Mathematical techniques in biology and medicine
Code  Completion  Credits  Range 

01MBM  Z,ZK  3  2+1 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

Spatially independent models; enzyme kinetics; excitable system; reactiondiffusion 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:

Basic courses on mathematical analysis and linear algebra, functional analysis, equations of mathematical physics (01MAN, 01MAA24, 01LAL, 01LAA2, 01RMF, 01FA1, 01FA2.)
 Syllabus of lectures:

1. Spatially independent models: single and multispecies interacting models including their analysis (discrete and continuous).
2. Enzyme kinetics (law of mass action) and nonequilibrium thermodynamics.
3. Excitable systems  a model for nerve pulses (FitzhughNagumo); theory of bifurcations and dynamical systems
(PDEs).
4. The influence of space (reactiondiffusion equations).
5. Diffusion equation  derivation, solution, possible modification, penetration depth, longrange diffusion.
6. Travelling waves.
7. Pattern formation  diffusiondriven 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 biological models.
 Study materials:

Key references:
1. L. EdelsteinKeshet  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.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Aplikovaná algebra a analýza (compulsory course in the program)
 Aplikované matematickostochastické metody (elective course)
 Matematické inženýrství (elective course)