Graduation Semester and Year
2020
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
Christopher Kribs
Abstract
The purpose of this dissertation is to use mathematical models to see how anthrax in the zebra population in Etosha National Park (ENP) interacts with scavenger populations and disease dynamics. First, we study if scavengers can save zebras from anthrax. Then we introduce a disease in the jackal population to see if anthrax in zebras can help propagate rabies in jackals. Finally, the last two models we develop describe the interaction between competing scavengers: jackals and vultures, with exploitative and interference competition. ENP is home to many different animals such as lions, jackals, hyenas, zebras, elephants, etc. Each year grazing animals are infected and die from anthrax caused by the bacteria Bacillus anthracis. This increases the number of carcasses, allowing scavengers such as jackals or vultures to feed off these carcasses. The first model uses a system of nonlinear differential equations to describe the population dynamics of how disease affects the populations of zebras, zebra carcasses, and scavengers. Standard qualitative analysis techniques distinguished outcomes (stable equilibria) using reproduction numbers as threshold quantities. We found that when scavengers feed on anthrax laden carcasses, the scavengers help the zebras reducing spread by orders of magnitude by eliminating potential infection zones for the zebras. We also identify conditions under which the presence of anthrax benefits the scavengers, in terms of death-to-birth ratios for zebras, scavengers, and anthrax. The zebra carcasses provide a location of conspecific interaction between jackals and may be a means of disease transmission among the jackals. We study how a disease in the zebra population may help to propagate a different disease (rabies) in the jackal population since the carcasses are providing a location of interaction between the jackals. We aim to answer the following research question: how do anthrax and rabies affect each other ability to spread? Using standard qualitative analysis, we found that rabies helps anthrax, and a little anthrax helps rabies invade, but a high level of anthrax prevents rabies by reducing the jackal population through its food source. There are multiple species of scavengers in ENP, and zebra carcasses provide a food source for facultative and obligate scavengers such as jackals and vultures, respectively. Since the jackals and vultures are competing for these carcasses, we study the research question: how does the presence of jackals affect the presence of vultures, in the exploitative model. Analysis verified that classical exploitative competition allows vultures to survive only when they are better competitors than jackals. In addition, we found conditions when the vultures are hurt by the presence of anthrax, and a condition under which the competitive interference caused by vultures' aerial quick access to carrion allows them to persist even when jackals are better competitors. In fact, this extended survival can also allow anthrax to persist when it should not.
Keywords
Non-linear Differential Equations, Limit cycles, Hopf Bifurcation, Anthrax-Rabies Interactions
Disciplines
Mathematics | Physical Sciences and Mathematics
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Mackey, Crystal Dawn, "Mathematical Modeling of Scavengers and Zebras on the African Savanna with Disease Dynamics" (2020). Mathematics Dissertations. 205.
https://mavmatrix.uta.edu/math_dissertations/205
Comments
Degree granted by The University of Texas at Arlington