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And your last Biology course was? when? Many of us have little (or woefully outdated) background in the biological sciences; however the list of enrollees in our courses will include students whose primary interests are in biology, environmental science, the allied health sciences, and so on. In the last decade or so, there has been a great deal of energy directed to the transformation of life science education through the integration of mathematical and computational content ? but how will we implement these approaches in the mathematics classroom? Can we find common ground (or common curriculum!) with our colleagues in the biological sciences?
The purpose of this short course is to introduce participants to a range of current topics in mathematical biology. Moreover, mathematical biology has exploded in recent years, developing new perspectives on both parent disciplines by combining biological and mathematical ideas and tools in sometimes unexpected ways. So we also hope that this short course will begin a continuing conversation on how we might integrate such modern applications into the undergraduate mathematics program.
This Short Course will consist of seven invited presentations taking place over two days. Each presenter will discuss his/her own research and offer suggestions as to how the topic might be included in various mathematics and/or biology courses. We will conclude with a group discussion of the challenges and opportunities of implementation in the undergraduate mathematics curriculum. All participants will be encouraged to suggest particular ways the short course topics might be incorporated into new or existing courses and how we might build a platform for further conversation with colleagues in the life sciences. We expect a lively discussion!
The thesis of my presentation is that student preparation in undergraduate math classes can be enhanced through the use of topics that the vast majority of students have already had personal experience with, but perhaps have not viewed as being connected to mathematics. It is natural to utilize biological examples that students already have some intuition about, that do not require a great deal of detailed biological knowledge to comprehend and yet about which we learn something new from the application of mathematical and computational thinking. Music is one such area, and as a starting point, I will illustrate how the basics of sound relate to student's everyday experiences, including the "art" of mixing various inputs to create a whole. This illustrates trigonometric functions in everyday experience, as well as transform methods for power spectra, and the non-linearity of human sensory response. Pharmacokinetics, the variety of arguments regarding the "cost" of sex and the evolution of recombination, and models for marriage "success' provide other illustrations that can be developed in courses at many levels of undergraduate mathematics.
Almost all of us have been connected to someone who has dealt with cancer. Many of us have wondered how we can help. We have raised money, run races, and supported family members. This presentation will focus on how undergraduates have been an integral part of several studies that use optimal control techniques to reduce cancerous tumors, at least theoretically. The work is associated with breast and ovarian cancers as well as neuroblastoma. Undergraduates who can differentiate can help investigate optimal control strategies to effect change in these and other medical situations.
One of the best ways to engage students in mathematics is to relate math to something that they?re already interested in. An even better way is to get them to make that connection. Mathematical neuroscience offers many stories that are easy to tell ? like why coffee keeps you awake and why your mom was right: you shouldn?t snack before dinner. I will discuss some accessible examples using math to explain how daily activities affect our brain and how such examples can be used to prompt students to raise questions from their own area of biological interest. Getting the students to offer the suggestion alleviates the need for the faculty member to have broad expertise in biology. It is particularly effective to then formulate the question into a project consisting of a handful of questions that the student will address, possibly in a group context. These questions can be tuned to an appropriate mathematical level, from calculus to differential equations to bifurcations. Challenging the student to work through a self-proposed connection can even convert a bio student into a math-bio student.
Phylogenetic tree reconstruction, ancestral trait estimation, and the modeling, analysis and reconstruction of biological networks (regulatory, metabolic, and signaling) pose fundamental biological problems that can provide real-life context for a host of mathematical topics commonly covered in a traditional undergraduate mathematics curriculum beyond calculus. ?Algebra? of many sorts arises, including Boolean algebra, linear algebra, polynomial algebra, and abstract algebra, mixed and mingled with geometry, combinatorics, probability and statistics, and additional topics in discrete mathematics. Motivated by current research questions associated to trees and networks, we will consider a sample of some classroom implementations of these ideas. This will include some joint work, with Raina Robeva, developing related curriculum modules with a project-based learning approach.
Mathematical biologists have a keen interest in the dynamics of networks, such as neuronal, biochemical, gene regulatory networks, or food webs. For example, the nodes in neuronal networks represent neurons and the arcs indicate synaptic connections. In food webs, the nodes represent species and the arcs indicate predator-prey relations.
In this presentation we will discuss the kind of choices one needs to make in modeling the dynamics of such networks. In particular, we will focus on two types of models, ODE models and difference equations with finite state spaces. The examples presented will be suitable for incorporation into the undergraduate classroom. Some of them will demonstrate that the same mathematical model can apply to several seemingly unconnected biological phenomena. Other examples will illustrate conditions under which coarse-grained difference equation models reliably represent the dynamics of more detailed ODE dynamics, which is the focus of the presenter's current research.
We focus on how to get students interested in the mathematical biology of games and on introducing them to solution methods that involve simple calculus and computational algorithms. With these goals in mind, I will provide a sampler of some recent and ongoing work using evolutionary game theory to address two categories of biological games: sex allocation and contests over resources. Under the sex allocation heading are (1) an analysis of the mating system of a simultaneous-hermaphrodite seabass, (2) sex ratios in a parasitoid wasp, and (3) effects of pollen limitation on sex distributions in small plant populations. Some contests over resources include (1) brothers and unrelated mice seeking access to a female, (2) public goods games between social groups of fish or insects, and (3) root competition that can generate a Tragedy of the Commons.
Recent increases in reported outbreaks of vector-borne diseases throughout the world have led to increased interest in understanding and controlling epidemics involving transmission vectors. Ticks have very unique life histories that create epidemics that differ from other vector-borne diseases. I will discuss two approaches for modeling for the lone star tick (Amblyomma americanum) and the spread of human monocytic ehrlichiosis (Ehrlichia chaffeensis). Additionally, I will discuss the development of experiments to identify model parameters and the results of such experiments.
All participants will be encouraged to suggest particular ways the short course topics might be incorporated into new or existing courses and how we might build a platform for further conversation with colleagues in the life sciences. We expect a lively discussion!