## Dissertations

2018

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

Wenzhang Huang

Shangbing Ai

Boris I. Kunin

David Halpern

Rudi Weikard

#### Subject(s)

Biology--Mathematical models., Lotka-Volterra equations.

#### Abstract

This dissertation consists of four chapters. In Chapter 1, I give an introduction of background, models to be investigated, and main goal of this dissertation. In Chapter 2, I study the space homogeneous model. The existence and local stability analysis of nonnegative equilibrium points in three cases, as well as the sufficient and necessary conditions for the existence of the positive interior equilibrium point are geiven in Section2.1. In Section 2.2, I perform a careful analysis to show the global stability of the boundary equilibrium point at which two prey coexist with extinction of predator for case 1. In Section 2.3, I construct a Lyapunov function to prove that the positive interior equilibrium point is globally stable under some assumptions. In Section 2.4, I prove that the boundary equilibrium point at which predator and prey 1 coexist but prey 2 is extinct is globally stable under certain assumptions by using another Lyapunov function for case 3. In Chapter 3, I consider a space variation or space diffusive predator-prey mode described by a reaction-diffusion system. The main puepose is to investigate the existence of the traveling wave solutions. My approach consists of two steps. First I show the existence of nonnegative semi-traveling wave solutions( i.e., nonnegative wave solutions that converge to an unstable equilibrium point as time goes to $-\infty$) for the models by a geometric method. Then I use the Lyapunov method to show that the traveling wave solutions converge to another equilibrium point as time goes to +$\infty$. Chapter 4 provides a couple of conclusion remarks that give brief descriptions of biological implication from the mathematical results obtained in this dissertation. Further research efforts, as a continuation of this dissertation has been addressed in Chapter 4.

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