Introductory course on applied mathematics methods
with emphasis on modeling of biological problems in terms of
differential equations and stochastic dynamical systems. Students
will be working in groups on several projects and will present them
in class in the end of the course.
1. Linear difference and differential equations in 1
dimension with applications to population dynamics. Second order
linear difference and differential equations. Nonlinear differential
equations and their phase diagrams.
populations with carrying capacity, infection transmission. Linear
differential equations in 2 dimensions. Solution via eigenvalues and
2. Nonlinear systems of differential equations.
species systems, epidemiology. First integrals and Lyapunov
functions. Applications: predator-prey systems, classical physics,
HIV transmission. Periodic orbits: Poincare-Bendixson method,
Bendixson-duLac Criterion, Hopf Bifurcation. Chaotic dynamics with
applications to population dynamics.
3. Nonlinear methods of empirical analysis:
distinguishing deterministic chaos from randomness.
data sets. Elements of statistical analysis. Methods of
4. Markov processes in biology and physics.
Stochastic dynamical systems. Application:
birth-death processes in population models.
Cellular Automata: definition, examples.
models over time and space, Game of Life, statistical physics.
Theory and simulation of one-dimensional cellular automata.
Applications: plant and animal growth.
Monte Carlo simulations in physics and biology. Examples
Nonlinear Dynamical Systems and Chaos with
Applications to Physics, Biology, Chemistry, and Engineering,
Steven H. Strogatz, Studies in Nonlinearity, Addison-Wesley
Publishing Company, 1994.
An Introduction to Stochastic Processes with
Applications to Biology, Linda J.S. Allen, Pearson Education,
Modeling Biological Populations in Space and Time,
Eric Renshaw, Cambridge
Studies in Mathematical Biology, Cambridge
University Press, 1995.