Fall 2020 – Summer 2021 – Abstracts

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Thursday, April 29, 2021.  Mathematics Graduate Seminar.

Matthew D’Angelo, Mathematics MA student.
“Evolutionary Game Theory and Population Dynamics”
4:00 – 4:50 pm, via videoconference

Abstract: The second talk in the series will demonstrate how to use any symmetric normal-form game as the basis for a model of population dynamics. The resulting model will be defined by a system of ordinary differential equations (ODEs), and the trajectories will represent the changing state of the population over time. The replicator equation is used to construct this ODE system, which incorporates the principles of natural selection and survival of the fittest. Concepts such as Nash equilibria and evolutionarily stable states can be defined in much the same way as before, and can provide insight into the dynamics of the model. Explicit connections will be made between the strategy space and the population state space, both of which will take the form of a simplex. We will illustrate this process using well-studied games such as Rock-Paper-Scissors and Hawk-and-Dove as examples. These examples will be accompanied by simulations of the phase portraits using MATLAB. Finally, several propositions and theorems will be discussed showing how game theory concepts correspond to important features of the dynamical system, such as rest points, Lyapunov stable points, and asymptotically stable points.

Wednesday, April 21, 2021.  Mathematics Thesis Defense.

Matthew Ackley, Mathematics MA student. (Advisor: Peter Stechlinski)
“Lexicographic sensitivity functions for nonsmooth models in mathematical biology”
3:30 – 4:20 pm, via videoconference

Abstract: In systems of ordinary differential equations (ODEs) whose right-hand-side rules are at least once continuously differentiable, classical sensitivity theory is well-established and allows the quantification of a given system’s most influential parameters. Recent developments in nonsmooth analysis, called lexicographic and lexicographic directional derivatives, have yielded analogous sensitivity theory for a much broader class of ODEs whose right-hand-side rules may not be continuously differentiable. Equipped with this lexicographic sensitivity theory, we investigate two models in mathematical biology which exhibit nonsmoothness (and hence invalidate any conclusions reached via classical sensitivity theory). The first is the “minimal model” of glucose-insulin kinetics, in which we investigate the sensitivity of the concentrations of both substances to parameters describing bodily functions in the glucose-insulin subsystem. In the original formulation of this model, nonsmoothness arises from pancreatic behavior as insulin begins to be secreted after a blood-glucose threshold is crossed; after analysis of this formulation, we consider two novel variants which are meant to model the kinetics in individuals afflicted with type 1 diabetes and contend with the appearance of more nonsmoothness. The second model we investigate describes the spread of riots, which is considered to be a “social contagion” that spreads in a wave similar to those associated with the spread of disease. In this model, nonsmoothness is introduced through the increased rate at which susceptible individuals join a riot after the riot grows to a threshold size. With aims of determining the most influential mechanisms within the dynamics of both of these models, we derive and simulate the lexicographic sensitivity functions associated with each and investigate the results.

Wednesday, April 14, 2021.  Mathematics Graduate Seminar.

Matthew D’Angelo, Mathematics MA student.
“An Introduction to Evolutionary Game Theory”
3:00 – 3:50 pm, via videoconference

Abstract: The first of a series of two talks will focus on introducing some basic game theory concepts.  The main object of study will be symmetric, normal-form games, which are games defined by a payoff matrix.  Pure and mixed strategies, represented as points on the simplex, will be explored, including special classes of strategies, such as Nash equilibria and evolutionarily stable strategies, and their relations.  We will illustrate these concepts using games such as Rock-Paper-Scissors or Hawk-and-Dove as examples.  Important results will be discussed, such as conditions guaranteeing the existence of a Nash equilibrium for a given game.  This talk sets the stage for the second talk in the series, which will use game theory concepts to build a framework for modeling population dynamics.

Wednesday, March 24, 2021.  Mathematics Graduate Seminar.

Matthew Ackley, Mathematics MA student.
“Quantifying parametric dependence in nonsmooth systems of ordinary differential equations”
4:00 – 4:50 pm, via videoconference (click to join)

Abstract: Systems of ordinary differential equations (ODEs) are a common framework for mathematical models in engineering and biology; however, different choices for parameter values can cause ODE solutions to exhibit drastically different behavior over time. Classical forward-parametric sensitivity functions are capable of providing a local measure of a model’s predicted reactivity to parametric perturbations about some particular reference values and are hence useful when comparing parametric influence. Even so, the existence of these functions requires the ODEs at hand to be at least once continuously differentiable, which is restrictive when trying to model phenomena exhibiting hybrid or switching behaviors. In this talk, we will discuss recently-developed lexicographic sensitivity functions, which are based on lexicographic calculus and analogous to the classical functions but applicable to a much larger class of functions which are not necessarily . Relevant background theory in nonsmooth analysis will be given and illustrative examples discussed.