This Nebraska REU in Applied Mathematics is an eight-week summer research opportunity for ten students offered by the Department of Mathematics at the University of Nebraska-Lincoln under a grant from the National Science Foundation. The university campus is right on the edge of the downtown area. Lincoln is a town of about 235,000 people that is home to the University of Nebraska and the state government. The University of Nebraska-Lincoln is the state's major research university with 22,000 students whose mission is research, teaching, and service. The Department of Mathematics has major research groups in applied mathematics, algebra, and analysis and a national reputation for excellence in education.
2007-2012
2012
Discrete Modeling of Biological Systems
Project Mentors
Professor Alan Veliz Cuba (Department of Mathematics) and Eric Eager (Graduate Student, Department of Mathematics)
Project Description
The REU students studied how the design of experiments affects the network inference problem, which is to use the data of a dynamical system to infer the network. The goal was to determine how the generation of the initial data affects the inferred network.
We compared two ways of generating data:
- by running the system as long as possible for few initializations (few long timecourses);
- running the system for short periods of time, but many initializations (many short timecourses).
We did this for different published models of gene regulatory networks.
The question is, given a fixed sample size, is it better to have few long timecourses or many short timecourses? If the sample size is small, then generating data using few long timecourses will produce a better inferred network. However, when the sample size increases, there is a point where generating data using many short timecourse is better. This is an interesting result, because it was not expected for the sample size to matter so much. The students presented a poster at the 2013 JMM.
Quantum Calculus
Project Mentors
Professor Allan Peterson (Department of Mathematics) and Tanner Auch (Graduate Student, Department of Mathematics)
Project Description
The REU students extended many results known in the q-calculus (quantum calculus, which has very important applications in quantum theory). This includes a q-Laplace transform, which has several similar properties to the continuous Laplace transform.
They have a large number of results on the properties of their q-Laplace transform. They have a clever definition of exponential growth of a function and with their definition are able to show regions of convergence of the q-Laplace transform in the numerous formulas that they derive. They use their q-Laplace transform to show how they can solve the so called $q$-difference equations. Finally, they use their q-Taylor monomials to obtain a Variation of Constants Formula.
The students presented a poster at the 2013 JMM.
Peridynamics Models in Heat Conduction and Elasticity
Project Mentors
Professor Petronela Radu (Department of Mathematics), Professor Mikil Foss (Department of Mathematics), and Solomon Akesseh (Graduate Student, Department of Mathematics)
Project Description
The students worked in the area of peridynamics, a theory introduced by S. Silling that has been successful in modeling phenomena in materials with discontinuities, or propagation of fractures. We establish connections between classical local operators and their nonlocal counterparts, and identify convergence rates for these models.
The students also derived a nonlinear diffusion model in the nonlocal framework of peridynamics, following ideas of Bobaru and Duangpanya. For the case when the conductivity is time dependent a fundamental solution for the nonlocal problem was derived, and they we proved an exponential decay rates by using energy methods and a nonlocal version of the Poincare’s inequality.
Both papers present numerical simulations that illustrate estimates for the solution in the quasilinear case as well as in the case of time dependent conductivity.
2011
Metapopulation Modeling and Analysis
Project Mentors
Professor Richard Rebarber (Department of Mathematics), Professor Brigitte Tenhumberg, and Eric Eager (Graduate Student, Department of Mathematics)
Project Description
This group learned about modeling demographic stochasticity with Markov models. They also learned about density dependent deterministic models. They constructed models for a metapopulation using both approaches, and compared the predictions for each model, in order to determine the effect of demographic stochasticity on the dynamics of the metapopulation. This involved mathematical analysis and numerical simulation.
The students presented a poster on their research at the Joint Meetings in 2012.
Discrete Modeling of Biological Systems
Project Mentors
Professor Alan Veliz Cuba (Department of Mathematics) and Michael Uhrig (Graduate Student, Department of Mathematics)
Project Description
The students studied a continuous model for Th-cell differentiation based on a previously published discrete model. They produced the most general (as of 2012) result about steady states of continuous of discrete models in biology. Roughly speaking, the results states that if a continuous model is "sigmoidal enough" and has the same "qualitative features" of a discrete model, then there is a one-to-one correspondence between their steady states.
The students produced code to transform a discrete model into a continuous model. The students used the code to create and analyze a new continuous model of Th-cell differentiation, and show that this continuous model does predict the three main types of Th-cells.
The students have presented this work at two conferences.
Fractional Calculus
Project Mentors
Professor Allan Peterson (Department of Mathematics) and Michael Holm (Graduate Student, Department of Mathematics)
Project Description
The students studied the discrete nabla fractional calculus. They defined the appropriate Laplace transformation and proved results for this Laplace transform. Included in these results are the formulas for the Laplace transforms of the fractional nabla differences of a function.
They used their results to verify laws concerning the composition of different order fractional derivatives and prove a generalized power rule. They also showed how their results can be used to solve fractional differential equations. Furthermore, they also defined the nabla fractional Taylor monomials, and found the discrete nabla Laplace transform of the discrete three-parameter nabla Mittag--Leffler function. They used these results to develop important variations of constants formulas for initial value problems for a fractional difference equation.
They presented a poster at the Joint Meetings in 2011.
2010
Life History of Plants
Project Mentor
Professor Glenn Ledder (Department of Mathematics)
Prerequisites
An introductory course in ODEs. Familiarity with a scientific programming language is also desirable. All relevant biology will be presented during the early weeks of the project.
Project Description
Various life history strategies are observed among plants. While most plants can be classified as annual or perennial, there are variations on these themes. Annual plants differ in timing, with some sprouting in the spring and others in the fall. Those that sprout in the fall may set seeds in the fall or overwinter and set seeds the following spring. A small number of annual plants flower in both the fall and the spring. Some perennials reproduce every year, while others save resources for occasional reproduction binges. The distribution of resources between roots and shoots can be adapted to the individual plant's micro-environment. Leaves can be delicate or sturdy, long-lived or short-lived, defended against herbivory by toxic chemicals or not. In all cases, the principle of natural selection suggests that a plant's actual life history is approximately the optimal life history for its environment and ecological niche, subject to limitations in genetic variation. Various mathematical models have been used to determine plant fitness in terms of life history parameters and schedules and to determine the optimal life history for different scenarios, but there is a lot of room for new work.
The students who work on this project will learn some of the models and methods that have already been developed for this area, identify a new feature to include in a model or a new scenario to study, construct an appropriate mathematical model, and analyze the model to see what biological phenomena it predicts. Analytical work may include methods of control theory, optimization and/or dynamic programming, with some scientific computation.
Calculus on Time Scales
Project Mentors
Professor Allan Peterson (Department of Mathematics)
Prerequisites
An introductory course in ODEs.
Project Description
The concept of time scales unifies and extends discrete time and continuous time. In this project the students will study calculus and dynamical systems on time scales, which is a natural extension of the more familiar calculus and differential equations in continuous time which the students have already studied. The focus of the project will be on fractional derivatives and fractional differential equations, and their generalization to time scales. Fractional differential equations have applications in numerous diverse fields, including electrical engineering, chemistry, mathematical biology, control theory and the calculus of variations. For instance, the fractional calculus may provide more mathematically accurate epidemic models.
The students will first learn about fractional derivatives and fractional differential equations in continuous time. They will then learn about the fractional calculus on arbitrary time scales. This is a very new topic, and hence has many new research areas to explore.
New Models for Heat Conduction and Elasticity in Structures with Cracks
Project Mentors
Professor Petronela Radu (Department of Mathematics) and Professor Florin Bobaru (Engineering Mechanics)
Prerequisites
An introductory course in differential equations; familiarity with a scientific programming language is also desirable.
Project Description
The classical equations for heat and mass transfer are not well suited for structures where discontinuities (like cracks) appear. An example of such a phenomenon would be the melting of an iceberg. Similarly, the classical theory cannot be applied directly in the formation of cracks in an elastic body subject to an external force.
The very new area of solid mechanics called Peridynamics offers a framework in which one can formulate and investigate mathematical models that take into account the breakage of bonds between different parts of a body.
Students will learn about mathematical modeling of such structures using the general theory of peridynamics, and then investigate analytically and numerically properties of a perdynamic model for a body with evolving material discontinues.
2009
Reconstruction Problems in Graph Theory
Project Mentor
Professor Stephen Hartke (Department of Mathematics) and Student Mentor, Derrick Stolee (Department of Mathematics)
Prerequisites
There aren't many prerequisites in terms of material studied, but students must have had a proof-based course. Background in discrete mathematics (such as graph theory and combinatorics) will be beneficial.
Project Description
Graph theory is a very broad area focusing on the mathematical study of networks. We will work on studying the question of graph reconstruction: Suppose that G is a fixed but unknown graph. What information about pieces of G would allow you to "reconstruct" G exactly? One famous conjecture is that G can be reconstructed from the collection of subgraphs formed by deleting one vertex from G. We will study variants of the question when extra information is also provided.
Differential/Difference Equations
Project Mentors
Professor Allan Peterson (Department of Mathematics) and Student Mentor, Chris Ahrendt (Department of Mathematics)
Prerequisites
A calculus sequence and a one-semester course in Differential Equations.
Project Description
We will be concerned with problems that come up in differential equations and difference equations. One of our interests will be to see how these two theories can be unified and extended. For example, calculus is very useful in differential equations and when one studies difference equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general time scales calculus (which for example also includes quantum calculus). Consequently we get a generalization of difference equations and differential equations to so-called dynamic equations on time scales. A simple application of dynamic equations on time scales is a population model which is discrete in season, dies out in winter, while their eggs are incubating or dormant, and, in season again, when hatching gives rise to a overlapping population. Other potential areas of applications include engineering, biology, economics and finance, and mathematics education. Currently there are about 300 researchers worldwide who have published about 500 research articles in the area of dynamic equations on time scales.
Research Activities
Previous REU teams have studied the so-called nabla exponential function (which is a generalization of the usual exponential function) oscillation of a Euler--Cauchy dynamic equation, oscillation of factored dynamic equations, first order dynamic equations, the Henstock--Kurzweil delta integral, stability theory, and existence and nonexistence of periodic points, and the asymptotic behavior and stability of the exponential function on certain periodic time scales. We will work on similar projects this summer. Some of our time will be devoted to studying difference equations -- no prior knowledge of difference equations is required. As part of our REU experience this year we will spend eleven days (travel, food, and lodging provided) at the University of Wyoming Rocky Mountain Mathematics Consortium Summer Conference, attending introductory lectures that will be useful for your research project.
Stabilization of Underactuated Mechanical Systems
Project Mentors
Professor Mikil Foss (Department of Mathematics) and Student Mentor, Joseph Geisbauer (Department of Mathematics)
Prerequisites
The equivalent of a three-semester sequence in college Calculus and a one-semester course in Differential Equations. Familiarity with a symbolic programming language such as MAPLE would be useful but is not a requirement.
Project Description
First, a bit of jargon, which will be illustrated by an example: A mechanical system is called underactuated if it has fewer actuators than degrees of freedom. A degree of freedom is called actuated if there is some device that can directly influence the system with respect to that degree of freedom. An example of an underactuated system is a person balancing a broom stick in the horizontal palm of their hand. This system has five degrees of freedom: three for the position of the hand in space and two that describe the angle of the broom relative to the horizontal palm. The person has direct control of the hand's position coordinates in space, so these coordinates are the actuated degrees of freedom. The angles for the broom stick, however, can only be indirectly influenced by the motion of the hand, and these angles are the unactuated degrees of freedom. Underactuated systems are quite prevalant; some other examples include spacecraft, aerial rockets, underwater vehicles, vertical takeoff aircraft,satellites, hovercraft, and ship-to-shore cargo transport cranes. A basic problem for these systems is to design a controller that can take the system from an initial state and steer the system to a state of equilibrium. Such a controller is called a stabilizing control law. In general, the system of differential equations governing the dynamics of these mechanical systems are nonlinear, and methods from nonlinear control theory are required for the controller design. The objective of this project is to introduce nonlinear systems and control theory and to develop and implement a general strategy for producing stabilizing control laws for underactuated mechanical systems. Some of the components of the project will involve deriving mathematical models and using numerical methods to design stabilizing control laws and simulate system responses.
2008
Mathematical Models of Neurons
Project Mentor
Professor Bo Deng (Department of Mathematics)
Prerequisites
Three-semester sequence in college Calculus, plus one-semester in Differential Equations. Familiarity with linear electrical circuits from any first course on ordinary differential equations is a plus but not required. Having some experience with Matlab is also a preferable option.
Project Description
Neurons are the fundamental building blocks of animal nervous systems. They are the basic units as information encoders and decoders in the communication system that we think the central nervous system is. In this project, students will learn how to construct a mathematical model for neurons, learn how to use phase plane analysis to understand some spiking mechanisms of the model, and learn to use numerical simulation to explore some known and unknown features of the model.
Matrix and Integral Models for Predicting Population Dynamics
Project Mentors
Professor Richard Rebarber (Department of Mathematics), Professor Brigitte Tenhumberg (School of Biological Sciences)
Prerequisites
Introductory course in matrix theory. Some introductory mathematical analysis and familiarity with Matlab or Maple would also be helpful, but are not required.
Project Description
Matrix and Integral Models: Many topics in ecology involve predicting the behavior of populations. A typical approach to modeling a population is to use Population Projection Matrices. In such models the population is grouped into a small number of stages, for instance categorized by age or by size. This defines a population vector, which is a function of time, where time is measured discretely. The life history parameters describing rates of survival, growth and reproduction are incorporated into the projection matrix. If these rates are mainly determined by a continuous variable such as size, the decomposition of the population into a small number of classes can be inadequate. In these cases it would be better to classify members of the population along a continuum of stages. The population vector is then replaced by a function, and the matrix system is replaced by an Integral Projection Model. The eigenvalues of this matrix (or integral operator) provide predictions for the long-term steady-state behavior of the population. Other properties of the matrix provide predictions for the transient behavior, which is the short-term deviation from this steady state.
Research Activities
You will learn how to derive matrix and integral model using basic theoretical principles. We will consider how to choose a model which best matches experimentally determined data sets; we are especially interested in how the models match the transient dynamics. We will perform numerical experiments to address the question of whether a matrix model is close to an integral model when the matrix model uses a large number of stages, and work to mathematically prove our conjectures. An important task in modern ecology is to predict what happens to the population dynamics when the transition rates are changed; this could happen due to uncertainty in estimating model parameters, a disturbance of the model (e.g. climate change), or population management strategies. We will address this question numerically and mathematically.
The direction this project takes will be partly determined by current research developments. This field is inherently interdisciplinary, and will be approached from both a mathematical and a biological perspective.
Differential/Difference Equations
Project Mentors
Professor Allan Peterson (Department of Mathematics) and Chris Ahrendt (Graduate Student, Department of Mathematics)
Prerequisites
Calculus sequence and one semester course in Differential Equations.
Project Description
We will be concerned with problems that come up in differential equations and difference equations. One of our interests will be to see how these two theories can be unified and extended. For example calculus is very useful in differential equations and when one studies difference equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general time scales calculus (which for example also includes quantum calculus). Consequently we get a generalization of difference equations and differential equations to so-called dynamic equations on time scales. A simple application of dynamic equations on time scales is a population model which is discrete in season, dies out in winter, while their eggs are incubating or dormant, and, in season again, when hatching gives rise to a overlapping population. Other potential areas of applications include engineering, biology, economics and finance, and mathematics education. Currently there are about 250 researchers worldwide who have published about 400 research articles in the area of dynamic equations on time scales.
Earlier REU teams have studied the so-called nabla exponential function which is a generalization of the usual exponential function, oscillation of a Euler--Cauchy dynamic equation, oscillation of factored dynamic equations, first order dynamic equations, the Henstock--Kurzweil delta integral, stability theory, and existence and nonexistence of periodic points.
Future projects similar to the ones mentioned above will be considered. Some of our time will be devoted to studying difference equations and no prior knowledge of difference equations is required.
2007
Modeling in Ecology
Professor Glenn Ledder, gledder@math.unl.edu
Prerequisites: a course in linear algebra, a lot of intellectual curiosity, and some background in one of the following: advanced calculus, real analysis, ecology, and/or programming with Maple, Matlab, or Mathematica. (It would be helpful to have an advanced calculus course and some experience with computational software such as Maple, Matlab, or Mathematica. Some background in ecology would also be helpful.)
Project Description: The simplest discrete model for growth of a population is the exponential growth model, in which the population at time t+1 is taken to be the product of the population at time t with a constant growth factor. Such a model can be made more realistic by making the growth factor decrease as the population becomes more dense. This makes the model nonlinear, however, which makes it more difficult to analyze.
A more realistic discrete model arises from the division of a population into age classes, with separate linear growth equations for each class. The resulting model is the vector equivalent of the scalar exponential growth model. One is again motivated to make the model more realistic by making one or more of the parameters dependent on the population density. As in the scalar case, this also makes the model nonlinear. We will study models of this general class and identify a research problem involving such a model.
Differential/Difference Equations on Time Scales
Professor Allan Peterson, apeterso@math.unl.edu
We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example, calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently, we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite. Details
Graph Theory
Professor Jamie Radcliffe, jradclif@math.unl.edu
Prerequisites: There aren't many prerequisites in terms of material studied, but it would be very beneficial to have had a proof based course. Background in linear algebra and discrete mathematics will be beneficial.
Project Description: Graph theory is the mathematical study of networks. This is an extremely broad area which has applications in a wide variety of networks. We will work on studying some graph polynomials. A graph polynomial is a polynomial that we associate with a graph that captures some important information about its structure. In particular, we will think about the interlace polynomial, a relative newcomer to the world of graph polynomials. There are several interesting open questions concerning the interlace polynomial that we will work on.
2002-2006
2006
Differential/Difference Equations on Time Scales:
Professor Allan Peterson, apeterso@math.unl.edu
We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example, calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently, we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite. Details
Effects of Cannibalism on Ecosystems:
Professor Glenn Ledder, gledder@math.unl.edu
Prerequisites: a thorough understanding of the material in an introductory course in differential equations and a basic course in linear algebra. (It would be helpful to have an elementary analysis course and some experience with computational software such as Maple, Matlab, or Mathematica).
Project Description: Simple differential equation models for two interacting species have been known since the 1920's. Recent advances in mathematical techniques and computer software have brought about the opportunity to study more complicated systems of interacting populations. Most of the research to date in this area provides more questions than answers. Many of these questions are fascinating and quite accessible using basic tools in dynamical systems and mathematical modeling.
We will begin by identifying an ecologically realistic scenario in which cannibalism appears to be present. One particularly interesting possibility is provided by bass, which under certain circumstances can subsist entirely on a diet of bass larvae, and whose biological characteristics are well known. There are also some kinds of snails in which the first ones to hatch enjoy rapid initial growth by eating an egg that has not yet hatched.
We'll spend our first week studying mathematical methods for population dynamics and doing a literature survey. After that, we will develop one or more mathematical models based on general biological principles that seem to be valid for our chosen scenario. Questions will arise as we attempt to characterize the behavior of the models. Ultimately, we will try to identify environmental and physiological reasons why cannibalism is viable for some species and not for others.
The Spreading of Information and Social Interactions:
Professor Steve Dunbar, sdunbar@math.unl.edu
Prerequisites: Introductory knowledge of probability and matrix algebra. Some knowledge of differential equations and difference equations would also be helpful, but is not strictly required. Familiarity with Matlab, Maple or Mathematica would also be helpful, but is not required.
Project Description: In this project we will apply techniques from the subject of Markov processes to model social dynamics and associated random models in populations. The emphasis will be on modeling, simulation, visualization and analysis of a stochastic process and quantities associated with the process. The mathematical analysis of the Markov processes will rely on "first-step analysis" and limiting arguments resulting in difference and differential equations. A population of individuals can randomly mix and mingle and exchange "information". Depending on how the members exchange "information", the result can model the transmission of a rumor, a social attitude, or an epidemic. A basic question is to deduce whether there are universal trends, such as whether there is always a fraction of the population which is untouched by the rumor, attitude or disease. Another basic question is to determine the distribution of important characteristics of the process, such as duration. Finally, even simple models contain rich information, and multiple views and statistics of the processes help to explain and enlighten.
2005
Control Theory Techniques Applied to Biological Population Problems
Advisors: Professors Richard Rebarber, Andrew Tyre, and Brigitte Tehnumbuerg
Prerequisites: Introductory course in matrix theory. Some introductory mathematical analysis and complex analysis would also be helpful, but are not strictly required. Familiarity with Matlab, Maple or Mathematica would also be helpful, but are not required.
Project Description: In this project we will apply techniques from the subject of mathematical control theory to problems in population dynamics. When analyzing how the population of an animal or a plant changes with time, the transition rates of survival, growth and reproduction can be incorporated into a population projection matrix. The eigenvalues of this matrix provide predictions for the behavior of the population. For instance, if the largest eigenvalue modulus, called the spectral radius, is less than 1, then the population decreases no matter what the initial conditions are. An important task in modern ecology is to predict what happens to the population dynamics when these transition rates are perturbed (that is, changed); the perturbations can be described by one or more parameters. This could happen due to uncertainty in the model, or these rates could be intentionally changed in order to get a desired result, such as maintaining an endangered species or eradicating a predator.
Differential/Difference Equations on Time Scales
Advisor: Professor Allan Peterson
Prerequisites: A first course in Differential Equations
We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example, calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently, we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite.
2004
Differential/Difference Equations
Advisor: Professor Allan Peterson apeterso@math.unl.edu
Prerequisite: A first course in Differential Equations.
We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite.
Elemental Dynamics in Biology
Advisor: Professor Irakli Loladze iloladze@math.unl.edu
Prerequisites: well developed calculus techniques together with a thorough understanding of introductory differential equations and basic linear algebra. Some experience with computational software will be helpful. Students will learn and use Mathematica. Some background in ecology is preferred, but most important is a strong interest in ecology and mathematical modeling.
Such diverse issues as interactions among competing populations, effects of globally rising carbon dioxide on plants, or molecular mechanisms of protein synthesis in cells, can be studied by using techniques of ecological stoichiometry - that is modeling mass balance, ratios and flows of chemical elements such as carbon, nitrogen and phosphorus. These elements are fundamental to all life forms.
The REU students will be studying ecological problems and will construct mathematical models that reflect the essence of the underlying biological dynamics. Students will carry out analytical analyses of the models searching for stable equilibria, limit cycles, or chaos, and running numerical simulations using Mathematica. All these mathematical techniques will not be used for their own sake, but instead will be geared toward understanding the biology of the problem.
Game Theory and Population Dynamics
Advisors: Professors Wendy Hines ghines@math.unl.edu, and Jamie Radcliffe jradclif@math.unl.edu
Prerequisites: Standard courses in matrix theory and differential equations.
Since John Maynard Smith's first introduction of the notion of an "evolutionary stable strategy" in 1973, there has been a productive synthesis of game theory and problems of population dynamics. Conflict between individuals in a population can be modeled as a game in which each individual has a preferred strategy that it follows over repeated plays of the game. The individuals' relative success at playing the game affects the makeup of the population in future generations. Sometimes there is a single winning strategy leading to a shared species characteristic, while other times the winning strategy involves a mixture of plays. The latter case leads one to consider dynamical systems based on payoff matrices of games.
In this project, the students will investigage the dynamics associated with a variety of non-zero sum games (the classic example being, of course, Prisoner's Dilemma). The students will need to develop realistic but tractable models of potential strategies for repeated play of these games and then investigage the mathematical and biological consequences.
2003
Differential/Difference Equations
Advisor: Professors Allan Peterson apeterso@math.unl.edu
Prerequisite: A first course in Differential Equations.
We will be concerned with problems that come up in Differential Equations and Difference Equations. One of our interests will be to see how these two theories can be unified. For example calculus is very useful in Differential Equations and when one studies Difference Equations one of the first things that is done is to study an analogous discrete calculus. The discrete calculus and the continuous calculus can be unified in such a way that it includes a more general calculus. Consequently we get a generalization of difference equations and differential equations. We study dynamic equations, where differential equations and difference equations are special cases. This leads to many interesting open questions which we will consider. Any student interested in Differential Equations should enjoy this experience. About a fourth of our time will be spent studying Difference Equations so knowledge of Difference Equations is not a prerequisite.
Dynamics of Interacting Populations
Advisors: Professors Bo Deng and Glenn Ledder bdeng@math.unl.edu, gledder@math.unl.edu
Prerequisite: a thorough understanding of the material in an introductory course in differential equations and a basic course in linear algebra. An elementary analysis course, considerable experience with computational software (Maple, Matlab, or Mathematica), and/or some background in population biology or ecology is preferred.
Simple differential equation models for two interacting species and for interaction of a host species with a disease organism have been known since the 1920's. Recent advances in mathematical techniques and computer software have brought about the opportunity to study more complicated systems of interacting populations. There are a lot of unsolved problems in the dynamics of interacting populations that are accessible to undergraduates armed with some basic tools in dynamical systems and mathematical modeling.
We will develop and study a mathematical model for a problem of current interest in population dynamics. This could be a problem with several interacting species in a food chain or a problem in which the predator-prey dynamics is influenced by a disease of either the predator or the prey. The REU students will be studying mathematical questions, such as whether or not there is a stable equilibrium state, a stable limit cycle, or chaos, and modeling questions, such as whether or not we can tell from current data what the final state of the system will be and how rapidly the system will approach that state.
A key feature of population dynamics is the importance of processes having very different time scales. These differences allow for the use of singular perturbation methods to identify simple components within complex models whose union gives approximately the behavior of the complex system. At the beginning of the project, we will provide tutorials in dynamical systems, (phase plane analysis and singular orbit analysis) and mathematical modeling (common predator-prey and epidemiology models and scaling). The students will apply these skills to some previously solved problems and then use them to study a new problem.
Sampled-Data Control
Advisor: Professor Richard Rebarber, rrebarbe@math.unl.edu
Prerequisites: Introductory courses in differential equations, partial differential equations, and linear algebra. Some complex analysis and some computer programming would also be helpful, but are not required.
Project Description: This project is in the area called control theory, which is a topic in both mathematics and engineering. In a control problem there is an equation which gives the relationship between two functions of time: w(t), which we call the state, and u(t), which we call the control. In the type of control problem we consider, called a stabilization problem, the job is to choose the control u(t) to make the state w(t) decay to 0 as t ! 1. This kind of problem would arise in applications where the goal is to stop a structure (or an electrical circuit) from vibrating.
If we can observe w(t) at all times t > 0, we might be able to do this job by choosing u(t) to be a linear continuous-time feedback of w(t). However, in many real-world applications we cannot observe w(t) for all times, but just at discrete times 0, h, 2h, 3h, . . . ., where h is called the sampling time. In this project we try to determine how to use this sampled data (instead of the continuous-time data) to choose the control u(t) to do the job. One way of doing this is to find a continuous-time feedback control which does the job, then apply a sample-and-hold process to this feedback to obtain a new controller which only uses the sampled data; we call this new feedback a sampled-data controller. We consider the following question: If the continuous-time feedback causes the state to be stable, will the sampled-data controller also cause the state to be stable, provided we use small enough sampling times h?
It seems reasonable that the answer to this question should be "yes", because we would expect that if we sample-and-hold fast enough, then the sampled data controller would look like the continuous time controller; roughly speaking, this is why compact discs and mp3's sound good as long as the sampling time is fast enough. In fact, the answer to this question is indeed yes when the relationship between the state and the controller is given by a linear ordinary differential equation.
When the state and control are related by a linear partial differential equation, the answer is "sometimes yes, sometimes no, often we have no idea." In this project we look at a case where we have no idea: when w satisfies a one-dimensional wave equation, and u is a boundary control. In this case we can easily find a stabilizing continuous-time feedback (essentially the common-sense solution of pushing one end of the wave up when you feel it go down, and pushing down when you feel it go up). We then apply a sample-and-hold to this feedback, and try to determine whether the sampled-data controller also helps stabilize the state, or possibly makes it worse.
2002
Time Scales
Advisors: Professors Allan Peterson and Lynn Erbe apeterso@math.unl.edu, lerbe@math.unl.edu
Prerequisite: a thorough understanding of the material in an introductory differential equations course.
The theory of time scales, which has recently received a great deal of attention because it unifies and extends calculus on discrete as well as continuous domains, is an ideal topic for undergraduate research. Many of the ideas of discrete and continuous calculus, and more generally dynamic equations, may be combined and unified via this more general structure. The notion of generalized exponential, logarithmic, trigonometric, and other special functions have already been given in an abstract setting. Consider the case of the exponential function. In the real case ek(t-a) may be regarded as the solution of the initial value problem x' = kx, x(a) = 1. More generally, the exponential function e(t,k,a) may be regarded as the solution of the dynamic initial value problem xD = kx, x(a) = 1, where xD is the so-called delta derivative of x; xD is a generalization of the standard derivative if the time scale is the real numbers, and is the forward difference operator if the time scale is the integers. If the time scale is the integers, then e(t,k,a) = (1+k)t-a is this exponential function. For a certain time scale this exponential function is a binomial coefficient, for another time scale it is e(t,k,a) = t(ln(a)-ln(t))/2ln(a). There are many other interesting examples to be explored. These exponential functions may arise in the modeling of the growth of certain plant species (e.g., when the time scale is the union of disjoint intervals). Students with a good background in calculus and elementary analysis would be prepared to consider other special cases.
Additional topics for study include derivation of explicit forms of certain well-known differential or integral inequalities, and a study of certain properties of the so-called Hilger complex plane. In this context generalized trigonometric functions may be defined and their properties investigated, in analogy with the real case. The chain rule does not hold for a general time scale. It would be interesting to find conditions on the time scale that ensure that some kind of a chain rule holds.
Chaos Theory in Food-Chain Models
Advisors: Professors Gwendolen Hines and Bo Deng ghines@math.unl.edu, bdeng@math.unl.edu
Prerequisite: a thorough understanding of the material in an introductory course in differential equations. An elementary analysis course or considerable to modest experience in some computational software (such as Maple, Mathematica, or Matlab), is preferred.
Food-chains involving three or more species are fundamental building blocks for ecosystems. While a particular ecosystem may be intractable, simple chains can be modeled by systems of ordinary differential equations. Chaos theory has been found to be an appropriate paradigm within which food-chain models can be analyzed, thus shedding new light on ecocomplexity in general. We have designed an REU project in which the students will discover chaos theory by exploring food-chain models, thus gaining greater appreciation for its usefulness in biocomplexity studies. In this project students will learn how to handle one-dimensional iterative systems, with an emphasis on the sensitivity of the results to initial conditions.
The REU students will work on specific food-chain chaos generation mechanisms. These models have parameters that are variable. The type of chaos, and the way in which it arises, depends upon the relative values of these parameters. The students will determine which ways of varying the parameters will lead to chaos. One way of measuring chaos and sensitivity is to measure the Lyapunov exponent of the system. This cannot be done analytically without analytical forms for the solutions, which are usually unavailable, but it can be done numerically. From a literature search the students will learn existing methods (including ones with which the advisors are currently unfamiliar), and figure out how to adapt these to particular problems. One goal is for the students to write a program that works for a general class of systems.
Since many dynamical systems, including food-chain models, have many different time scales, the students will develop multi-time analysis skills on singularly perturbed systems of equations. These skills are mostly rudimentary and are geometric in nature. Geometric singular perturbation analysis on food-chain models will inevitably lead the students to discover that there are many different types of one-dimensional maps embedded in the models, and that these largely chaotic maps can be analyzed by techniques they have already mastered.
During the project, we will assign readings, including but not limited to the following publications: Introduction to Dynamical Systems, by Alligood, Sauer and Yorke, Complex dynamics and phase synchronization in spatially extended ecological systems, by Bluaius, Huppert and Stone, Yield and dynamics of tritrophic food chains by De Feo and Rinaldi, and Slow-fast limit cycles in predator-prey models, by Rinaldi and Muratori.
Problems in Mathematical Modeling
Advisor: Professor Glenn Ledder, gledder@math.unl.edu
Prerequisites: a thorough grounding in the material in a standard first course in differential equations, a basic course in linear algebra (sometimes called matrix theory), and college-level background in at least one of biology, chemistry, engineering, geology, or physics.
We will develop and study a mathematical model for some problem of current interest in natural science, physical science, or engineering. The specific topic for the mathematical modeling project will be made shortly after the students for the project have been selected. This will allow the students to participate in the topic selection process and to have about two months to do some background reading on the topic. Two broad categories of topics will be considered:
- Population models from ecology or epidemiology, and
- Models of processes involving diffusion, including such diverse areas as heat flow, fluid mechanics, flow in porous media, pharmacokinetics, and digestion processes in animals.
Example topics include:
Many aquatic parasites have a complicated life history that could include stages in which they live in snails as well as stages that live in fish and free-swimming stages. Typically, biologists have a qualitative description of the population dynamics, but no quantitative mathematical model. Based on the qualitative description and ecological data, we could build a mathematical model that should assist biologists in better understanding the population dynamics.
Dynamic energy budget models are used in biology to describe the evolution of physiological state variables such as biomass and stored energy for the life cycle of individual members of a population. It is possible to couple such models to ecological models that describe the interactions between populations in order to construct a general population model based on both physiological and ecological principles. In this project, the student will develop and study a mathematical model to predict variations in the production of grasshopper eggs under a range of environmental conditions including weather differences and differences in predator populations.
The National Weather Service has just implemented a new system for measuring the effect of wind speed on the level of chill experienced by people. The new system is based on a complicated model selected from a variety of proposed models. While it is undoubtedly more accurate than the old system, it is still subject to questions about how wind chill effects ought to be defined. It is likely that equally accurate results could be obtained from a relatively simple model, with the advantage that the simple model would make it easier to examine the consequence of the choice of definition of the wind chill effect.