Evolution equations arising in the modelling of life science problems is a fascinating subject area. It is at the very heart of the understanding of many important problems arising in biology, medicine, ecology, physics and mechanics. The goal of this course is to convince the audience of this and to show the important role of applying mathematics to real world situations, and explain, in particular, experimental findings.

I hope that this course will create additional interest in life-science problems among young researchers. Moreover, we aim to show how modelling real world situations (life science problems) has a crucial impact on mathematics, creating new challenging situations for mathematics itself, in particular for infinite-dimensional dynamical systems and their attractors.

Overview

1. Introduction and course overview

2. Mathematical modelling

2.1 What is mathematical biology?

2.2 Multiscale biological processes

2.3 Convergence of sciences: system biology, system medicine and mathematics

2.4 Verification of biological systems with diffusion, convection and interaction of species

3. Functional spaces

3.1 Embeddings theorem

3.2 Fractal dimension

3.3 Kolmogorov entropy and its asymptotics

4. Infinite dimensional dynamical systems

4.1 Global attractor and its generalisations

4.2 Reduction principle in dynamical systems

4.3 Biology enforce  reduction principle to  fail

5. Mathematics itself benefits from biological modeling

5.1 An example

5.2 New classes of degenerate parabolic type equations

5.3 Effect of degeneracy on dynamical behavior of biological systems

Useful References:

1. A.V. Babin and M.I.Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, vol.25, North-Holland, 1992.

2. M.A.Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics, Gakkotoscho International Series, Tokyo, vol.33 , 2010.

3.  M.A.Efendiev, Evolution Equations Arising in the Modelling of Life Sciences, ISNM Series, Birkhäuser/Springer, vol.163, 2013.

4. M.A.Efendiev, Attractors for Degenerate Parabolic type Equations, American Mathematical Society; Serie Mathematical Surveys and Monographs, vol.192, 2013.

5. R.Temam , Infinite-Dimensional Dynamical Systems in Mechanics and in Physics, Springer, Applied Mathematical Sciences,68, 1988.

6. J.Wu, Theory and Applications of Partial Functional Differential Equations, Springer, Applied Mathematical Sciences, 119, 2011.