__Course
plan. Mathematical modeling and applied biomathematics____spring
2005__

Teaching will include lectures, exercises, assignments to be done at home, and a larger project done in a group of 3-4 people.

We will study during 7 weeks and have lectures and exercises on Mondays at 10.00 in FL74 and on Tuesdays at 13.15. One meeting per week with working groups for discussing the projects will take place at the Mathematical Center at a useful time.

The week seven (VII) will be used for presentation of projects 30 minutes for every group. A working group can consist of maximum 3people. -------------------------------------------------------------------------------------------------------------------------------------

**week 14 (I)** (4 - 10 April)

·
Monday 10.00:
Introduction to modeling by ordinary differential equations, PDE and
stochastic processes. Dimensions analysis and scaling. Equations in
dimensionless variables. Notes
after the first lecture.

Sal: FL74

·
Tuesday 13:15: Ordinary
differential equations. Existens and uniqueness of solutions. Phase
portrait. Linear systems of ordinary differential equations.

Sal: FL63

**week 15 (II)** (11-17 April)

·
Monday 10.00: Stability
of stationary points and solutions of differential equations. On
numerics. Exercises on stability.
Home assignment N1

Sal:FL74

·
Tuesday 13:15: : Parameter
dependence of models. Bifurcations. Bifurcation diagrams.
Introduction to cell cycle.
**OBS!!! ** Important to have a plan for the
project.

Sal:FL63

**week 16 (III)** (18-24
april)

·
Monday 10:00: Modeling
cell cycle - continuation. Periodical
solutions.

Sal:FL74

·
Tuesday 13:15:Examples
of periodical solutions in population dynamic, cell biology.
electronics.

Sal:FL63

**week 17 (IV) **(25 April-1 may)

· Monday 10:00: Poincaré-Bendixson theorem. Hopf bifurcation. Home assignment N2

Sal:FL74

·
Tuesday 13:15: A
Monte Carlo method for integrals. Modeling of stochastic variables.
Importance sampling. A general stochastic model for particles. Monte
Carlo methods for integral equations. Monte Carlo methods for
integral equations.

Sal:FL63

**week 18 (V)** (2-8
may)

·
Monday 10:00: Monte
Carlo methods for a stochastic
model for particles. General and direct modeling.
Non-stationary stochastic models.
Master equation. Monte Carlo methods for master equation. Home
assignment N3

Sal:FL74

· Tuesday 13:15: Master equation for chemical kinetics by Monte Carlo method. The Gillespie method. Birth and death type processes.

Sal:FL63

**week 19 (VI)** (9-15
may)

·
Monday 10:00: Fokker-Planck
equation. Modeling of transport processes by PDE. Calcium waves in
cells.

Sal:FL74

·
Tuesday 13:15: Reserve
time.Asymptotic methods. Method
with two time scales. Mihaelis-Menthen dynamics in biology.Averaging
method.

Sal:FL63

**week 20 (VII) **(17-22 may)

·
Monday 10:00: Projects presentations

Sal: FL74

·
Tuesday 13:15: Projects presentations
.

Sal: FL63

· Extra day: Consultation before examination.

Examination consists on 70% of project and on 30% of a written test. Students will also get 3 home assignments over different parts of the course. The written test will go during 2 hours.

Alexei Heintz<heintz@math.chalmers.se>