Date of Award

8-1-2018

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Xu, Dashun

Abstract

In this thesis, we study the population dynamics of predator-prey interactions described by mathematical models with age/stage structures. We first consider fixed development times for predators and prey and develop a stage-structured predator-prey model with Holling type II functional response. The analysis shows that the threshold dynamics holds. That is, the predator-extinction equilibrium is globally stable if the net reproductive number of the predator $\mathcal{R}_0$ is less than $1$, while the predator population persists if $\mathcal{R}_0$ is greater than $1$. Numerical simulations are carried out to demonstrate and extend our theoretical results. A general maturation function for predators is then assumed, and an age-structured predator-prey model with no age structure for prey is formulated. Conditions for the existence and local stabilities of equilibria are obtained. The global stability of the predator-extinction equilibrium is proved by constructing a Lyapunov functional. Finally, we consider a special case of the maturation function discussed before. More specifically, we assume that the development times of predators follow a shifted Gamma distribution and then transfer the previous model into a system of differential-integral equations. We consider the existence and local stabilities of equilibria. Conditions for existence of Hopf bifurcation are given when the shape parameters of Gamma distributions are $1$ and $2$.

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