642:611:02 Selected Topics in Applied Mathematics:

Math Foundations for Biology

Instructor: Natalia Komarova, office hours Tuesday 1:00-2:00 pm, Hill 426; email: komarova@math

Place: Hill Center 525

Time: T3; 11:30-12:50, Th 5; 2:50-4:10

Main text: Advanced Engineering Mathematics by Erwin Kreyszig. The course will be loosely based on this, but other materials will also be used. Some copies of this book will be available for shared use.

Other texts:

  • Lectures on ordinary differential equations by W. Hurewitz. This book has been put on reserve in the Hill Center Library for students attending the class. It is helpful for part I of the course.
  • An introduction to stochastic processes with applications to biology by Linda Allen. This is the main text for part IV of the course.
  • A First Course in Stochastic Processes by Samuel Karlin.

    Prerequisites: calculus, some undergrad exposure to ODE's, linear algebra, and basic probability.

    Goal: This course is primarily intended for students in the BioMaPS program, as well as other students from Engineering, Computer Science, Statistics, and other departments who are interested in biological applications but whose undergraduate background did not include basic differential equations and stochastic processes.

     

    Syllabus:

    I. Ordinary Differential Equations (8 lectures)

  • Review of separation of variables and 2nd order homogeneous linear systems

  • 2D systems: phase plane, nullclines, steady states, stability, parameter dependence and the concept of bifurcation

  • Periodic orbits, limit cycles, Poincare'-Bendixon

  • n-dim linear time-invariant systems: matrix exponentials, forced systems; for more materials, click here

  • Laplace transforms: definition, basic properties, application to 2nd-order forced systems, computing inverses

    II. Fourier Series and Transforms (4 lectures)

  • Basic properties (linearity, convolutions), computation of expansions of simple examples.

  • FT as continuous limits of FT, relate to Laplace transform, mention briefly basic properties, and inverse transform.

  • FFT: numerical practice using applets or MATLAB code. To learn more about FFT, read chapters 12.0, 12.1 and 12.2 of Numerical Recipes.

    III. Diffusion PDE's (3 lectures)

  • Introduction, separation of variables solutions in one-dimension (boundary value problems; use of FS for initial conditions)

  • Gaussian kernels on infinite domain; speed of diffusion. See this link for Gaussian kernels.

    IV. Probability

  • Review: events, independence, conditional probability, Bayes' theorem.

  • Densities and distributions, including Poisson and exponential (and how they relate), binomial, normal, etc

  • From binomial to normal (see notes).

  • Random walks in 1 dimension; relate to diffusion PDE

  • Finite Markov chains: equilibrium distributions, Perron-Frobenious

  • Continuous-time, finite-space processes - derive ODE for probabilities

  • Birth and death (queuing) as an example of a countable continuous-time process

  • Linear stochastically forced systems dx = Ax dt + dw: intuition as limit of discrete-time continuous-state process, calculation of spectrum of x.

     

    Homework Assignments:

  • Homework 1. Also, look at this tutorial and play with equation ay''+by'+c=0. You can change the coefficients a,b,c and one of the initial conditions (see the manual). Watch the solution to go from exponential to oscillatory as you change coefficients.

  • Homework 2. Also, look at this paper for an example of a Hopf bifurcation in biology.

  • Homework 3. For this, you will need this ODE applet . Also, if you are interested, you can look at this paper.

  • Homework 4. For this, you will need this Fourier series applet .

  • Homework 5. For this, you will need this applet and then this applet .

  • Homework 6. For this, you will need this applet and then this one . For your info, here is an interesting article about the log-normal distribution in nature.

  • Take-home exam. You will need this applet, this applet, and this applet .