Partial Differential Equations Commons^™

Inhomogeneous Branching Random Walks: Incorporating Genealogy And Density Effects, Lauren Ajax, Beatrice Durham, Pratima Hebbar, Cade Johnston, Jiayi Zhang

Spora: A Journal of Biomathematics

We introduce a novel framework using inhomogeneous branching random walks (BRWs) to model biological processes, specifically by introducing genealogy-dependence in branching rates and displacement distributions to model bacterial colony growth. Current stochastic models often either assume independent and identical behavior of individual agents or incorporate only spatiotemporal inhomogeneity, ignoring the effect of genealogy-based inhomogeneity on the long-time behavior of these processes. Such asymptotics are of independent mathematical interest and are crucial in understanding the emergence of patterns. We propose several inhomogeneous BRW models in 2D space where displacement distributions and branching rates vary with time, space, and genealogy. A combined …

A Mathematical Frameworks For Singular, Nonlinear Phenomena: Applications To Nematocyst Firing And Inhomogeneous Nls With Coulomb Potential, Abdulrahman Alharbi

Theses and Dissertations

Nematocysts are specialized cellular organelles found in all cnidarians, including corals and jellyfish, as well as in some single-celled protists such as dinoflagellates. These organelles display remarkable diversity in morphology and function, playing roles in prey capture and defense. The firing of a nematocyst is one of the fastest accelerations in nature, yet the underlying physical mechanisms remain not fully understood. In this work, we address key questions: how sufficient force is generated to overcome the fluid boundary layer, whether fluid–structure interaction models can reproduce observed dynamics, and what mechanisms trigger discharge.

Our research investigates models based on osmotic pressure …

Mathematical Model Of Ovarian Cancer Tumor Growth In Mice, Jessica A. Hoffman

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Individual And Collective Properties Of Tunable Photochemical Belousov-Zhabotinsky Micro-Reactors, Kudakwashe Benedict Shumba

Graduate Theses, Dissertations, and Problem Reports (ETD)

Cell-like model chemical systems are powerful tools that can be used to explore the role of intercellular coupling on population level behaviors in communities of biological cells. Firstly, we present a new method for fabricating such micro-reactors using the photosensitive Belousov–Zhabotinsky (BZ) reaction system employed in silica microparticles. These BZ micro-reactors have a tunable response to photochemical coupling, varying from a fully excitatory response to a fully inhibitory response. Their response can be tuned through variations in either the reactive mixture or, on an individual micro-reactor level, by changes in the synthesis temperature used during the fabrication of the silica …

A Model Of Oocyte Population Dynamics For Fish Oogenesis, Louis Fostier, Frédérique Clément, Romain Yvinec, Violette Thermes

Biology and Medicine Through Mathematics Conference

No abstract provided.

Reaction-Diffusions System Simulated On Irregular Shapes And Surfaces Model Petal Spot Patterns In Monkeyflower Hybrids, Emily Simmons, Arielle M. Cooley, Joshua R. Puzey, Gregory D. Conradi Smith

Biology and Medicine Through Mathematics Conference

No abstract provided.

Exploring The Evolution Of Altruistic Punishment Using A Pde Model For Multilevel Selection, Daniel Cooney

Biology and Medicine Through Mathematics Conference

No abstract provided.

Identifiability For Pde Models Of Fluorescence Microscopy Experiments, Veronica Ciocanel

Biology and Medicine Through Mathematics Conference

No abstract provided.

Chemoattractant Distribution In The Drosophila Egg Chamber, Lara Scott

Biology and Medicine Through Mathematics Conference

No abstract provided.

Sperm-Egg Interaction For Fertilization Success, Prajakta P. Bedekar

Biology and Medicine Through Mathematics Conference

No abstract provided.

Tools For Biomolecular Modeling And Simulation, Xin Yang

Mathematics Theses and Dissertations

Electrostatic interactions play a pivotal role in understanding biomolecular systems, influencing their structural stability and functional dynamics. The Poisson-Boltzmann (PB) equation, a prevalent implicit solvent model that treats the solvent as a continuum while describes the mobile ions using the Boltzmann distribution, has become a standard tool for detailed investigations into biomolecular electrostatics. There are two primary methodologies: grid-based finite difference or finite element methods and body-fitted boundary element methods. This dissertation focuses on developing fast and accurate PB solvers, leveraging both methodologies, to meet diverse scientific needs and overcome various obstacles in the field.

Effects Of A Protection Zone In A Reaction-Diffusion Model With Strong Allee Effect., Isaac Johnson

Electronic Theses and Dissertations

A protection zone model represents a patchy environment with positive growth over the protection zone and strong Allee effect growth outside the protection zone. Generally, these models are considered through the corresponding eigenvalue problem, but that has certain limitations. In this thesis, a general protection zone model is considered. This model makes no assumption on the direction of the traveling wave solution over the Strong Allee effect patch. We use phase portrait analysis of this protection zone model to draw conclusions about the existence of equilibrium solutions. We establish the existence of three types of equilibrium solutions and the necessary …

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

Master's Theses

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

Pde Model For Protocell Evolution And The Origin Of Chromosomes Via Multilevel Selection, Daniel B. Cooney, Fernando W. Rossine, Dylan H. Morris, Simon A. Levin

Biology and Medicine Through Mathematics Conference

No abstract provided.

Reaction-Diffusion System On Irregular Boundaries Reproduces Multiple Generations Of Petal Spot Patterns In Monkeyflower Hybrids, Emily Simmons

Biology and Medicine Through Mathematics Conference

No abstract provided.

Helices In Fluids And Their Applications, Eva M. Strawbridge

Biology and Medicine Through Mathematics Conference

No abstract provided.

(R2020) Dynamical Study And Optimal Harvesting Of A Two-Species Amensalism Model Incorporating Nonlinear Harvesting, Manoj Kumar Singh, Poonam .

Applications and Applied Mathematics: An International Journal (AAM)

This study proposes a two-species amensalism model with a cover to protect the first species from the second species, with the assumption that the growth of the second species is governed by nonlinear harvesting. Analytical and numerical analyses have both been done on this suggested ecological model. Boundedness and positivity of the solutions of the model are examined. The existence of feasible equilibrium points and their local stability have been discussed. In addition, the parametric conditions under which the proposed system is globally stable have been determined. It has also been shown, using the Sotomayor theorem, that under certain parametric …

Modeling Of Population Dynamics In Spatially Continuous And Heterogeneous Environments Using Partial Differential Equations, Ryan T. St. Clair

Masters Theses & Specialist Projects

This thesis provides an expansion of existing reaction diffusion population ecology models into patchy landscapes with greater spatial heterogeneity. The Sturm-Loiville eigenvalue problem for a heterogeneous 3 patch landscape is solved implicitly and the population growth rate and migration dynamics on such a landscape are thoroughly discussed. In addition a system of interface equations is found whose greatest solution is the dominant eigenvalue of an N-patch landscape. The results of this model agree well with previously published results and show the mathematical equivalence of two published approaches for incorporating organism behavior at habitat interfaces. Increased modeling capabilities in heterogeneous environments …

De-Coupling Cell-Autonomous And Non-Cell-Autonomous Fitness Effects Allows Solution Of The Fokker-Planck Equation For The Evolution Of Interacting Populations, Steph J. Owen

Biology and Medicine Through Mathematics Conference

No abstract provided.

Mathematical Modeling Of Brain Cancer Growth Using A Level-Set Method, Gbocho M. Terasaki

Biology and Medicine Through Mathematics Conference

No abstract provided.

Spatial Patterning Of Predator-Prey Distributions Arising From Prey Defense, Evan C. Haskell, Jonathan Bell

Biology and Medicine Through Mathematics Conference

No abstract provided.

Genetically Explicit Model May Explain Multigenerational Control Of Emergent Turing Patterns In Hybrid Mimulus, Emily Simmons

Biology and Medicine Through Mathematics Conference

No abstract provided.

Ecological Dynamics On Large Metapopulation Graphs, Daniel Cooney

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Qualitative Analysis Of A Modified Leslie-Gower Predator-Prey Model With Weak Allee Effect Ii, Manoj K. Singh, B. S. Bhadauria

Applications and Applied Mathematics: An International Journal (AAM)

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for …

An Examination Of Fontan Circulation Using Differential Equation Models And Numerical Methods, Vanessa Maybruck

Honors Student Research

Certain congenital heart defects can lead to the development of only a single pumping chamber, or ventricle, in the heart instead of the usual two ventricles. Individuals with this defect undergo a corrective, three-part surgery, the third step of which is the Fontan procedure, but as the patients age, their cardiovascular health will likely deteriorate. Using computational fluid dynamics and differential equations, Fontan circulation can be modeled to investigate why the procedure fails and how Fontan failure can be maximally prevented. Borrowing from well-established literature on RC circuits, the differential equation models simulate systemic blood flow in a piecewise, switch-like …

A New Mathematical Theory For The Dynamics Of Large Tumor Populations, A Potential Mechanism For Cancer Dormancy & Recurrence And Experimental Observation Of Melanoma Progression In Zebrafish, Adeyinka A. Lesi

Dissertations and Theses

Cancer, a family of over a hundred disease varieties, results in 600,000 deaths in the U.S. alone. Yet, improvements in imaging technology to detect disease earlier, pharmaceutical developments to shrink or eliminate tumors, and modeling of biological interactions to guide treatment have prevented millions of deaths. Cancer patients with initially similar disease can experience vastly different outcomes, including sustained recovery, refractory disease or, remarkably, recurrence years after apparently successful treatment. The current understanding of such recurrences is that they depend on the random occurrence of critical mutations. Clearly, these biological changes appear to be sufficient for recurrence, but are they …

The Impact Of Nonlinear Harvesting On A Ratio-Dependent Holling-Tanner Predator-Prey System And Optimum Harvesting, Manoj Kumar Singh, B. S. Bhadauria

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a Holling-Tanner predator-prey model with ratio-dependent functional response and non-linear prey harvesting is analyzed. The mathematical analysis of the model includes existence, uniqueness and boundedness of positive solutions. It also includes the permanence, local stability and bifurcation analysis of the model. The ratio-dependent model always has complex dynamics in the vicinity of the origin; the dynamical behaviors of the system in the vicinity of the origin have been studied by means of blow up transformation. The parametric conditions under which bionomic equilibrium point exist have been derived. Further, an optimal harvesting policy has been discussed by using …

Spreading Mechanics And Differentiation Of Astrocytes During Retinal Development, Tracy Stepien, Timothy W. Secomb

Biology and Medicine Through Mathematics Conference

No abstract provided.

Attraction-Repulsion Taxis Mechanisms In A Predator-Prey Model, Evan C. Haskell

Biology and Medicine Through Mathematics Conference

No abstract provided.

Mathematical Modeling Of Gliding Motility And Its Regulation In Myxococcus Xanthus, Yirui Chen

Biology and Medicine Through Mathematics Conference

No abstract provided.