Approximation of Dynamical Systems: The Importance of Small Parameters
John A. Burns
Interdisciplinary Center for Applied Mathematics
The development of practical numerical methods for simulation of and control of partial differential equations leads to problems of convergence, accuracy (in time and space) and efficiency. Verification of a computational algorithm consist in part of establishing a convergence theory for the discretized equations. Convergence is often based on some form of the Lax Equivalence Theorem, or its functional analysis version, the Trotter-Kato Theorem. It is well known that the long time behavior of a system may not be captured even by "convergent" approximating methods and additional requirements must be placed on the scheme to ensure the discretized equations capture the correct asymptotic behavior. Even on finite intervals, there are always uncertainties in the problem data that can be a source of difficulty for nonlinear problems. This includes uncertainty in parameters, initial data, boundary conditions and forcing terms. These uncertainties in the problem data lead to uncertainty in the computed results and should be considered as part of the verification step. In this talk we discuss these issues and illustrate how sensitivities to "small" parameters can impact accuracy and predictability of numerical models constructed from discretizing PDE systems.
Dynamics of Some Nonlinear Discontinuous Discrete Population Models
Vlajko L. Kocic
Xavier University of Louisiana
New Orleans, Louisiana
We investigate the dynamics of some classes of discontinuous discrete population models including piecewise linear and nonlinear population models. In particular we focus on the study of oscillatory character of solutions, their semicycles, the existence and stability of periodic orbits, existence of invariant intervals, attractivity, and bifurcations. The brief survey of literature about discrete discontinuous population models is presented and several open problems are formulated.
Self-Excited Vibrations for One-Dimensional Semilinear Damped Wave Equations
University of South Alabama
Over the last fifty years a huge effort has been devoted to the study of periodic response solutions of forced wave equations. On the other hand, many vibrations arising in mechanical engineering problems are 'self-excited', meaning that they arise from time delay due to sampling time. Such a mechanism can destabilize the trivial steady state and lead to vibrations. We discuss some C^infinity theory in this direction.
Darwinian dynamics and matrix models for structured populations
J. M. Cushing
Department of Mathematics
The University of Arizona
I will talk about some theorems concerning the asymptotic dynamics of evolutionary game theoretic versions of (discrete time) matrix models for structured populations. I'll look at some specific models motivated by observations from marine bird colonies (specifically the glaucous-winged gull) on Protection Island National Wildlife Refuge in the Strait Juan de Fuca, Washington, and by changes in their behavior and life history strategies that are correlated with climate change in the region. The analysis centers on bifurcations, stability, and Allee effects and on conditions under which these changes are adaptive and promote long term population survival. Collaborators are James Hayward, Shandelle Henson and Amy Veprauskas.
Euler-Lagrange Poisson PDEs Based Active Contour and Hessian Matrix, Features Extraction and Matching
Nikolay Metodiev Sirakov
Department of Mathematics
Department of Computer Science
Texas A&M University Commerce
In the beginning of this speech the author will brief the audience with the main concepts he has introduced, during the 2011 and 2012 talks at the same Conference: active contour (AC) on the Heat partial differential equation (PDE); region based convex AC; and Support Vector Machine.
The main body of the talk will present two approaches: the 1st one is a new AC model to extract objects boundaries from images [1, 2, 3]; the 2nd one detects from image regions salient points to be used for matching. The AC model is developed on the Euler-Lagrange and Poisson PDEs. The Poison equation with a right side equal to the Euclidian norm of the image gradient is solved on the image. This way a new image is created and its gradient generates a vector field [1, 2]. The approximate solution of the Euler-Lagrange PDE develops and algorithm which evolves the AC following the vector field trough the image [2, 3]. The approximate solution  uses a half step numerical scheme whose matrix is invertible . The AC evolution halts its motion on the objects boundaries using gradient based penalty function [1, 2]. A software tool is designed on the theory in a Matlab environment. Experimental results with this tool will validate the advantages and the bottlenecks of the method. The second approach will introduce robust features (SURF) detection in images . SURF are used objects description and matching . The method applies the Hessian matrix of a Gaussian convolved on an image. Thus, the salient point of the image regions are detected and a number of features are extracted for every point. The vectors associated with the salient points are rotationally and scaling invariant. The speaker will demonstrate experimental results on objects detection and matching in cluttered scenes using a MatLab developed tool.
- A. Bowden, N. M. Sirakov, Applications of The Euler – Lagrange Poisson Active Contour in Vector Fields, Overcoming Noise, and Line Integrals, J. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, Watam Press, 23 (2016) 59-73
- A. Bowden, M. D. Todorov and N. M. Sirakov, Implementation of the Euler - Lagrange and Poisson Equations to Extract One Connected Region, American Institute of Physics (AIP) 1629, pp.400-407, 2014, doi: 10.1063/1.4902301
- M.Todorov, N.M.Sirakov, S.Suh, 2013, Fast Splitting Scheme to Minimize New Energy Functional Containing Schroedinger Equation Solution, Proc. AMiTANS2013, In M. Todorov Ed., American Institute of Physics, V.1561, 378 (2013), DOI: http://dx.doi.org/10.1063/1.4827250
- C. I. Christov, Gaussian elimination with pivoting for multidiagonal systems, Internal Report 4, University of Reding, UK, (1994)
- H. Bay, A. Ess, T. Tuytelaars, L. Van Gool, Speeded-Up Robust Features (SURF), Computer Vision and Image Understanding, V. 110, Issue 3, June 2008, pp 346–359, (2008
An Innovative Approach and Fundamental Properties of Levy-type Nonlinear Stochastic Dynamic Models
G. S. Ladde
Department of Mathematics and Statistics
University of South Florida
We consider a prototype stochastic dynamic model for dynamic processes in biological, chemical, economic financial, medical, military, physical and technological sciences. The dynamic model is described by Levy-type nonlinear stochastic differential equation. The model validation is established by using Lyapunov-like function. The basic innovative idea is to transform a Levy-type nonlinear stochastic differential into a simpler stochastic differential equation that is easily tested for the existence and uniqueness of theorem. Using the nature of Lyapunov-like function, the existence and uniqueness of solution of the original Levy-type nonlinear stochastic differential equation is established. The main idea of the proof is based on the property of the one-to-one and onto transformation. As the byproduct of the analysis, it is shown that the closed form implicit solution of transformed stochastic differential equation is a positive martingale. Furthermore, using the change of measure, a Girsanov-type theorem for Levy-type nonlinear stochastic dynamic model is established.
This work is supported by the Mathematical Sciences Division, US Army Research Office, Grant Number: W911NF-15-1-0182
Mathematical and Statistics Techniques for Optimal Gene Regulatory Networks
Department of Mathematics, Physics and SMED
Southern University and A&M College
Baton Rouge, Louisiana
Inferring Gene Regulatory Networks (GRNs) is fundamental to decipher the complexity of transcriptional mechanisms within the cell. Inferring a GRN can be accomplished with the use of multiple time-course microarray or RNA-seq datasets from different conditions. The available methods for GRN construction fall into two main categories, model-based approaches and machine learning-based approaches. In the model-based methods, chemical reactions of transcription and translation are described as linear or non-linear differential equations, in which the parameters represent the regulation strengths of the regulators (e.g. multiple linear regression, singular value decomposition method, network component analysis, and linear programming). In 2010, a model-based method (TIGRE) was developed from a Gaussian process prior distribution to identify potential targets of a transcription factor (TF) with limited data. For the machine learning-based approaches, the network is inferred via measuring the dependences between TFs and target genes (e.g. partial correlation coefficient and Bayesian network analysis). Other popular methods that use gene regulatory networks are mutual information (MI) and conditional mutual information (CMI). A recent method, using noise and redundancy reduction technology combining ordinary differential equations, based recursive optimization and MI (NARROMI), combines model-based and machine-learning-based methods to improve the accuracy of GRN inference from gene expression data. This talk will highlight the main parts of an efficient data analysis tool, integrating mathematical algorithms from model-based and machine learning-based approaches with graph theory, producing optimal GRNs for the analysis of large available experimental data.
Converting $L^p$ estimates into $L^\infty$ for nonlinear diffusion models with memory at the boundary
Indiana University-Purdue University Fort Wayne
Fort Wayne, Indiana
Drawing motivation from models of tumor-induced capillary growth, we initiated the study of diffusion models with boundary flux governed by memory around 5 years ago. Considering instances of power laws where supersolution comparison methods are not generally applicable, it turns out to be possible to estimate $L^p$ norms of the solution for any large value of $p$. Although constants in the estimates do not permit passage to the limit for obtaining an $L^\infty$ estimate, one may instead apply an integral form of the maximum principle that has frequently been applied in the case of porous medium equations. We provide an adaptation of the method to also apply equally well in so-called fast diffusion cases. Combined with results on blow-up in finite time, this allows a complete analysis of global solvability for such memory flux models which exactly parallels known results for corresponding localized nonlinear flux models.
Structured Multi-Scale Models for Daphnia magna Population Dynamics
Center for Research in Scientific Computation
N.C. State University
In this lecture we discuss statistical validation techniques to verify density-dependent mechanisms hypothesized for populations of Daphnia magna. We develop structured population models that exemplify specific mechanisms, and use multi-scale experimental data in order to test their importance. We show that fecundity and survival rates are affected by both time-varying density-independent factors, such as age, and density-dependent factors, such as competition. We perform uncertainty analysis and show that our parameters are estimated with a high degree of confidence. Further, we perform a sensitivity analysis to understand how changes in fecundity and survival rates affect population size and age-structure.
Analysis of a System of Parabolic Conservation Laws Arising from Biology
Department of Mathematics
New Orleans, Louisiana
In contrast to random diffusion without orientation, chemotaxis is the biased movement of organisms toward the region that contains higher concentration of beneficial or lower concentration of unfavorable chemicals. Chemotaxis has been advocated as a leading mechanism to account for the morphogenesis and self-organization of a variety of biological coherent structures. In this talk, I will present a group of recent results concerning the rigorous analysis of a system of parabolic conservation laws arising from the study of the interaction between the cellular aggregation of vascular endothelial cells (VEC) and chemical degradation of vascular endothelial growth factor (VEGF) associated with tumor angiogenesis. In particular, global well-posedness, long-time asymptotic behavior, diffusion limit and boundary layer formation of classical solutions will be discussed. The global well-posedness and long-time behavior results show that constant equilibrium states are globally asymptotically stable, which indicates that supplying cellular density to chemical degradation effectively prevents the cell population from blowing up (aggregation). The results on diffusion limit and boundary layer formation demonstrate that the chemically diffusive model is consistent with the non-diffusive model under certain boundary conditions, which validates one of the basic assumptions in the original development of the model.
Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge
Department of Mathematics and Statistics
Texas Tech University
In this talk, we introduce a discrete host-parasitoid model with a host refuge. The model is built upon a modified Nicholson-Bailey system by assuming that in each generation a constant proportion of the host is free from parasitism. We investigate population coexistence and possible bifurcations. Both populations can persist under some restrictions on the model parameters. The system is able to undergo a supercritical and then a subcritical Neimark-Sacker bifurcation or the system only exhibits a supercritical Neimark-Sacker bifurcation. It is illustrated numerically that a constant proportion of host refuge can stabilize the host-parasitoid interaction. This work is joint with Yunshyong Chow, Institute of Mathematics, Academia Sinica, Taiwan
Modeling the effect of increased chest wall compliance on respiratory dynamics
Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Increased compliance (flexibility) of the chest wall in very premature infants due to bone undermineralization in early gestation results in progressive lung collapse as the forces needed to open airspaces after each exhalation become insufficient. The survival rate for these infants is increasing, but they remain at significant risk for developing chronic lung disease related to treatment with mechanical ventilation. We present a mathematical model of respiratory dynamics in infants that accounts for changes in chest wall compliance and the effects on lung compliance and volume. Our long-term objective is to develop a supportive exterior treatment that will stiffen the chest wall to facilitate ventilation.
Mathematical Modeling and Dynamics of Interactive Wild and Sterile Mosquito Populations and Release Strategies
University of Alabama in Huntsville
In this talk, I will present several mathematical models for the interactive wild and sterile mosquitoes. We incorporate different strategies for the releases of sterile mosquitoes in the models, including impulsive releases. We investigate the model dynamics and illustrate how these different release strategies are applied to the control of mosquito populations. We also compare the impacts of these control measures.
Asymptotics of greedy energy sequences
University of South Alabama
Given a compact set K in the Euclidean space, a greedy energy sequence on K is a sequence whose points are generated by means of a greedy algorithm in which a certain energy functional is minimized (at each step of the algorithm). A classical example of such sequences are the Leja sequences on compact subsets of the complex plane. When studying these sequences, one is typically interested in the asymptotic distribution and the asymptotic behavior of the energy of the first n points of the sequence. In this talk I will discuss some recent results concerning these problems, in the context of logarithmic and Riesz potentials in the Euclidean space.
Modeling the spread of epidemics from the viewpoint of the infected population
New Orleans, Louisiana
Most mathematical models are created and analyzed from the viewpoint of the force OF infection on susceptible population. That is, the focus is on the risk of a susceptible person being infected, and number of susceptible people who are infected each day. I will describe the advantages of creating the model from the viewpoint of the force FROM the infected population, where the focus is on the number of people an infected person infects each day. The two viewpoints are equivalent for simple models, but there are significant advantages of using infected viewpoint in heterogeneous populations with complex social structures and behavior changes. I will describe the basic methodology for creating and analyzing these models for the spread of Ebola and vector-borne diseases.
Vector-borne pathogen and host evolution in a structured immuno-epidemiological system
Arizona State University
Vector-borne disease transmission is a common dissemination mode used by many pathogens to spread in a host population. Similar to directly transmitted diseases, the within-host interaction of a vector-borne pathogen and a host's immune system influences the pathogen’s transmission potential between hosts via vectors. Yet there is much less theoretical studies on virulence-transmission tradeoffs and evolution in vector-borne pathogen-host systems. Here we consider an immuno-epidemiological model that links the within-host dynamics to between-host circulation of a vector-borne disease. On the immunological scale, the model mimics antibody-pathogen dynamics for arbovirus diseases, such as Rift Valley Fever and West Nile Virus. The within-host dynamics govern transmission and host mortality and recovery in an age-since-infection structured host-vector-borne pathogen epidemic model. By considering multiple pathogen strains and multiple competing host populations differing in their within-host replication rate and immune response parameters, respectively, we derive evolutionary optimization principles for both pathogen and host. Invasion analysis shows that the R_0 maximization principle holds for the vector-borne pathogen. For the host, we prove that evolution favors minimizing case fatality ratio (CFR). These results are utilized to compute host and pathogen evolutionary trajectories, and to determine how model parameters affect evolution outcomes. We find that increasing the vector inoculum size increases the pathogen R_0, but can either increase or decrease the pathogen virulence (the host CFR), suggesting that vector inoculum size can contribute to virulence of vector-borne diseases in distinct ways.
Anomalous Diffusion in Biological Fluids
Scott A McKinley
New Orleans, Louisiana
The last twenty years have seen a revolution in tracking data of biological agents across unprecedented spatial and temporal scales. An important observation from these studies is that path trajectories of living organisms can appear random, but are often poorly described by classical Brownian motion. The analysis of this data can be controversial because practitioners tend to rely on summary statistics that can be produced by multiple, distinct stochastic process models. Furthermore, these summary statistics inappropriately compress the data, destroying details of non-Brownian characteristics that contain vital clues to mechanisms of transport and interaction. In this talk, I will survey the mathematical and statistical challenges that have arisen from recent work on the movement of foreign agents, including viruses, antibodies and synthetic microparticle probes, in human mucus.