Ordinary Differential Equations and Applied Dynamics Commons^™

Population Dynamics And Bifurcations In Predator-Prey Systems With Allee Effect, Yanni Zeng

Electronic Thesis and Dissertation Repository

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 …

Study Of Behaviour Change And Impact On Infectious Disease Dynamics By Mathematical Models, Tianyu Cheng

Electronic Thesis and Dissertation Repository

This thesis uses mathematical models to study human behaviour changes' effects on infectious disease transmission dynamics. It centers on two main topics. The first concerns how behaviour response evolves during epidemics and the effects of adaptive precaution behaviour on epidemics. The second topic is how to build general framework models incorporating human behaviour response in epidemiological modelling.

In the first project, based on the fact that a fraction of the epidemiologically susceptible population is actually susceptible due to precautions, we present a novel perspective on understanding the infection force, incorporating human protection behaviours. This view explains many existing infection force …

Data-Driven Exploration Of Coarse-Grained Equations: Harnessing Machine Learning, Elham Kianiharchegani

Electronic Thesis and Dissertation Repository

In scientific research, understanding and modeling physical systems often involves working with complex equations called Partial Differential Equations (PDEs). These equations are essential for describing the relationships between variables and their derivatives, allowing us to analyze a wide range of phenomena, from fluid dynamics to quantum mechanics. Traditionally, the discovery of PDEs relied on mathematical derivations and expert knowledge. However, the advent of data-driven approaches and machine learning (ML) techniques has transformed this process. By harnessing ML techniques and data analysis methods, data-driven approaches have revolutionized the task of uncovering complex equations that describe physical systems. The primary goal in …

Complex-Valued Approach To Kuramoto-Like Oscillators, Jacqueline Bao Ngoc Doan

Electronic Thesis and Dissertation Repository

The Kuramoto Model (KM) is a nonlinear model widely used to model synchrony in a network of oscillators – from the synchrony of the flashing fireflies to the hand clapping in an auditorium. Recently, a modification of the KM (complex-valued KM) was introduced with an analytical solution expressed in terms of a matrix exponential, and consequentially, its eigensystem. Remarkably, the analytical KM and the original KM bear significant similarities, even with phase lag introduced, despite being determined by distinct systems. We found that this approach gives a geometric perspective of synchronization phenomena in terms of complex eigenmodes, which in turn …

On The Spatial Modelling Of Biological Invasions, Tedi Ramaj

Electronic Thesis and Dissertation Repository

We investigate problems of biological spatial invasion through the use of spatial modelling. We begin by examining the spread of an invasive weed plant species through a forest by developing a system of partial differential equations (PDEs) involving an invasive weed and a competing native plant species. We find that extinction of the native plant species may be achieved by increasing the carrying capacity of the forest as well as the competition coefficient between the species. We also find that the boundary conditions exert long-term control on the biomass of the invasive weed and hence should be considered when implementing …

Lecture Note On Delay Differential Equation, Wenfeng Liu

Undergraduate Student Research Internships Conference

Delay differential equation is an important field in applied mathematics since it concerns more situations than the ordinary differential equation. Moreover, it makes the equations more applicable to the object's movement in real life.

My project is the lecture note on the delay differential equation provides a basic introduction to the delay differential equation, its application in real life, improving the ordinary differential equation, the primary method and definition for solving the delay differential equation and the use of the way in the ordinary differential equation to estimate the periodic solution to the delay differential equation.

Coevolution Of Hosts And Pathogens In The Presence Of Multiple Types Of Hosts, Evan J. Mitchell

Electronic Thesis and Dissertation Repository

How will hosts and pathogens coevolve in response to multiple types of hosts? I study this question from three different perspectives. First, I model a scenario in which hosts are categorized as female or male. Hosts invest resources in maintaining their immune system at a cost to their reproductive success, while pathogens face a trade-off between transmission and duration of infection. Importantly, female hosts are also able to vertically transmit an infection to their newborn offspring. The main result is that as the rate of vertical transmission increases, female hosts will have a greater incentive to pay the cost to …

Population And Evolution Dynamics In Predator-Prey Systems With Anti-Predation Responses, Yang Wang

Electronic Thesis and Dissertation Repository

This thesis studies the impact of anti-predation strategy on the population dynamics of predator-prey interactions. This work includes three research projects.

In the first project, we study a system of delay differential equations by considering both benefit and cost of anti-predation response, as well as a time delay in the transfer of biomass from the prey to the predator after predation. We reveal some insights on how the anti-predation response level and the biomass transfer delay jointly affect the population dynamics; we also show how the nonlinearity in the predation term mediated by the fear effect affects the long term …

Mathematical Modelling Of Prophage Dynamics, Tyler Pattenden

Electronic Thesis and Dissertation Repository

We use mathematical models to study prophages, viral genetic sequences carried by bacterial genomes. In this work, we first examine the role that plasmid prophage play in the survival of de novo beneficial mutations for the associated temperate bacteriophage. Through the use of a life-history model, we determine that mutations first occurring in a plasmid prophage are far more likely to survive drift than those first occurring in a free phage. We then analyse the equilibria and stability of a system of ordinary differential equations that describe temperate phage-host dynamics. We elucidate conditions on dimensionless parameters to determine a parameter …

Abelian Integral Method And Its Application, Xianbo Sun

Electronic Thesis and Dissertation Repository

Oscillation is a common natural phenomenon in real world problems. The most efficient mathematical models to describe these cyclic phenomena are based on dynamical systems. Exploring the periodic solutions is an important task in theoretical and practical studies of dynamical systems.

Abelian integral is an integral of a polynomial differential 1-form over the real ovals of a polynomial Hamiltonian, which is a basic tool in complex algebraic geometry. In dynamical system theory, it is generalized to be a continuous function as a tool to study the periodic solutions in planar dynamical systems. The zeros of Abelian integral and their distributions …

Phage-Bacteria Interaction And Prophage Sequences In Bacterial Genomes, Amjad Khan

Electronic Thesis and Dissertation Repository

In this investigation, we examined the interaction of phages and bacteria in bacterial biofilm colonies, the evolution of prophages (viral genetic material inserted into the bacterial genome) and their genetic repertoire. To study the synergistic effects of lytic phages and antibiotics on bacterial biofilm colonies, we have developed a mathematical model of ordinary differential equations (ODEs). We have also presented a mathematical model consisting of a partial differential equation (PDEs), to study evolutionary forces acting on prophages. We fitted the PDE model to three publicly available databases and were able to show that induction is the prominent fate of intact …

Modelling Non-Linear Functional Responses In Competitive Biological Systems., Nickolas Goncharenko

Western Research Forum

One of the most versatile and well understood models in mathematical biology is the Competitive Lotka Volterra (CLV) model, which describes the behaviour of any number of exclusively competitive species (that is each species competes directly with every other species). Despite it's success in describing many phenomenon in biology, chemistry and physics the CLV model cannot describe any non-linear environmental effects (including resource limitation and immune response of a host due to infection). The reason for this is the theory monotone dynamical systems, which was codeveloped with the CLV model, does not apply when this non-linear effect is introduced. For …

Bifurcation Analysis Of Two Biological Systems: A Tritrophic Food Chain Model And An Oscillating Networks Model, Xiangyu Wang

Electronic Thesis and Dissertation Repository

In this thesis, we apply bifurcation theory to study two biological systems. Main attention is focused on complex dynamical behaviors such as stability and bifurcation of limit cycles. Hopf bifurcation is particularly considered to show bistable or even tristable phenomenon which may occur in biological systems. Recurrence is also investigated to show that such complex behavior is common in biological systems.

First we consider a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. Main attention is focused on the sta- bility and bifurcation of equilibria when the prey has a …

Ecology And Evolution Of Dispersal In Metapopulations, Jingjing Xu

Electronic Thesis and Dissertation Repository

Dispersal plays a key role in the persistence of metapopulations, as the balance between local extinction and colonization is affected by dispersal. Herein, I present three pieces of work related to dispersal. The first two are devoted to the ecological aspect of delayed dispersal in metapopulations. The first one focuses on how dispersal may disrupt the social structure on patches from which dispersers depart. Examinations of bifurcation diagrams of the dynamical system show a metapopulation will, in general, be either in the state of global extinction or persistence, and dispersal only has a limited effect on metapopulation persistence. The second …

Real Solution Of Dae And Pdae System, Zahra Mohammadi, Greg Reid

Western Research Forum

General systems of differential equations don't have restrictions on the number or type of equations. For example, they can be over or under-determined, and also contain algebraic constraints (e.g. algebraic equations such as in Differential-Algebraic equations (DAE) and Partial differential algebraic equations (PDAE). Increasingly such general systems arise from mathematical modeling of engineering and science problems such as in multibody mechanics, electrical circuit design, optimal control, chemical kinetics and chemical control systems. In most applications, only real solutions are of interest, rather than complex-valued solutions. Much progress has been made in exact differential elimination methods, which enable characterization of all …

Modelling Walleye Population And Its Cannibalism Effect, Quan Zhou

Electronic Thesis and Dissertation Repository

Walleye is a very common recreational fish in Canada with a strong cannibalism tendency, such that walleyes with larger sizes will consume their smaller counterparts when food sources are limited or a surplus of adults is present. Cannibalism may be a factor promoting population oscillation. As fish reach a certain age or biological stage (i.e. biological maturity), the number of fish achieving that stage is known as fish recruitment. The objective of this thesis is to model the walleye population with its recruitment and cannibalism effect. A matrix population model has been introduced to characterize the walleye population into three …

On Honey Bee Colony Dynamics And Disease Transmission, Matthew I. Betti

Electronic Thesis and Dissertation Repository

The work herein falls under the umbrella of mathematical modeling of disease transmission. The majority of this document focuses on the extent to which infection undermines the strength of a honey bee colony. These studies extend from simple mass-action ordinary differential equations models, to continuous age-structured partial differential equation models and finally a detailed agent-based model which accounts for vector transmission of infection between bees as well as a host of other influences and stressors on honey bee colony dynamics. These models offer a series of predictions relevant to the fate of honey bee colonies in the presence of disease …

The Survival Probability Of Beneficial De Novo Mutations In Budding Viruses, With An Emphasis On Influenza A Viral Dynamics, Jennifer Ns Reid

Electronic Thesis and Dissertation Repository

A deterministic model is developed of the within-host dynamics of a budding virus, and coupled with a detailed life-history model using a branching process approach to follow the fate of de novo beneficial mutations affecting five life-history traits: clearance, attachment, eclipse, budding, and cell death. Although the model can be generalized for any given budding virus, our work was done with a major emphasis on the early stages of infection with influenza A virus in human populations. The branching process was then interleaved with a stochastic process describing the disease transmission of this virus. These techniques allowed us to predict …

Study Of Infectious Diseases By Mathematical Models: Predictions And Controls, Sm Ashrafur Rahman

Electronic Thesis and Dissertation Repository

The aim of this thesis is to understand the spread, persistence and prevention mechanisms of infectious diseases by mathematical models. Microorganisms that rapidly evolve pose a constant threat to public health. Proper understanding of the transmission machinery of these existing and new pathogens may facilitate devising prevention tools. Prevention tools against transmissions, including vaccines and drugs, are evolving at a similar pace. Efficient implementation of these new tools is a fundamental issue of public health. We primarily focus on this issue and explore some theoretical frameworks.

Pre-exposure prophylaxis (PrEP) is considered one of the promising interventions against HIV infection as …

Bifurcation Of Limit Cycles In Smooth And Non-Smooth Dynamical Systems With Normal Form Computation, Yun Tian

Electronic Thesis and Dissertation Repository

This thesis contains two parts. In the first part, we investigate bifurcation of limit cycles around a singular point in planar cubic systems and quadratic switching systems. For planar cubic systems, we study cubic perturbations of a quadratic Hamiltonian system and obtain 10 small-amplitude limit cycles bifurcating from an elementary center, for which up to 5th-order Melnikov functions are used. Moreover, we prove the existence of 12 small-amplitude limit cycles around a singular point in a cubic system by computing focus values. For quadratic switching system, we develop a recursive algorithm for computing Lyapunov constants. With this efficient algorithm, we …

Understanding Recurrent Disease: A Dynamical Systems Approach, Wenjing Zhang

Electronic Thesis and Dissertation Repository

Recurrent disease, characterized by repeated alternations between acute relapse and long re- mission, can be a feature of both common diseases, like ear infections, and serious chronic diseases, such as HIV infection or multiple sclerosis. Due to their poorly understood etiology and the resultant challenge for medical treatment and patient management, recurrent diseases attract much attention in clinical research and biomathematics. Previous studies of recurrence by biomathematicians mainly focus on in-host models and generate recurrent patterns by in- corporating forcing functions or stochastic elements. In this study, we investigate deterministic in-host models through the qualitative analysis of dynamical systems, to …

Study Of Virus Dynamics By Mathematical Models, Xiulan Lai

Electronic Thesis and Dissertation Repository

This thesis studies virus dynamics within host by mathematical models, and topics discussed include viral release strategies, viral spreading mechanism, and interaction of virus with the immune system.

Firstly, we propose a delay differential equation model with distributed delay to investigate the evolutionary competition between budding and lytic viral release strategies. We find that when antibody is not established, the dynamics of competition depends on the respective basic reproduction numbers of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for …

On Evolution Dynamics And Strategies In Some Host-Parasite Models, Liman Dai

Electronic Thesis and Dissertation Repository

In this thesis, we use mathematical models to study the problems about the evolution of hosts and parasites. Firstly, we study a within-host age-structured model with mutation and back mutation which is in the form of partial differential equations with double-infections by two strains of viruses. For the case when the production rates of viruses are gamma distributions, the PDE model can be transferred into an ODE one. Then, we analyze our model in two cases: one is without mutation, and the other is with mutation. In the first case, we prove that the two strains of viruses without mutation …

Modeling Leafhopper Populations And Their Role In Transmitting Plant Diseases., Ji Ruan

Electronic Thesis and Dissertation Repository

This M.Sc. thesis focuses on the interactions between crops and leafhoppers.

Firstly, a general delay differential equations system is proposed, based on the infection age structure, to investigate disease dynamics when disease latencies are considered. To further the understanding on the subject, a specific model is then introduced. The basic reproduction numbers $\cR_0$ and $\cR_1$ are identified and their threshold properties are discussed. When $\cR_0 < 1$, the insect-free equilibrium is globally asymptotically stable. When $\cR_0 > 1$ and $\cR_1 < 1$, the disease-free equilibrium exists and is locally asymptotically stable. When $\cR_1>1$, the disease will persist.

Secondly, we derive another general delay differential equations system to examine how different life stages of leafhoppers affect crops. The basic reproduction numbers $\cR_0$ is determined: when …

Bifurcations And Stability In Models Of Infectious Diseases, Bernard S. Chan

Electronic Thesis and Dissertation Repository

This work is concerned with bifurcation and stability in models related to various aspects of infections diseases.

First, we study the dynamics of a mathematical model on primary and secondary cytotoxic T-lymphocyte responses to viral infections by Wodarz et al. This model has three equilibria and the stability criteria of them are discussed. We analytically show that periodic solutions may arise from the third equilibrium via Hopf bifurcation. Numerical simulations of the model agree with the theoretical results. These dynamical behaviours occur within biologically realistic parameter range.

After studying the single-strain model, we analyze the bifurcation dynamics of an …