Applied Mathematics Seminar
The Applied Mathematics Seminar has talks on a wide range of topics, including but not limited to approximation theory and practice, numerical linear algebra, numerical optimization, numerical aspects of computer science, theoretical and applied partial differential equations and their numerical solutions, and mathematical biological models.
For more information contact Hayriye Gulbudak.
Spring 2019
For the Spring 2019 semester we will meet at 3:30 pm on Tuesdays in 208 Maxim Doucet Hall.

22 January 2019
Special seminar in Oliver Hall 
29 January 2019
Understanding species persistence under reoccurring and interacting disturbances
Amy Vebruaskas
UL Lafayette
Abstract: An important focus for management and conservation is determining whether a species or a system of interacting species can sustain itself. This question becomes increasingly important as populations are exposed to various disturbances, both natural and anthropogenic, such as hurricanes, habitat fragmentation, toxicants, and invasive species. Here, we first develop a model to consider the effect of a reoccurring disturbance on population persistence. Disturbances are assumed to occur stochastically according to a twostate Markov chain with their frequency depending on the average length of effect of a disturbance and the average time between disturbances. Using this model, we derive an approximation for the stochastic growth rate that allows us to consider how changes in the parameters describing the disturbance may impact species persistence. This information can be used to help inform management decisions. We then consider case where populations are exposed to multiple types of disturbances. In particular, we examine how the interaction between disturbance types, that is whether the combined effect of the disturbances is greater or less than their additive effects, may alter persistence conditions. 
5 February 2019
Competing Interactions, Patterns, and Traveling Waves in Discrete Systems
Aijum Zhang
Abstract: We consider bistable lattice differential equations with competing first and second nearest neighbor interactions. We construct heteroclinic orbits connecting the stable zero equilibrium state with stable spatially periodic orbits of period p=2,3,4 using transform techniques and a bilinear bistable nonlinearity. We investigate the existence, global structure, and multiplicity of such traveling wave solutions. For smooth nonlinearities an abstract result on the persistence of traveling wave solutions is presented and applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions. 
12 February 2019
open 
19 February 2019
SIMM seminar in Oliver Hall 
26 February 2019
Paul Salceanu 
5 March 2019
Mardi Gras break  no meeting 
12 March 2019
Anna Jolles, Associate Prof. at Oregon State University 
19 March 2019
Subedi Subhash 
26 March 2019
SIMM seminar in Oliver Hall 
2 April 2019
Peter Hinow, Prof. at UW Milwaukee. 
9 April 2019
Jorge Macias Diaz, Prof at Universidad Autonoma de Aguascalientes 
16 April 2019
spring break  no meeting 
23 April 2019
SIMM seminar in Oliver Hall
Fall 2018
For the Fall 2018 semester we will meet at 3:30 pm on Tuesdays in 208 Maxim Doucet Hall.

25 September 2018
Global dynamics of a disease model with both direct and indirect transmissions
XiangSheng Wang
Abstract: We study the global dynamics of a disease model with both direct and indirect transmissions. Though the model system is nonlinear and it couples two transmission mechanisms, the basic reproduction number exhibits a nice linear property: it is simply the sum of two basic reproduction numbers for direct and indirect disease transmissions respectively. We further demonstrate that the basic reproduction number is a threshold parameter which characterizes the local and global dynamics of the model system. The key ideas (as well as main difficulties) in the proof will be explained in detail, and the presentation should be accessible to graduate and undergraduate students who have some basic knowledge of differential equations. 
2 October 2018
Population Dynamics of FisherKPP Equations with Nonlocal Dispersal in Spatially Periodic Environment
Aijum Zhang
Abstract: This talk deals with front propagation dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. We show that a general spatially periodic monostable equation with nonlocal dispersal has a unique spatially periodic positive stationary solution and has a spreading speed in every direction. In this talk, we also show that a spatially periodic nonlocal monostable equation with certain spatial homogeneity or small nonlocal dispersal distance has a unique stable periodic traveling wave solution connecting its unique spatially periodic positive stationary solution and the trivial solution in every direction for all speeds greater than the spreading speed in that direction. In the end, we will discuss some open problems in population dynamics. 
9 October 2018
Picard’s Iterative Method for Caputo Fractional Differential Equations
Rainey Lyons
Abstract: With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz condition. As an application of our method, we have provided several numerical examples. 
16 October 2018
October SIMM in Oliver Hall 
23 October 2018
Continuous Data Assimilation from Scattered Spatial Observations in TimeDependent PDEs
Tong Wu
Tulane University
Abstract: Data assimilation is the task of combining mathematical models with observational data. In this work, we developed a new nonlinear data assimilation algorithm that extends the earlier approaches introduced by Azouani, Olson, and Titi (AOT) based on feedback control penalty for the solution of a system of partial differential equations. We improved the existing AOT algorithm by introducing a dynamic control process. Instead of using a constant parameter for the feedback control, we choose the parameter based on the current local state, the physical dynamic from the partial differential equations and the error estimate of the algorithm, which provides better feedback locally and leads a faster convergence rate comparing with the fixed feedback control. Also, we construct the interpolation from the discrete spatial data using a robust weighted least square interpolation method. The weight for each observation is determined by the distance to the observation location, which can be easily implemented over nonstructured data in higher dimensions. As a proofofconcept for these models, we have tested our algorithm on a number of test problems, including 1D KPPBurgers' equation, 1D KuramotoSivashinsky equation, and 2D shallow water equations. 
30 October 2018
Title to be announced
Diana Paola LizarraldeBejarano
EAFIT University, Medellin, Colombia 
6 November 2018
Prey evolution of toxicant resistance and its effect on the predatorprey dynamics
Istiaq Hossain
Abstract: Continuous exposure to a toxicant may result in the evolution of toxicant resistance in relatively shortlived species. In this study, we investigate the effect of such an evolution of toxicant resistance in the prey population on the overall dynamics of a predatorprey system. We first derive and analyze a discrete time predatorprey model. We establish conditions for the existence and stability (local or global) of various equilibria, as well as conditions for the persistence of both the predator and prey populations. We then extend this model to an evolutionary model by applying the Darwinian evolutionary game theory methodology. This methodology couples the population dynamics with the dynamics of an evolving phenotypic trait, which we assume provides a measure for the level of toxicant resistance developed by the prey. The predator is impacted by prey evolution indirectly, through changes in prey density, and directly, through an assumed tradeoff between toxicant resistance and the ability of the prey to escape predation. We study the dynamics of this model and establish conditions for when the prey is able to evolve toxicant resistance. In particular, we show that the evolution of toxicant resistance may allow both the predator and prey to persist when, without the evolution, both may go extinct. 
13 November 2018
November SIMM in Oliver Hall 
20 November 2018
Interval analysis for the treatment of uncertainty in epidemiological models based on systems of ordinary differential equations
Diana Paola LizarraldeBejarano
EAFIT University, Medellin, Colombia
Abstract: In epidemiological models based on ordinary differential equations systems (henceforth ODEs), knowledge and available information about the model parameters and the initial conditions are limited. This is especially true for models that simulated the transmission of infectious diseases. Also, there is inherent uncertainty in any measurement process. We propose to consider such uncertainty by defining parameters and initial conditions as closed real intervals. After that, we will use the VSPODE (Verifying Solver for Parametric ODEs) solver for parametric ODEs, which produce guaranteed bounds on the solutions of nonlinear dynamic systems with intervalvalued initial states and parameters. On the other hand, to understand the meaning of model fit to interval data, we present the concept of strong compatibility between interval data and the parameters and initial conditions of the nonlinear system. Finally, given a numerical solution of our system and the initial interval data, we formulate a strategy and an optimization problem to find the set of parameters and initial conditions which produce the best model fit to the interval data.
Spring 2018
For the Spring 2018 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

23 January 2018
Changes in population outcomes resulting from phenotypic evolution and environmental disturbances
Amy Veprauskas
Abstract: We develop an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. For this evolutionary model, we use bifurcation analysis to establish the existence and stability of a branch of positive equilibria that bifurcates from the extinction equilibrium when the inherent growth rate passes through one. We then present an application to a daphnia model to demonstrate how the evolution of resistance to a toxicant may change persistence scenarios. We show that if the effects of a disturbance are not too large, then it is possible for a daphnia population to evolve toxicant resistance whereby it is able to persist at higher levels of the toxicant than it would otherwise. These results highlight the complexities involved in using surrogate species to examine toxicity. Time permitting, we will also consider a nonautonomous matrix model to examine the possible longterm effects of environmental disturbances, such as oils spills, floods, and fires, on population recovery. We focus on population recovery following a single disturbance, where recovery is defined to be the return to the predisturbance population size. Using methods from matrix calculus, we derive explicit formulas for the sensitivity of the recovery time with respect to properties of the population or the disturbance. 
6 February 2018
Modeling Distinct Virus Infection Strategies in VirusMicrobe Systems
Hayriye Gulbudak
Abstract: Viruses of microbes, including bacterial viruses (phage), archaeal viruses, and eukaryotic viruses, can influence the fate of individual microbes and entire populations. Here, we model distinct modes of virushost interactions and study their impact on the abundance and diversity of both viruses and their microbial hosts. We consider two distinct viral populations infecting the same microbial population via two different strategies: lytic and chronic. A lytic strategy corresponds to viruses that exclusively infect and lyse their hosts to release new virions. A chronic strategy corresponds to viruses that infect hosts and then continually release new viruses via a budding process without cell lysis. The chronic virus can also be passed on to daughter cells during cell division. The longterm association of virus and microbe in the chronic mode drives differences in selective pressures with respect to the lytic mode. We utilize invasion analysis of the corresponding nonlinear differential equation model to study the ecology and evolution of heterogenous viral strategies. We first investigate stability of equilibria, and characterize oscillatory and bistable dynamics in some parameter regions. Then, we derive fitness quantities for both virus types and investigate conditions for competitive exclusion and coexistence. In so doing we find unexpected results, including a regime in which the chronic virus requires the lytic virus for survival and invasion. 
20 February 2018
Overcoming the addedmass instability for coupling incompressible flows and elastic beams
Longfei Li
Abstract: A new partitioned algorithm is described for solving fluidstructure interaction (FSI) problems coupling incompressible flows with elastic structures undergoing finite deformations. The new algorithm, referred to as the AddedMass Partitioned (AMP) scheme, overcomes the addedmass instability that has for decades plagued partitioned FSI simulations of incompressible flows coupled to light structures. Within a FiniteDifference framework, the AMP scheme achieves fully secondorder accuracy and remains stable, without subtimestep iterations, even for very light structures when addedmass effects are strong. The stability and accuracy of the AMP scheme is validated through mode analysis and numerical experiments. Aiming to extend the AMP scheme to an FiniteElement framework, we also develop an accurate and efficient FiniteElement Method for solving the Incompressible NavierStokes Equations with highorder accuracy upto the boundary. 
6 March 2018
No seminar 
20 March 2018
Rainey Lyons 
27 March 2018
Estimation of Distributed Delays
Temitope Gaudet
Abstract: Delay differential equations have been studied for several decades as they arise in many applications. A common approach is to transform a distributed delay system to a related ordinary differential equation system via the ‘linear chain trick’. This is due to the fact that the term involv ing the distributed delay can be replaced by a state variable that is coupled to other state variables in a linear system of ODEs. This transformation relies on the assumption that the delay follows a gamma distribution. We try to determine when one correctly or incorrectly assumes a gamma distribution and the implications of such assumption by estimating the parameters associated with the distribution followed by a time delay. The results when performing the estimations in the ODE system (this is equivalent to the delay system if a gamma distribution is assumed) are compared to the results in the delay system. 
10 April 2018
XiangSheng Wang 
17 April 2018
Models of Dynamic Virus and Immune Response Networks
Cameron Browne
Abstract: The dynamics of virus and immune response within a host can be viewed as a complex and evolving ecological system. For example, during HIV infection, an array of CTL immune response populations recognize specific epitopes (viral proteins) presented on the surface of infected cells to effectively mediate their killing. However HIV can rapidly evolve resistance to CTL attack at different epitopes, inducing a dynamic network of viral and immune response variants. We consider models for the network of virus and immune response populations. Our analysis provides insights on viral immune escape from multiple epitopes. In the “binary allele” setting, we prove that if the viral fitness costs for gaining resistance to each of n epitopes are equal and multiplicative, then the system of 2^n virus strains converges to a “perfectly nested network” with less than n+1 persistent virus strains. Overall, our results suggest that immunodominance is the most important factor determining viral escape pathway of HIV against multiple CTL populations. To conclude, I briefly discuss ongoing collaborative work to connect the models with intrahost SIV/immune response data and to extend analysis to coevolving virus/antibody populations. 
24 April 2018
To be announced
Fall 2017
For the Fall 2017 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

26 September 2017
An Introduction to Bernstein polynomials
Dun Liu
Abstract: Bernstein polynomials have been playing a crucial role in approximation theory since the early 20th century. They were used to prove the Weierstrass theorem by S. Bernstein in 1910s, and later became the basis of Bézier Curves, which are now widely used in computer graphics to model smooth curves. With Bernstein polynomials, we can approximate a function over a finite domain and refine the approximation to any desired precision. In this presentation, we will look into the definition of Bernstein polynomials, some important properties of the polynomials, the Weierstrass approximation theorem, and Bézier Curves. 
10 October 2017
The dynamics of an ion channel model
XiangSheng Wang
Abstract: We study a time dependent PoissonNernstPlanck system which arises from the model of ion channels. The objective is to understand how the ion concentrations are distributed in the channel if there are ion fluxes on the channel boundaries. Assuming that the Debye length is small relative to the channel length, we derive an asymptotic formula for the dynamic solution by matching outer and Debye layer solutions. It is interesting to note that for the timedependent problem, the outer solution has a boundary layer that does not occur in timeindependent problems. 
31 October 2017
Competitive Exclusion through Discrete Time Models
Paul Salceanu 
7 November 2017
Numerical Algorithms for PDEConstrained Optimization Problems
Jun Liu
Southern Illinois University Edwardsville
Abstract: PDEConstrained optimization problems arise in many different scientiffic and engineering applications. In this talk, we will first present several efficient optimizethendiscretize algorithms for iteratively solving the firstorder optimality KKT system from both parabolic and wave PDEConstrained optimal control problems. Second, we will discuss some interesting numerical issues regarding the discretizethenoptimize algorithms, which are also widely used in practice. Numerical results will be shown to illustrate the effectiveness of our proposed numerical algorithms.
Spring 2017
For the Spring 2017 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

31 January 2017
PoissonNernstPlanck system with multiple ions
XiangSheng Wang
Abstract: We study the PoissonNernstPlanck (PNP) system with an arbitrary number of ion species with arbitrary valences in the absence of fixed charges. Assuming point charges and that the Debye length is small relative to the domain size, we derive an asymptotic formula for the steadystate solution by matching outer and boundary layer solutions. The case of two ionic species has been extensively studied, the uniqueness of the solution has been proved, and an explicit expression for the solution has been obtained. However, the case of three or more ions has received significantly less attention. Previous work has indicated that the solution may be nonunique and that even obtaining numerical solutions is a difficult task since one must solve complicated systems of nonlinear equations. By adopting a methodology that preserves the symmetries of the PNP system, we show that determining the outer solution effectively reduces to solving a single scalar transcendental equation. Due to the simple form of the transcendental equation, it can be solved numerically in a straightforward manner. Despite the fact that for three ions, previous studies have indicated that multiple solutions may exist, we show that all except for one of these solutions are unphysical and thereby prove the existence and uniqueness for the threeion case. 
22 February 2017 (WEDNESDAY 2:30 Maxim Doucet Hall 208)
Directional Statistics for High Volatility and Big Data Science
Ashis SenGupta
Indian Statistical Institute, Kolkata, West Bengal, INDIA and
Augusta University, Augusta, Georgia
Abstract: In this era of emerging complex problems, both small and big data – linear and nonlinear, exhibit challenging characteristics which need to be carefully modelled. Thus, multidisciplinary research in mathematical sciences has become indispensable. Marked presence of asymmetry, multimodality, high volatility, long and fat tails, nonlinear dependency, etc. are common features of contemporary data. Notwithstanding pitfalls, ideas from several disciplines do enrich the contribution of the research work. Directional statistics is one such scientific “key technology” as which on one hand is developed from the conglomeration of the inductive logic of statistics, objective rigor of mathematics and the skills of numerical analysis of computer science. On the other hand, it possesses the richness to handle the need for providing statistical inference to a wide and emerging arena of applied sciences. Directional data (DD) in general refer to multivariate observations on variables with possibly linear, axial, circular and spherical components. Circular random variables are usually those which pertain to observations on directions, angles, orientations, etc. Data on periodic occurrences can also be cast in the arena of circular data. In general, DD may be mapped to smooth manifolds, e.g. circle, hypersphere, hypertoroid, hypercylinder, or to axial and hyperdisc also. Analysis of such data sets differs markedly from those for linear ones due to the disparate topologies between the line and the circle. First, some methods of construction of probability distributions and statistical models for DD on smooth manifolds are presented. The need for applications of these abound for data in a variety of applied sciences. To illustrate this fact, we take up two important problems, one for linear and the other for directional data. With linear data, we take up the problem of obtaining probability distributions for modelling high volatility. The work of Mandelbrot has shown the appropriateness of the stable families of distributions for high volatility. However, in general, these families do not possess any analytical closed form for their probability density functions. This leads to the complexity of inference involving the parameters of such distributions. We overcome this problem of modelling high volatility data by appealing to the area of probability distributions for directional data. A new family of possibly multimodal, asymmetric and heavytail distribution is presented. The usual fattail, Cauchy and t, distributions are encompassed by this family and it has even tails comparable to that of the stable family. The second problem deals with data, possibly Big Data, on smooth manifolds. It is first noted that such data invariably exhibit multimodality and hence the possibility of underlying multiple component distributions. Thus, it would be prudent to “Divide and Conquer”, prior to proceeding for drawing statistical inference on such data. Here we deal with this problem by developing Hierarchical Unsupervised Learning or statistical Clustering techniques for manifold data. We illustrate our approach by a reallife example based on agricultural insurance data. 
4 April 2017 (208 Maxim Doucet Hall)
Estimation of Distributed Delays
Temi Gaudet
University of Louisiana at Lafayette
Abstract: Delay differential equations have been studied for several decades as they arise in many applications. A common approach is to transform a distributed delay system to a related ordinary differential equation system via the ‘linear chain trick’. This is due to the fact that the term involving the distributed delay can be replaced by a state variable that is coupled to other state variables in a linear system of ODEs. This transformation relies on the assumption that the delay follows a gamma distribution. We try to determine when one correctly or incorrectly assumes a gamma distribution and the implications of such assumption by estimating the parameters associated with the distribution followed by a time delay. The results when performing the estimations in the ODE system (this is equivalent to the delay system if a gamma distribution is assumed) are compared to the results in the delay system.
Fall 2016
For the Fall 2016 semester we will meet at 3:30 pm on Tuesdays in 201 Maxim Doucet Hall.

13 September 2016
Traveling wave solutions of a diffusive epidemic model
XiangSheng Wang
Mathematics Department
University of Louisiana at Lafayette
We study the traveling wave solutions of a diffusive epidemic model with standard incidence. The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and a minimal wave speed. Our proof is based on Schauder fixed point theorem and Laplace transform. 
27 September 2016
The Dynamics of VectorBorne Relapsing Diseases
Cody Palmer
Mathematics Department
University of Louisiana at Lafayette
Motivated by TickBorne Relapsing Fever (TBRF) we will be investigating the dynamics of various models for the spread of a relapsing disease by a vector. In particular we quantify the effect that relapses have on the disease spread and the how the number of relapses influence control strategies for the disease. 
4 October 2016
Modeling MultiEpitope HIV/CTL Immune Response Dynamics and Evolution
Cameron Browne
Mathematics Department
University of Louisiana at Lafayette
The CTL (Cytotoxic T Lymphocyte) immune response plays a large role in controlling HIV infection. CTL immune effectors recognize epitopes (viral proteins) presented on the surface of infected cells to mediate their killing. The immune system has an extensive repertoire of CTLs, however HIV can evolve resistance to attack at different epitopes. The ensuing arms race creates an evolving network of viral strains and CTL populations with variable levels of epitope resistance. Motivated by this, we formulate a general ODE model of multiepitope virusimmune response dynamics. Some special cases for the HIV/CTL interaction network are considered, the case of a nested network and the general twoepitope case. We characterize the persistent viral strains and immune response in terms of system parameters and prove global properties of solutions via Lyapunov functions. The results are interpreted in the context of withinhost HIV/CTL evolution and numerical simulations are provided. To conclude, we discuss extensions of the model to a PDE system which incorporates cellinfection age structure. 
11 October 2016
Synchrony and the dynamic dichotomy in a class of matrix population models
Amy Veprauskas
Mathematics Department
University of Louisiana at Lafayette
In this talk, I will discuss the dynamics of a class of discretetime structured population models called synchrony models. Synchrony models are characterized by the simultaneous bifurcation of a branch of positive equilibria and a branch of synchronous 2cycles from the extinction equilibrium. These models exhibit a dynamic dichotomy in which the two steady states have opposite stability properties that are determined by the relative levels of competition in the population. I will also present an application that is motivated by observations of a population of cannibalistic gulls.