#### Date of Award

Spring 2019

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics, Statistics and Computer Science

#### Program

Computational Sciences

#### First Advisor

Merrill, Stephen J.

#### Second Advisor

Factor, Kim

#### Third Advisor

Sanders, Rebecca

#### Abstract

Merrill (2010) described bifurcation in a Markov chain by examining the eigenvalues of the associated probability matrix. The bifurcation point is that point where the dynamics of the system’s structure changes. He recognized a change in the dynamics of a sample path in a Markov chain when the nature of its eigenvalues changes. We built upon this work and found that not all changes in Markov chain dynamics are accompanied by change in the nature of the eigenvalues. And we introduce other measures that will recognize a change in dynamics. This was applied to solve the problem of evaluating the effectiveness of an ecological corridor. This was also used as a measure to examine bifurcation in metapopulation dynamics.Ovaskeinan and Hanski (2003) gave four definitions of patch value (contribution of a patch to metapopulation dynamics and persistence). One of them denotes a patch value as W_i, the contribution of patch i to colonization in the patch network. It is the left leading eigenvector of matrix B whose entries, b_ij=(p_j c_ij)/(∑▒〖p_k c_ik 〗). This is a Markov chain, where p_i is the probability that patch i is occupied, c_ij is the contribution that occupied patch j makes to the colonization rate of empty patch i. This matrix is in the family of coperiodic cospectral, which will be introduced in this dissertation. Therefore, it could be an effective tool in studying metapopulation dynamics. The goal is to evaluate the effectiveness of corridor introduction on species persistence, richness, and ecosystem dynamics. We focused our application on available data from the Osceola-Ocala black bears in Florida.