Matching Items (20)
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- All Subjects: Applied Mathematics
- Creators: Gardner, Carl
Description
A general continuum model for simulating the flow of ions in the salt baths that surround and fill excitable neurons is developed and presented. The ion densities and electric potential are computed using the drift-diffusion equations. In addition, a detailed model is given for handling the electrical dynamics on interior membrane boundaries, including a model for ion channels in the membranes that facilitate the transfer of ions in and out of cells. The model is applied to the triad synapse found in the outer plexiform layer of the retina in most species. Experimental evidence suggests the existence of a negative feedback pathway between horizontal cells and cone photoreceptors that modulates the flow of calcium ions into the synaptic terminals of cones. However, the underlying mechanism for this feedback is controversial and there are currently three competing hypotheses: the ephaptic hypothesis, the pH hypothesis and the GABA hypothesis. The goal of this work is to test some features of the ephaptic hypothesis using detailed simulations that employ rigorous numerical methods. The model is first applied in a simple rectangular geometry to demonstrate the effects of feedback for different extracellular gap widths. The model is then applied to a more complex and realistic geometry to demonstrate the existence of strictly electrical feedback, as predicted by the ephaptic hypothesis. Lastly, the effects of electrical feedback in regards to the behavior of the bipolar cell membrane potential is explored. Figures for the ion densities and electric potential are presented to verify key features of the model. The computed steady state IV curves for several cases are presented, which can be compared to experimental data. The results provide convincing evidence in favor of the ephaptic hypothesis since the existence of feedback that is strictly electrical in nature is shown, without any dependence on pH effects or chemical transmitters.
ContributorsJones, Jeremiah (Author) / Gardner, Carl (Committee member) / Baer, Steven (Committee member) / Crook, Sharon (Committee member) / Kostelich, Eric (Committee member) / Ringhofer, Christian (Committee member) / Arizona State University (Publisher)
Created2013
Description
Recent experimental and mathematical work has shown the interdependence of the rod and cone photoreceptors with the retinal pigment epithelium in maintaining sight. Accelerated intake of glucose into the cones via the theoredoxin-like rod-derived cone viability factor (RdCVF) is needed as aerobic glycolysis is the primary source of energy production. Reactive oxidative species (ROS) result from the rod and cone metabolism and recent experimental work has shown that the long form of RdCVF (RdCVFL) helps mitigate the negative effects of ROS. In this work I investigate the role of RdCVFL in maintaining the health of the photoreceptors. The results of this mathematical model show the necessity of RdCVFL and also demonstrate additional stable modes that are present in this system. The sensitivity analysis shows the importance of glucose uptake, nutrient levels, and ROS mitigation in maintaining rod and cone health in light-damaged mouse models. Together, these suggest areas on which to focus treatment in order to prolong the photoreceptors, especially in situations where ROS is a contributing factor to their death such as retinitis pigmentosa (RP). A potential treatment with RdCVFL and its effects has never been studied in mathematical models. In this work, I examine an optimal control with the treatment of RdCVFL and mathematically illustrate the potential that this treatment might have for treating degenerative retinal diseases such as RP. Further, I examine optimal controls with the treatment of both RdCVF and RdCVFL in order to mathematically understand the potential that a dual treatment might have for treating degenerative retinal diseases such as RP. The RdCVFL control terms are nonlinear for biological accuracy but this results in the standard general theorems for existence of optimal controls failing to apply. I then linearize these models to have proof of existence of an optimal control. Both nonlinear and linearized control results are compared and reveal similarly substantial savings rates for rods and cones.
ContributorsWifvat, Kathryn (Author) / Camacho, Erika (Thesis advisor) / Wirkus, Stephen (Thesis advisor) / Gardner, Carl (Committee member) / Fricks, John (Committee member) / Kawski, Matthias (Committee member) / Arizona State University (Publisher)
Created2022
Description
\begin{abstract}The human immunodeficiency virus (HIV) pandemic, which causes the syndrome of opportunistic infections that characterize the late stage HIV disease, known as the acquired immunodeficiency syndrome (AIDS), remains a major public health challenge to many parts of the world. This dissertation contributes in providing deeper qualitative insights into the transmission dynamics and control of the HIV/AIDS disease in Men who have Sex with Men (MSM) community. A new mathematical model (which is relatively basic), which incorporates some of the pertinent aspects of HIV epidemiology and immunology and fitted using the yearly new case data of the MSM population from the State of Arizona, was designed and used to assess the population-level impact of awareness of HIV infection status and condom-based intervention, on the transmission dynamics and control of HIV/AIDS in an MSM community. Conditions for the existence and asymptotic stability of the various equilibria ofthe model were derived. The numerical simulations showed that the prospects for the effective control and/or elimination of HIV/AIDS in the MSM community in the United States are very promising using a condom-based intervention, provided the condom efficacy is high and the compliance is moderate enough. The model was extended in Chapter 3 to account for the effect of risk-structure, staged-progression property of HIV disease, and the use of pre-exposure prophylaxis (PrEP) on the spread and control of the disease. The model was shown to undergo a PrEP-induced \textit{backward bifurcation} when the associated control reproduction number is less than one. It was shown that when the compliance in PrEP usage is $50%(80%)$ then about $19.1%(34.2%)$ of the yearly new HIV/AIDS cases recorded at the peak will have been prevented, in comparison to the worst-case scenario where PrEP-based intervention is not implemented in the MSM community. It was also shown that the HIV pandemic elimination is possible from the MSM community even for the scenario when the effective contact rate is increased by 5-fold from its baseline value, if low-risk individuals take at least 15 years before they change their risky behavior and transition to the high-risk group (regardless of the value of the transition rate from high-risk to low-risk susceptible population).
ContributorsTollett, Queen Wiggs (Author) / Gumel, Abba (Thesis advisor) / Crook, Sharon (Committee member) / Fricks, John (Committee member) / Gardner, Carl (Committee member) / Nagy, John (Committee member) / Arizona State University (Publisher)
Created2023
Description
A description of numerical and analytical work pertaining to models that describe the growth and progression of glioblastoma multiforme (GBM), an aggressive form of primary brain cancer. Two reaction-diffusion models are used: the Fisher-Kolmogorov-Petrovsky-Piskunov equation and a 2-population model that divides the tumor into actively proliferating and quiescent (or necrotic) cells. The numerical portion of this work (chapter 2) focuses on simulating GBM expansion in patients undergoing treatment for recurrence of tumor following initial surgery. The models are simulated on 3-dimensional brain geometries derived from magnetic resonance imaging (MRI) scans provided by the Barrow Neurological Institute. The study consists of 17 clinical time intervals across 10 patients that have been followed in detail, each of whom shows significant progression of tumor over a period of 1 to 3 months on sequential follow up scans. A Taguchi sampling design is implemented to estimate the variability of the predicted tumors to using 144 different choices of model parameters. In 9 cases, model parameters can be identified such that the simulated tumor contains at least 40 percent of the volume of the observed tumor. In the analytical portion of the paper (chapters 3 and 4), a positively invariant region for our 2-population model is identified. Then, a rigorous derivation of the critical patch size associated with the model is performed. The critical patch (KISS) size is the minimum habitat size needed for a population to survive in a region. Habitats larger than the critical patch size allow a population to persist, while smaller habitats lead to extinction. The critical patch size of the 2-population model is consistent with that of the Fisher-Kolmogorov-Petrovsky-Piskunov equation, one of the first reaction-diffusion models proposed for GBM. The critical patch size may indicate that GBM tumors have a minimum size depending on the location in the brain. A theoretical relationship between the size of a GBM tumor at steady-state and its maximum cell density is also derived, which has potential applications for patient-specific parameter estimation based on magnetic resonance imaging data.
ContributorsHarris, Duane C. (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric J. (Thesis advisor) / Preul, Mark C. (Committee member) / Crook, Sharon (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2023
Description
The most advanced social insects, the eusocial insects, form often large societies in which there is reproductive division of labor, queens and workers, have overlapping generations, and cooperative brood care where daughter workers remain in the nest with their queen mother and care for their siblings. The eusocial insects are composed of representative species of bees and wasps, and all species of ants and termites. Much is known about their organizational structure, but remains to be discovered.
The success of social insects is dependent upon cooperative behavior and adaptive strategies shaped by natural selection that respond to internal or external conditions. The objective of my research was to investigate specific mechanisms that have helped shaped the structure of division of labor observed in social insect colonies, including age polyethism and nutrition, and phenomena known to increase colony survival such as egg cannibalism. I developed various Ordinary Differential Equation (ODE) models in which I applied dynamical, bifurcation, and sensitivity analysis to carefully study and visualize biological outcomes in social organisms to answer questions regarding the conditions under which a colony can survive. First, I investigated how the population and evolutionary dynamics of egg cannibalism and division of labor can promote colony survival. I then introduced a model of social conflict behavior to study the inclusion of different response functions that explore the benefits of cannibalistic behavior and how it contributes to age polyethism, the change in behavior of workers as they age, and its biological relevance. Finally, I introduced a model to investigate the importance of pollen nutritional status in a honeybee colony, how it affects population growth and influences division of labor within the worker caste. My results first reveal that both cannibalism and division of labor are adaptive strategies that increase the size of the worker population, and therefore, the persistence of the colony. I show the importance of food collection, consumption, and processing rates to promote good colony nutrition leading to the coexistence of brood and adult workers. Lastly, I show how taking into account seasonality for pollen collection improves the prediction of long term consequences.
The success of social insects is dependent upon cooperative behavior and adaptive strategies shaped by natural selection that respond to internal or external conditions. The objective of my research was to investigate specific mechanisms that have helped shaped the structure of division of labor observed in social insect colonies, including age polyethism and nutrition, and phenomena known to increase colony survival such as egg cannibalism. I developed various Ordinary Differential Equation (ODE) models in which I applied dynamical, bifurcation, and sensitivity analysis to carefully study and visualize biological outcomes in social organisms to answer questions regarding the conditions under which a colony can survive. First, I investigated how the population and evolutionary dynamics of egg cannibalism and division of labor can promote colony survival. I then introduced a model of social conflict behavior to study the inclusion of different response functions that explore the benefits of cannibalistic behavior and how it contributes to age polyethism, the change in behavior of workers as they age, and its biological relevance. Finally, I introduced a model to investigate the importance of pollen nutritional status in a honeybee colony, how it affects population growth and influences division of labor within the worker caste. My results first reveal that both cannibalism and division of labor are adaptive strategies that increase the size of the worker population, and therefore, the persistence of the colony. I show the importance of food collection, consumption, and processing rates to promote good colony nutrition leading to the coexistence of brood and adult workers. Lastly, I show how taking into account seasonality for pollen collection improves the prediction of long term consequences.
ContributorsRodríguez Messan, Marisabel (Author) / Kang, Yun (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Kuang, Yang (Committee member) / Page Jr., Robert E (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2018
Description
Rabies is an infectious viral disease. It is usually fatal if a victim reaches the rabid stage, which starts after the appearance of disease symptoms. The disease virus attacks the central nervous system, and then it migrates from peripheral nerves to the spinal cord and brain. At the time when the rabies virus reaches the brain, the incubation period is over and the symptoms of clinical disease appear on the victim. From the brain, the virus travels via nerves to the salivary glands and saliva.
A mathematical model is developed for the spread of rabies in a spatially distributed fox population to model the spread of the rabies epizootic through middle Europe that occurred in the second half of the 20th century. The model considers both territorial and wandering rabid foxes and includes a latent period for the infection. Since the model assumes these two kinds of rabid foxes, it is a system of both partial differential and integral equations (with integration
over space and, occasionally, also over time). To study the spreading speeds of the rabies epidemic, the model is reduced to a scalar Volterra-Hammerstein integral equation, and space-time Laplace transform of the integral equation is used to derive implicit formulas for the spreading speed. The spreading speeds are discussed and implicit formulas are given for latent periods of fixed length, exponentially distributed length, Gamma distributed length, and log-normally distributed length. A number of analytic and numerical results are shown pertaining to the spreading speeds.
Further, a numerical algorithm is described for the simulation
of the spread of rabies in a spatially distributed fox population on a bounded domain with Dirichlet boundary conditions. I propose the following methods for the numerical approximation of solutions. The partial differential and integral equations are discretized in the space variable by central differences of second order and by
the composite trapezoidal rule. Next, the ordinary or delay differential equations that are obtained this way are discretized in time by explicit
continuous Runge-Kutta methods of fourth order for ordinary and delay differential systems. My particular interest
is in how the partition of rabid foxes into
territorial and diffusing rabid foxes influences
the spreading speed, a question that can be answered by purely analytic means only for small basic reproduction numbers. I will restrict the numerical analysis
to latent periods of fixed length and to exponentially
distributed latent periods.
The results of the numerical calculations
are compared for latent periods
of fixed and exponentially distributed length
and for various proportions of territorial
and wandering rabid foxes.
The speeds of spread observed in the
simulations are compared
to spreading speeds obtained by numerically solving the analytic formulas
and to observed speeds of epizootic frontlines
in the European rabies outbreak 1940 to 1980.
A mathematical model is developed for the spread of rabies in a spatially distributed fox population to model the spread of the rabies epizootic through middle Europe that occurred in the second half of the 20th century. The model considers both territorial and wandering rabid foxes and includes a latent period for the infection. Since the model assumes these two kinds of rabid foxes, it is a system of both partial differential and integral equations (with integration
over space and, occasionally, also over time). To study the spreading speeds of the rabies epidemic, the model is reduced to a scalar Volterra-Hammerstein integral equation, and space-time Laplace transform of the integral equation is used to derive implicit formulas for the spreading speed. The spreading speeds are discussed and implicit formulas are given for latent periods of fixed length, exponentially distributed length, Gamma distributed length, and log-normally distributed length. A number of analytic and numerical results are shown pertaining to the spreading speeds.
Further, a numerical algorithm is described for the simulation
of the spread of rabies in a spatially distributed fox population on a bounded domain with Dirichlet boundary conditions. I propose the following methods for the numerical approximation of solutions. The partial differential and integral equations are discretized in the space variable by central differences of second order and by
the composite trapezoidal rule. Next, the ordinary or delay differential equations that are obtained this way are discretized in time by explicit
continuous Runge-Kutta methods of fourth order for ordinary and delay differential systems. My particular interest
is in how the partition of rabid foxes into
territorial and diffusing rabid foxes influences
the spreading speed, a question that can be answered by purely analytic means only for small basic reproduction numbers. I will restrict the numerical analysis
to latent periods of fixed length and to exponentially
distributed latent periods.
The results of the numerical calculations
are compared for latent periods
of fixed and exponentially distributed length
and for various proportions of territorial
and wandering rabid foxes.
The speeds of spread observed in the
simulations are compared
to spreading speeds obtained by numerically solving the analytic formulas
and to observed speeds of epizootic frontlines
in the European rabies outbreak 1940 to 1980.
ContributorsAlanazi, Khalaf Matar (Author) / Thieme, Horst R. (Thesis advisor) / Jackiewicz, Zdzislaw (Committee member) / Baer, Steven (Committee member) / Gardner, Carl (Committee member) / Kuang, Yang (Committee member) / Smith, Hal (Committee member) / Arizona State University (Publisher)
Created2018
Description
Factory production is stochastic in nature with time varying input and output processes that are non-stationary stochastic processes. Hence, the principle quantities of interest are random variables. Typical modeling of such behavior involves numerical simulation and statistical analysis. A deterministic closure model leading to a second order model for the product density and product speed has previously been proposed. The resulting partial differential equations (PDE) are compared to discrete event simulations (DES) that simulate factory production as a time dependent M/M/1 queuing system. Three fundamental scenarios for the time dependent influx are studied: An instant step up/down of the mean arrival rate; an exponential step up/down of the mean arrival rate; and periodic variation of the mean arrival rate. It is shown that the second order model, in general, yields significant improvement over current first order models. Specifically, the agreement between the DES and the PDE for the step up and for periodic forcing that is not too rapid is very good. Adding diffusion to the PDE further improves the agreement. The analysis also points to fundamental open issues regarding the deterministic modeling of low signal-to-noise ratio for some stochastic processes and the possibility of resonance in deterministic models that is not present in the original stochastic process.
ContributorsWienke, Matthew (Author) / Armbruster, Dieter (Thesis advisor) / Jones, Donald (Committee member) / Platte, Rodrigo (Committee member) / Gardner, Carl (Committee member) / Ringhofer, Christian (Committee member) / Arizona State University (Publisher)
Created2015
Description
Cancer is a major health problem in the world today and is expected to become an even larger one in the future. Although cancer therapy has improved for many cancers in the last several decades, there is much room for further improvement. Mathematical modeling has the advantage of being able to test many theoretical therapies without having to perform clinical trials and experiments. Mathematical oncology will continue to be an important tool in the future regarding cancer therapies and management.
This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically.
As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice.
Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis.
This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically.
As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice.
Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis.
ContributorsRutter, Erica Marie (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric J (Thesis advisor) / Frakes, David (Committee member) / Gardner, Carl (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Arizona State University (Publisher)
Created2016
Description
Chapter 1 introduces some key elements of important topics such as; quantum mechanics,
representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´
tivistic wave equations that will play an important role in the work to follow. In Chapter 2,
a complex covariant form of the classical Maxwell’s equations in a moving medium or at
rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum
tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its
connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´
netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.
Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s
equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell
and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´
operators of the Poincare group. A connection between the spin of a particle/field and ´
consistency of the corresponding overdetermined system is emphasized in the massless
case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which
is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨
evolution of exact wave functions of the generalized harmonic oscillators is determined
in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is
shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem
for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the
methods introduced in Chapter 5 a model for the quantization of an electromagnetic field
in a variable media is analyzed. The concept of quantization of an electromagnetic field
in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode
of radiation for this model is used to find time-dependent photon amplitudes in relation
to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the
uncertainty relation, are explicitly given in terms of the Ermakov-type system.
representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´
tivistic wave equations that will play an important role in the work to follow. In Chapter 2,
a complex covariant form of the classical Maxwell’s equations in a moving medium or at
rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum
tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its
connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´
netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.
Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s
equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell
and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´
operators of the Poincare group. A connection between the spin of a particle/field and ´
consistency of the corresponding overdetermined system is emphasized in the massless
case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which
is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨
evolution of exact wave functions of the generalized harmonic oscillators is determined
in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is
shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem
for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the
methods introduced in Chapter 5 a model for the quantization of an electromagnetic field
in a variable media is analyzed. The concept of quantization of an electromagnetic field
in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode
of radiation for this model is used to find time-dependent photon amplitudes in relation
to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the
uncertainty relation, are explicitly given in terms of the Ermakov-type system.
ContributorsLanfear, Nathan A (Author) / Suslov, Sergei (Thesis advisor) / Kotschwar, Brett (Thesis advisor) / Platte, Rodrigo (Committee member) / Matyushov, Dmitry (Committee member) / Kuiper, Hendrik (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2016
Description
In recent decades, marine ecologists have conducted extensive field work and experiments to understand the interactions between bacteria and bacteriophage (phage) in marine environments. This dissertation provides a detailed rigorous framework for gaining deeper insight into these interactions. Specific features of the dissertation include the design of a new deterministic Lotka-Volterra model with n + 1 bacteria, n
+ 1 phage, with explicit nutrient, where the jth phage strain infects the first j bacterial strains, a perfectly nested infection network (NIN). This system is subject to trade-off conditions on the life-history traits of both bacteria and phage given in an earlier study Jover et al. (2013). Sufficient conditions are provided to show that a bacteria-phage community of arbitrary size with NIN can arise through the succession of permanent subcommunities, by the successive addition of one new population. Using uniform persistence theory, this entire community is shown to be permanent (uniformly persistent), meaning that all populations ultimately survive.
It is shown that a modified version of the original NIN Lotka-Volterra model with implicit nutrient considered by Jover et al. (2013) is permanent. A new one-to-one infection network (OIN) is also considered where each bacterium is infected by only one phage, and that phage infects only that bacterium. This model does not use the trade-offs on phage infection range, and bacterium resistance to phage. The OIN model is shown to be permanent, and using Lyapunov function theory, coupled with LaSalle’s Invariance Principle, the unique coexistence equilibrium associated with the NIN is globally asymptotically stable provided that the inter- and intra-specific bacterial competition coefficients are equal across all bacteria.
Finally, the OIN model is extended to a “Kill the Winner” (KtW) Lotka-Volterra model
of marine communities consisting of bacteria, phage, and zooplankton. The zooplankton
acts as a super bacteriophage, which infects all bacteria. This model is shown to be permanent.
+ 1 phage, with explicit nutrient, where the jth phage strain infects the first j bacterial strains, a perfectly nested infection network (NIN). This system is subject to trade-off conditions on the life-history traits of both bacteria and phage given in an earlier study Jover et al. (2013). Sufficient conditions are provided to show that a bacteria-phage community of arbitrary size with NIN can arise through the succession of permanent subcommunities, by the successive addition of one new population. Using uniform persistence theory, this entire community is shown to be permanent (uniformly persistent), meaning that all populations ultimately survive.
It is shown that a modified version of the original NIN Lotka-Volterra model with implicit nutrient considered by Jover et al. (2013) is permanent. A new one-to-one infection network (OIN) is also considered where each bacterium is infected by only one phage, and that phage infects only that bacterium. This model does not use the trade-offs on phage infection range, and bacterium resistance to phage. The OIN model is shown to be permanent, and using Lyapunov function theory, coupled with LaSalle’s Invariance Principle, the unique coexistence equilibrium associated with the NIN is globally asymptotically stable provided that the inter- and intra-specific bacterial competition coefficients are equal across all bacteria.
Finally, the OIN model is extended to a “Kill the Winner” (KtW) Lotka-Volterra model
of marine communities consisting of bacteria, phage, and zooplankton. The zooplankton
acts as a super bacteriophage, which infects all bacteria. This model is shown to be permanent.
ContributorsKorytowski, Daniel (Author) / Smith, Hal (Thesis advisor) / Gumel, Abba (Committee member) / Kuang, Yang (Committee member) / Gardner, Carl (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2016