A Course in Mathematical Biology

William R. Derrick
University of Montana



Mathematical Biology is a fast growing, although not completely defined, 
area of mathematics that arose initially in the late 1920's with the work 
of Lotka and Volterra in population models and Kermack and McKendrick 
in epidemiology. At the University of Montana, we have developed a senior-level 
interdisciplinary course in Mathematical Biology, that serves the needs 
of our applied mathematics emphasis as well as the modelling needs of 
students in the Wildlife Biology and Chemistry programs.

Many of the tools used in Mathematical Biology involve the analysis of 
nonlinear difference and differential equations (and systems). Although 
these topics may seem to be quite advanced for non-majors, most of these 
topics can be approached using graphical techniques easily mastered by 
upper-division students, even non-majors. Further, these tools provide 
genuine insight into the application of mathematics in the world around 
us. The models show how certain processes work in biology and predict 
outcomes that give insight into the biological mechanisms.

Much of the course involves the use of eigenvalues in analyzing local 
behavior, a task also within the reach of interdisciplinary students. 
We discuss primarily 2-dimensional systems, but some mention is also made 
of behaviors in higher dimensions.

A third aspect of the course is its use of technology: we use some simple 
TRUE BASIC programs to introduce certain models of population growth (Leslie 
model), and to discuss the chaotic behavior of logistic models. Students 
are then taught to use PHASE-PLANE, as a tool for finding numerical solutions 
of differential and discrete systems.

This one-semester course includes aspects of chaotic motions, phase-plane 
analysis, and bifurcation theory, but virtually all of the theoretical 
material require only a basic understanding of sophomore calculus.