Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Yu, Pei

Abstract

This thesis investigates a series of nonlinear predator-prey systems incorporating the Allee effect using differential equations. The main goal is to determine how the Allee effect affects population dynamics. The stability and bifurcations of the systems are studied with a hierarchical parametric analysis, providing insights into the behavioral changes of the population within the systems. In particular, we focus on the study of the number and distribution of limit cycles (oscillating solutions) and the existence of multiple stable states, which cause complex dynamical behaviors. Moreover, including the prey refuge, we examine how our method benefits the low-density animals and affects their population dynamics.

Summary for Lay Audience

Predator-prey relationships are an essential topic in ecology and mathematical biology. In recent years, considerable conservation efforts have been devoted to studying the interactions between low-density prey and their predators. The Allee effect refers to the behavior in which individual fitness is affected at low population sizes or densities. Cooperative behavior, mating difficulties, and various biological mechanisms can lead to this effect. Studies have shown empirical evidence of the Allee effects on marine animals and insects, but the theoretical aspects are ignored. In this thesis, we use mathematical methods to study multiple predator-prey models that incorporate the Allee effect. The long-term population dynamics of predator-prey are studied when their interaction takes in different functional responses (Holling type, ratio type). A comprehensive study on the stability and bifurcation analysis of the systems is presented based on the relevant parameters. Our research shows that the population tends to be extinct under a certain threshold, and the dynamics may exhibit a multistable structure, with a particular focus on the potential emergence of multiple periodic solutions. The study of periodic solutions corresponds to the limit cycles in dynamical systems. Mathematical techniques such as normal form theory and the Abelian integrals are used to study oscillating behaviors. The results provide insights into endangered species conservation and human intervention, offering multiple co-existence possibilities.

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