CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2023/2024

# Mathematical Methods in Biology and Medicine

Code Completion Credits Range
01MBI KZ 3 2+1
Garant předmětu:
Václav Klika
Lecturer:
Václav Klika
Tutor:
Václav Klika
Supervisor:
Department of Mathematics
Synopsis:

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

Requirements:

Course of Calculus, Linear Algebra, The equations of mathematical physics. Further, functional analysis is recommended. (In the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01MMF or 01RMF, 01FA held at the FNSPE CTU in Prague).

Syllabus of lectures:

(ODEs)

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, chaos

(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

Syllabus of tutorials:

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).

Study Objective:

Knowledge:

To gain deeper insight into acquired knowledge and concepts from the whole study by their usage in constructing and analysis of models in biology.

Skills:

deeper insight into acquired knowledge and terms from study; formulation and analysis of models

Study materials:

Key references:

[1] L. Edelstein-Keshet - Mathematical Models in Biology, SIAM, 2005

[2] F. Maršík - Biotermodynamika, Academia, 1998

[3] G. de Vries, T. Hillen, M. Lewis, J. Muller, B. Schonfisch - A Course in Mathematical Biology, SIAM, 2006

[4] J D Murray - Mathematical Biology: I. An Introduction, Springer, 2002

[5] J D Murray - Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2014

[6] J Crank - The mathematics of diffusion. Oxford university press, 1979.

Recommended references:

[1] J. Keener, J. Sneyd - Mathematical Physiology, I: Cellular Physiology, Springer, 2009

[2] W. Rudin - Analyza v komplexním a reálném oboru, Academia, Praha 2003

Note:
Time-table for winter semester 2023/2024:
Time-table is not available yet
Time-table for summer semester 2023/2024:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2024-08-12
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