#### Date of Award

2019

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### Department

Mathematics

#### First Advisor

Orlando Merino

#### Abstract

This dissertation investigates the local and global behavior of some monotone systems of difference equations. In each study, general results are provided as well as specific examples.

In Manuscript 2 it is shown that locally asymptotically equilibria of planar cooperative or competitive maps have basin of attraction *B* with relatively simple geometry. The boundary of each component of *B* consists of the union of two unordered curves, and the components of *B* are not comparable as sets. The curves are Lipschitz if the map is of class *C*^{1}. Further, if a periodic point is in ∂*B*, then ∂*B* is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

In Manuscript 3 Sufficient conditions are given for planar cooperative maps to have the qualitative global dynamics determined solely on local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of class *C*^{1} on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points, and also the minimal period-two points are ordered in a sense. An application is included.

In Manuscript 4 we give a characterization of monotone discrete systems of equations in terms of associated signature matrix and give some properties of certain invariant surfaces of codimension 1, which often give the boundary of attraction of some fixed points. We present several examples that illustrate our results in the case of k dimensional systems where k ≥ 3.

#### Recommended Citation

Marcotte, James, "GLOBAL DYNAMICS OF DISCRETE MONOTONE MAPS IN THE PLANE AND IN R^{N}" (2019). *Open Access Dissertations.* Paper 1101.

https://digitalcommons.uri.edu/oa_diss/1101