Mathematical methods of biological processes: Math 484/684

This course provides an introduction to mathematical methods in biology. In particular, the elements of the dynamical systems theory will be presented. The focus is on development of the pertinent mathematical theory, and the biological applications are used mainly to illustrate mathematical concepts.

Mathematical modeling in biology requires a wide variety of mathematical tools. One of the main such tools is the dynamical systems theory. Therefore, the main objective of the course is to learn the basics of the mathematical theory of deterministic dynamical systems. The biological applications will serve to motivate and illustrate mathematical techniques presented in the course. I tentatively plan to include the following topics (the last two if time permits):

Mathematical models of the population growth. Quantitative (or geometric) theory of ordinary differential equations (ODE). Phase space, orbits, equilibria, stability. Elementary bifurcations. Biological application: Why do insects have population outbreaks?
Mathematical models of interacting species. Phase portraits. Stability by linearization. Types of ecological interactions. Stability and Lyapunov functions. Limit cycles. Bendixson-Poincare theory. Biological application: Why do predators and prey sometimes exhibit periodic oscillations?
Discrete dynamical systems. Maps from R to R. Discrete dynamical system. Orbits, equilibria, periodic points. Logistic equation and chaos. Bifurcations. Lyapunov exponents. Discrete and continuous dynamical systems. Biological application: Do insect populations exhibit chaos?
Modeling the age heterogeneities. Linear discrete dynamical systems. Perron-Frobenius theory and nonnegative matrices. Biological application: Do we have a fundamental theorem in demography?
Evolutionary game theory. Classical game theory, Nash equilibria. Evolutionary game theory and the replicator equation. Biological application: How do altruists appear?
Modeling the space heterogeneities. Random walk and diffusion equation. Traveling wave solutions. Reaction-diffusion systems. Biological application: How big does an island have to be to support a population?

Fall 2015:

Classes: MWF 2:00pm-2:50pm, NDSU Ladd Hall Room 114
Office hours: MWF 9:00am-10:00am (Minard 408E22)
Syllabus
Detailed lecture notes will be provided.
Two very good, but non necessary sources of the material are
- Linda J. S. Allen. An Introduction to Mathematical Biology, Pearson, 2007 (Amazon)
- Steven H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, 2014, Second edition (the first edition will also work) (Amazon)

Lecture notes:

00. Review of ODE
01. Least square method. Models of population growth
02. Analysis of first order autonomous ODE
03. Bifurcation analysis
04. Insect outbreak model
05. Population cycles, Alfred Lotka, and Vito Volterra
06. General properties of an autonomous system of two first order ODE
07. Linear planar ODE. Classification of the equilibria
08. Linear system in d dimensions
09. Power law and scale-free distributions
10. Near equilibria. Outside of equilibria
11. Predator-prey interactions
12. Other types of ecological interaction
13. A thousand and one ecological models
14. Limit cycles
15. Limit sets. Lyapunov functions
16. Gause predator-prey model
17. Poincare-Andronov-Hopf bifurcation
18. Biological models with discrete time. Cobweb diagrams
19. Stability of fixed points. Bifurcations.
20. Periodic solutions. First encounter with chaos
21. Definition of a chaotic orbit. Lyapunov exponents
22. Two dimensional discrete dynamical systems
23. Structured populations. Leslie's model (see the textbook)
24. Chaos in biological populations. The LPA model (see the textbook)
25. The diffusion equation. Traveling wave solutions of the Fisher equation (see the textbook)

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