannot exhibit a Hopf Bifurcation. However, in a series of theorems, it is shown that the system can exhibit a very rich array of flip (period-doubling) bifurcations, which are of codimension one, two or three. A key to proving this result is that the hierarchical nature of the system makes it essentially equivalent to a sequence of one-dimensional systems when it comes to several properties of the dynamics. This hierarchical principle is then used to prove chaos for the system in the limit of a period-doubling cascade, and also in terms of shift map behavior on an invariant two-component Cantor set for systems containing a climax component. Bifurcation diagrams and Lyapunov diagrams are computed to further illustrate the chaotic dynamics. Finally, the concept of a 3-dimensional horseshoe type map is also used to prove the existence of chaos in an approximate graphical manner.

Included in

Mathematics Commons

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