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.