Description:

Spatially independent models; enzyme kinetics; excitable system; reaction-diffusion equations; travelling waves; pattern formation; conditions for Turing instability, the effect of domain size; the concept of stability in PDEs, spectrum of a linear operator, semigroups.

Contents:

1. Spatially independent models: single and multispecies interacting models including their analysis (discrete and continuous).
2. Enzyme kinetics (law of mass action) and non-equilibrium thermodynamics.
3. Excitable systems - a model for nerve pulses (Fitzhugh-Nagumo); theory of bifurcations and dynamical systems
(PDEs).
4. The influence of space (reaction-diffusion equations).
5. Diffusion equation - derivation, solution, possible modification, penetration depth, long-range diffusion.
6. Travelling waves.
7. Pattern formation - diffusion-driven instability (Turing instability), the effect of domain size.
8. Concept of stability in evolution equations in form of partial differential equations, connection to spectrum and brief touching upon theory of semigroups.

Seminar contents:

Outline of excercises follows outline of the course. For analysis of models and eventual plotting of results and solutions, symbolic mathematical programs will be used (as Mathematica, Maple).

Recommended literature:

Key references:
1. L. Edelstein-Keshet - Mathematical Models in Biology, SIAM, 2005.
2. G. de Vries, T. Hillen, M. Lewis, J. Muller, B. Schonfisch - A Course in Mathematical Biology, SIAM, 2006.
3. J. D. Murray - Mathematical Biology: I. An Introduction, Springer, 2002.
4. J. D. Murray - Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2014.
5. J. Crank - The mathematics of diffusion. Oxford university press, 1979.

Recommended references:
6. J. Keener, J. Sneyd - Mathematical Physiology, I: Cellular Physiology, Springer, 2009.
7. W. Rudin - Analyza v komplexním a reálném oboru, Academia, Praha 2003.

Keywords:

Mathematical biology, discrete models, continuous models, spatial models; reaction-diffusion models, Turing instability.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar