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- All Subjects: Applied Mathematics
- Creators: Kang, Yun
- Creators: Armbruster, Dieter
Description
Signaling cascades transduce signals received on the cell membrane to the nucleus. While noise filtering, ultra-sensitive switches, and signal amplification have all been shown to be features of such signaling cascades, it is not understood why cascades typically show three or four layers. Using singular perturbation theory, Michaelis-Menten type equations are derived for open enzymatic systems. When these equations are organized into a cascade, it is demonstrated that the output signal as a function of time becomes sigmoidal with the addition of more layers. Furthermore, it is shown that the activation time will speed up to a point, after which more layers become superfluous. It is shown that three layers create a reliable sigmoidal response progress curve from a wide variety of time-dependent signaling inputs arriving at the cell membrane, suggesting that natural selection may have favored signaling cascades as a parsimonious solution to the problem of generating switch-like behavior in a noisy environment.
ContributorsYoung, Jonathan Trinity (Author) / Armbruster, Dieter (Thesis advisor) / Platte, Rodrigo (Committee member) / Nagy, John (Committee member) / Baer, Steven (Committee member) / Taylor, Jesse (Committee member) / Arizona State University (Publisher)
Created2013
Description
Bacteriophage (phage) are viruses that infect bacteria. Typical laboratory experiments show that in a chemostat containing phage and susceptible bacteria species, a mutant bacteria species will evolve. This mutant species is usually resistant to the phage infection and less competitive compared to the susceptible bacteria species. In some experiments, both susceptible and resistant bacteria species, as well as phage, can coexist at an equilibrium for hundreds of hours. The current research is inspired by these observations, and the goal is to establish a mathematical model and explore sufficient and necessary conditions for the coexistence. In this dissertation a model with infinite distributed delay terms based on some existing work is established. A rigorous analysis of the well-posedness of this model is provided, and it is proved that the susceptible bacteria persist. To study the persistence of phage species, a "Phage Reproduction Number" (PRN) is defined. The mathematical analysis shows phage persist if PRN > 1 and vanish if PRN < 1. A sufficient condition and a necessary condition for persistence of resistant bacteria are given. The persistence of the phage is essential for the persistence of resistant bacteria. Also, the resistant bacteria persist if its fitness is the same as the susceptible bacteria and if PRN > 1. A special case of the general model leads to a system of ordinary differential equations, for which numerical simulation results are presented.
ContributorsHan, Zhun (Author) / Smith, Hal (Thesis advisor) / Armbruster, Dieter (Committee member) / Kawski, Matthias (Committee member) / Kuang, Yang (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2012
Description
A functioning food web is the basis of a functioning community and ecosystem. Thus, it is important to understand the dynamics that control species behaviors and interactions. Alterations to the fundamental dynamics can prove detrimental to the future success of our environment. Research and analysis focus on the global dynamics involved in intraguild predation (IGP), a three species subsystem involving both competition and predation. A mathematical model is derived using differential equations based on pre-existing models to accurately predict species behavior. Analyses provide sufficient conditions for species persistence and extinction that can be used to explain global dynamics. Dynamics are compared for two separate models, one involving a specialist predator and the second involving a generalist predator, where systems involving a specialist predator are prone to unstable dynamics. Analyses have implications in biological conservation tactics including various methods of prevention and preservation. Simulations are used to compare dynamics between models involving continuous time and those involving discrete time. Furthermore, we derive a semi-discrete model that utilizes both continuous and discrete time series dynamics. Simulations imply that Holling's Type III functional response controls the potential for three species persistence. Complicated dynamics govern the IGP subsystem involving the white-footed mouse, gypsy moth, and oak, and they ultimately cause the synchronized defoliation of forests across the Northeastern United States. Acorn mast seasons occur every 4-5 years, and they occur simultaneously across a vast geographic region due to universal cues. Research confirms that synchronization can be transferred across trophic levels to explain how this IGP system ultimately leads to gypsy moth outbreaks. Geographically referenced data is used to track and slow the spread of gypsy moths further into the United States. Geographic Information Systems (GIS) are used to create visual, readily accessible, displays of trap records, defoliation frequency, and susceptible forest stands. Mathematical models can be used to explain both changes in population densities and geographic movement. Analyses utilizing GIS softwares offer a different, but promising, way of approaching the vast topic of conservation biology. Simulations and maps are produced that can predict the effects of conservation efforts.
ContributorsWedekin, Lauren (Author) / Kang, Yun (Thesis advisor) / Green, Douglas (Committee member) / Miller, William (Committee member) / Arizona State University (Publisher)
Created2012
Description
Dividing the plane in half leaves every border point of one region a border point of both regions. Can we divide up the plane into three or more regions such that any point on the boundary of at least one region is on the border of all the regions? In fact, it is possible to design a dynamical system for which the basins of attractions have this Wada property. In certain circumstances, both the Hénon map, a simple system, and the forced damped pendulum, a physical model, produce Wada basins.
ContributorsWhitehurst, Ryan David (Author) / Kostelich, Eric (Thesis director) / Jones, Donald (Committee member) / Armbruster, Dieter (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2013-05
Description
The retina is the lining in the back of the eye responsible for vision. When light photons hits the retina, the photoreceptors within the retina respond by sending impulses to the optic nerve, which connects to the brain. If there is injury to the eye or heredity retinal problems, this part can become detached. Detachment leads to loss of nutrients, such as oxygen and glucose, to the cells in the eye and causes cell death. Sometimes the retina is able to be surgically reattached. If the photoreceptor cells have not died and the reattachment is successful, then these cells are able to regenerate their outer segments (OS) which are essential for their functionality and vitality. In this work we will explore how the regrowth of the photoreceptor cells in a healthy eye after retinal detachment can lead to a deeper understanding of how eye cells take up nutrients and regenerate. This work uses a mathematical model for a healthy eye in conjunction with data for photoreceptors' regrowth and decay. The parameters for the healthy eye model are estimated from the data and the ranges of these parameter values are centered +/- 10\% away from these values are used for sensitivity analysis. Using parameter estimation and sensitivity analysis we can better understand how certain processes represented by these parameters change within the model as a result of retinal detachment. Having a deeper understanding for any sort of photoreceptor death and growth can be used by the greater scientific community to help with these currently irreversible conditions that lead to blindness, such as retinal detachment. The analysis in this work shows that maximizing the carrying capacity of the trophic pool and the rate of RDCVF, as well as minimizing nutrient withdrawal of the rods and the cones from the trophic pool results in both the most regrowth and least cell death in retinal detachment.
ContributorsGoldman, Miriam Ayla (Author) / Camacho, Erikia (Thesis director) / Wirkus, Stephen (Committee member) / School of Mathematical and Natural Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2017-05
Description
Variation in living systems and how it cascades across organizational levels is central to biology. To understand the constraints and amplifications of variation in collective systems, I mathematically study how group-level differences emerge from individual variation in eusocial-insect colonies, which are inherently diverse and easily observable individually and collectively. Considering collective processes in three species where increasing degrees of heterogeneity are relevant, I address how individual variation scales to colony-level variation and to what degree it is adaptive. In Chapter 2, I introduce a Markov-chain decision model for stochastic individual quorum-based recruitment decisions of rock-ant workers during house hunting, and how they determine collective speed--accuracy balance. Differences in the average threshold-dependent response characteristics of workers between colonies cause collective differences in decision-making. Moreover, noisy behavior may prevent drastic collective cascading into poor nests. In Chapter 3, I develop an ordinary differential equation (ODE) model to study how cognitive diversity among honey-bee foragers influences collective attention allocation between novel and familiar resources. Results provide a mechanistic basis for changes in foraging activity and preference with group composition. Moreover, sensitivity analysis reveals that the main individual driver for foraging allocation shifts from recruitment (communication) to persistence (independent effort) as colony composition changes. This might favor specific degrees of heterogeneity that best amplify communication in wild colonies. Lastly, in Chapter 4, I consider diversity in size, age, and task for nest defense in stingless bees. To better understand how these dimensions of diversity interact to balance defensive demands with other colony needs, I study their effect on colony size and task allocation through a demographic Filippov ODE model. Along each dimension, variation is beneficial in a certain range, outside of which colony adaptation and survival are compromised. This work elucidates how variation in collective properties emerges from nonlinear interactions between varying components in eusocial insects, but it can be generalized to other biological systems with similar fundamental characteristics but less empirical tractability. Moreover, it has the potential of inspiring algorithms that capitalize on heterogeneity in engineered systems where simple components with limited information and no central control must solve complex tasks.
ContributorsNavas Zuloaga, Maria Gabriela (Author) / Kang, Yun (Thesis advisor) / Smith, Brian H (Thesis advisor) / Pavlic, Theodore P (Committee member) / Arizona State University (Publisher)
Created2022
Description
This research focuses on the intricate dynamical systems of eusocial insects, particularly ants, and honey bees, known for their highly organized colonies and cooperative behaviors. Research on eusocial insects contributes to understanding of animal and social behavior and promises to help agriculture and have huge economic impacts. Collaborating closely with ecologists, I construct diverse mathematical models tailored to different environmental contexts. These models encompass individual stochastic (Agent-based model), Ordinary Differential Equation (ODE), non-autonomous, and Delay Differential Equation (DDE) models, rigorously validated with experimental data and statistical methods. Employing dynamical theory, bifurcation analysis, and numerical simulations, I gain deeper insights into the adaptive behaviors exhibited by these insects at both colony and individual levels. Our investigation addresses pivotal questions: 1) What mechanisms underlie spatial heterogeneity within social insect colonies, influencing the spread of information and pathogens through their intricate social networks?2) How can I develop accurate mathematical models incorporating age structures, particularly for species like honeybees, utilizing delayed differential equations?
3) What is the influence of seasonality on honeybee population dynamics in the presence of parasites, as explored through non-autonomous equations?
4) How do pesticides impact honeybee population dynamics, considering delayed equations and seasonality? Key findings highlight:1) The spatial distribution within colonies significantly shapes contact dynamics, thereby influencing the dissemination of information and the allocation of tasks.
2) Accurate modeling of honeybee populations necessitates the incorporation of age structure, as well as careful consideration of seasonal variations.
3) Seasonal fluctuations in egg-laying rates exert varying effects on the survival of honeybee colonies.
4) Pesticides wield a substantial influence on adult bee mortality rates and the consumption ratios of pollen. This research not only unveils the intricate interplay between intrinsic and environmental factors affecting social insects but also provides broader insights into social behavior and the potential ramifications of climate change.
ContributorsChen, Jun (Author) / Kang, Yun (Thesis advisor) / DeGrandi-Hoffman, Gloria (Committee member) / Fewell, Jeniffer (Committee member) / Harrison, Jon (Committee member) / Towers, Sherry (Committee member) / Arizona State University (Publisher)
Created2023
Description
Balancing temporal shortages of renewable energy with natural gas for the generation of electricity is a challenge for dispatchers. This is compounded by the recent proposal of blending cleanly-produced hydrogen into natural gas networks. To introduce the concepts of gas flow, this thesis begins by linearizing the partial differential equations (PDEs) that govern the flow of natural gas in a single pipe. The solution of the linearized PDEs is used to investigate wave attenuation and characterize critical operating regions where linearization is applicable. The nonlinear PDEs for a single gas are extended to mixtures of gases with the addition of a PDE that governs the conservation of composition. The gas mixture formulation is developed for general gas networks that can inject or withdraw arbitrary time-varying mixtures of gases into or from the network at arbitrarily specified nodes, while being influenced by time-varying control actions of compressor units. The PDE formulation is discretized in space to form a nonlinear control system of ordinary differential equations (ODEs), which is used to prove that homogeneous mixtures are well-behaved and heterogeneous mixtures may be ill-behaved in the sense of monotone-ordering of solutions. Numerical simulations are performed to compute interfaces that delimit monotone and periodic system responses. The ODE system is used as the constraints of an optimal control problem (OCP) to minimize the expended energy of compressors. Moreover, the ODE system for the natural gas network is linearized and used as the constraints of a linear OCP. The OCPs are digitally implemented as optimization problems following the discretization of the time domain. The optimization problems are applied to pipelines and small test networks. Some qualitative and computational applications, including linearization error analysis and transient responses, are also investigated.
ContributorsBaker, Luke Silas (Author) / Armbruster, Dieter (Thesis advisor) / Zlotnik, Anatoly (Committee member) / Herty, Michael (Committee member) / Platte, Rodrigo (Committee member) / Milner, Fabio (Committee member) / Arizona State University (Publisher)
Created2023
Description
A pneumonia-like illness emerged late in 2019 (coined COVID-19), caused by SARSCoV-2, causing a devastating global pandemic on a scale never before seen sincethe 1918/1919 influenza pandemic. This dissertation contributes in providing deeper
qualitative insights into the transmission dynamics and control of the disease in the
United States. A basic mathematical model, which incorporates the key pertinent
epidemiological features of SARS-CoV-2 and fitted using observed COVID-19 data,
was designed and used to assess the population-level impacts of vaccination and face
mask usage in mitigating the burden of the pandemic in the United States. Conditions
for the existence and asymptotic stability of the various equilibria of the model were
derived. The model was shown to undergo a vaccine-induced backward bifurcation
when the associated reproduction number is less than one. Conditions for achieving
vaccine-derived herd immunity were derived for three of the four FDA-approved vaccines (namely Pfizer, Moderna and Johnson & Johnson vaccine), and the vaccination
coverage level needed to achieve it decreases with increasing coverage of moderately and highly-effective face masks. It was also shown that using face masks as a singular
intervention strategy could lead to the elimination of the pandemic if moderate or
highly-effective masks are prioritized and pandemic elimination prospects are greatly
enhanced if the vaccination program is combined with a face mask use strategy that
emphasizes the use of moderate to highly-effective masks with at least moderate coverage. The model was extended in Chapter 3 to allow for the assessment of the
impacts of waning and boosting of vaccine-derived and natural immunity against
the BA.1 Omicron variant of SARS-CoV-2. It was shown that vaccine-derived herd
immunity can be achieved in the United States via a vaccination-boosting strategy
which entails fully vaccinating at least 72% of the susceptible populace. Boosting
of vaccine-derived immunity was shown to be more beneficial than boosting of natural immunity. Overall, this study showed that the prospects of the elimination of
the pandemic in the United States were highly promising using the two intervention
measures.
ContributorsSafdar, Salman (Author) / Gumel, Abba (Thesis advisor) / Kostelich, Eric (Committee member) / Kang, Yun (Committee member) / Fricks, John (Committee member) / Espanol, Malena (Committee member) / Arizona State University (Publisher)
Created2023
DescriptionUnderstanding the evolution of opinions is a delicate task as the dynamics of how one changes their opinion based on their interactions with others are unclear.
ContributorsWeber, Dylan (Author) / Motsch, Sebastien (Thesis advisor) / Lanchier, Nicolas (Committee member) / Platte, Rodrigo (Committee member) / Armbruster, Dieter (Committee member) / Fricks, John (Committee member) / Arizona State University (Publisher)
Created2021