Date of Award

2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Orlando Merino

Abstract

Discrete dynamical systems are widely used in biological and entomological applications to model interacting populations. The manuscripts included in this thesis present global dynamic results for three different population models. Manuscript 1 presents basic concepts and definitions for general systems of difference equations in order to lay the theoretical foundation for the remaining sections.

Manuscript 2 discusses competitive systems of difference equations with the form

Xn+1 = b1 xn / α1 + xn + c1yn, yn+1 = b2yn / α2 + c2xn + yn n = 0, 1, 2, …,

where the parameters b1, b2 are positive real numbers and α12, c1, c2 and the initial conditions x0, y0 are arbitrary nonnegative numbers. In particular, the special cases when α1 = α2 = 0 and when α1 = 0 and α2 = 0 are investigated. The global behavior of the system in these cases is fully characterized. Global results are also established for general competitive systems of difference equations that have a particular orientation of equilibria and certain local stability characteristics.

In Manuscript 3, the system of different equations

xn+1 = α Xn / 1+ βyn , yn+1 = γ xn yn / xn + δ yn, n = 0, 1, 2, …,

is analyzed, where α, β, γ, δ, x0, y0 are positive real numbers. The system was formulated by P. H. Leslie in 1948 and models a host-parasite type of prey-predator interaction. Manuscript 3 provides the most complete dynamical analysis to date of this classic model. A boundedness and persistence result along with global attractivity results for various parameter regions are established. Numerical evidence of chaotic behavior is also presented for particular solutions of the system.

Finally, Manuscript 4 discusses structured models of difference with the forms:

Yn+1 = M (f1(y(1)n),…, fk(y(k)n))t, n = 0, 1, 2,…, y0 ∈ ℝk+, (I)

and

xn+1 = A xnkl=1 fl(clxn) bl, n = 0, 1, 2, ..., x0 X+, (II)

In (I) and (II), M +kxk, A is a bounded, linear operator on an ordered Banach space X with positive cone X+, and for each l ∈ {1,..., k}, blX+, cl is a positive, bounded linear functional on X, and fl : [0, ∞) --> [0, ∞) is a continuous function with fl(0) = 0. Conditions are established under which there is a oneto- one correspondence between positive equilibrium points (persistence states) of (I) and (II). Under these conditions, and when X = ℝm, the stability type of the zero equilibrium (extinction state) of (I) is shown to be the same as that for (II). Particular attention is given to the case when k = 2. The utility of this analysis is that the dynamics of model (II) on a high dimensionality state space X can be reduced to model (I), where the dimension of the state space is the same as the number k of nonlinearities that appear in (II).

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