Math 437 (Mathematical Biology). Spring 2023

Instructor: Steve Krone

Office: 421 Brink Hall
Office Hours: by appointment

Class Time: 2:30 MWF
Place: TLC 140

Exam 1:

Prerequisite: Math 310 (Ordinary Differential Equations), or some other exposure to differential equations and consent of instructor.

Text: Fred Brauer and Carlos Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2nd ed.

This course combines ideas from two subjects: mathematics (mostly differential equations) and biology. This is an exciting and very active area, with contributions coming from both mathematicians and biologists. We will primarily study qualitative issues, such as modeling (deterministic) biological phenomena, stability of solutions to differential equations, etc. This semester, we will focus on models in ecology and population biology although the same mathematics applies to models in physiology and molecular biology. There is no biology prerequisite and you won't have to dissect any frogs.

The course is suitable for students of mathematics, biology, ecology, physics, and any of the other disciplines that use things like differential equations (see topics below). We focus mostly on nonlinear models; these are the ones that arise in real biological systems. The book for the course is very well written and is full of interesting biological models.

Grading Scheme

Homework ... 25%
Exam 1 ......... 25%
Exam 2 ......... 25%
Project .......... 25%

Homework is due at the beginning of class on the due date.

The "project" involves picking a research article in mathematical biology, working through it (with help from me or others), then telling me about it. This basically means that you will teach me what's in the paper during an informal chat in my office. Plan on a 10 minute presentation and 5 minutes of questions. A good source of references is your text; I can help you find something appropriate. When you have picked one or more possibilities, you must get the paper approved by me (to be sure it has a reasonable amount of math and biology in it). Try to get this done some time in March so you have time to work through it. Your "final discussion" with me should be before THE END OF FINALS WEEK. I encourage you to get this done earlier rather than later; the end of the semester is hectic, as you know. One of the exciting things about this course is that it will quickly prepare you to read a sizeable portion of the research literature.

Course Outline:

1. Continuous-Time Models and Ordinary Differential Equations

  • nonlinear ODE's, steady states, eigenvalues, stability, isoclines, phase plane, limit cycles, oscillations, bifurcation;
  • single-species models (Malthus model, logistic growth, allee effect,...)
  • multiple species (Lotka-Volterra systems, predator-prey, competition, infectious diseases, metapopulations, viruses, chemostats, ...)
  • epidemics

2. Spatial Models and Partial Differential Equations

  • spatial diffusion and PDE's, steady states, spatially uniform steady states, traveling waves;
  • Fisher's equation (spread of an advantageous allele), spatial spread of epidemics.

Additional References:

  • Leah Edelstein-Keshet, Mathematical Models in Biology, McGraw-Hill. (1988)
  • Alan Hastings, Population Biology: Concepts and Models, Springer. (1998)
  • J.D. Murray, Mathematical Biology, Springer. (1989)

Some Mathematical Biology Journals (You might need to be on campus to access some of these online.)