Course on Introduction to Bio-Mathematics Erasmus Mundus M.Sc. MathMods - 3rd semester University of L'Aquila - Fall 2012-2013

Aim of the course is to provide basic and more advanced tools for studying differential-equation models for a selected set of applications arising from population biology, epidemiology, chemistry, cell biology, etc… Moreover, the course aims at introducing the mathematical methods to partly solve the proposed models, or rather to detect their qualitative behavior. Rigorous analytical results are complemented with numerical simulations and appropriate physical interpretations.

First Part. Continuous time-dependent models. lectures held by Chiara SIMEONI

Prerequisite topics : ordinary differential equations (existence and uniqueness, stability and qualitative behavior, numerical methods and simulation softwares for ODEs); continuous dynamical systems, linear dynamical systems (series of matrices, diagonalizable matrices, complex eigenvalues, nilpotent matrices), qualitative theory (method of linearization and Lyapunov's method)

Continuous population models for single species :

Lecture 1. continuous growth models;

Lecture 2. insect outbreak model - spruce budworm, delay models;

Lecture 3. numerical laboratory (introduction to numerical methods for ODEs and Scilab, linear and logistic growth models);

Lecture 4. linear analysis of delay population models - periodic solutions, harvesting a single natural population.

Continuous models for interacting populations :

Lecture 5. predator-prey models - Lotka-Volterra systems (complexity and stability), realistic predator-prey models;

Lecture 10. enzyme kinetics - basic enzyme reaction, transient time estimates and non-dimensionalization;

Lecture 11. Michaelis-Menten quasi-steady state analysis.

Lecture 12. basics on the dynamics of infectious diseases - epidemic models and AIDS.

Second Part. Time-space dependent models: PDEs in biology. lectures held by Monika TWAROGOWSKA

Introduction to the mathematical modeling of transport and reaction effects.

Diffusion equations : diffusion-reaction processes and Fick's law, simple random walk and derivation of the diffusion equation, the Gaussian distribution, smoothing and decay properties of the diffusion operator, nonlinear diffusion.

Reaction-diffusion models for one single species : diffusive Malthus equation and critical patch size, asymptotic stability of constant states for general nonlinear reaction diffusion equations, traveling waves, an example of traveling wave - the Fisher-Kolmogorov equations.

Reaction-diffusion systems : multi-species waves in predator-prey systems, Turing instability and spatial patterns, a sufficient condition for stability.

Chemotaxis : diffusion versus chemotaxis - stability versus instability, diffusion versus chemotaxis - stability versus blow-up, chemotaxis with nonlinear diffusion, models with maximal density.

Structured population dynamics : an example in ecology - competition for resources, continuous traits, evolutionary stable strategy in a continuous model.

Bibliographical references.

J.D. Murray, Mathematical Biology, Springer, 3rd Edition in 2 volumes : Mathematical Biology I. An Introduction, 2002 Mathematical Biology II. Spatial Models and Biomedical Applications, 2003

L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, 2004

D. Barnes and D. Chu, Introduction to Modelling for Biosciences, Springer, 2010

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 2nd Edition, Texts in Applied Mathematics, Vol. 37, Springer, 2007

A. Quarteroni and F. Saleri, Scientific Computing with Matlab and Octave, 3rd Edition, Texts in Computational Science and Engineering, Vol. 2, Springer, 2010

The course is based on the Lecture Notes "Mathematical models in life sciences" by M. Di Francesco that are downloadable at http://mat.uab.es/~difrancesco/teaching.htm (see also the lectures notes at www.math.ust.hk/~machas/mathematical-biology.pdf)

For each part of the course, few lectures are devoted to the numerical simulation of the mathematical models described above by using the software environments Scilab at www.scilab.org/ and FreeFem++ at www.freefem.org/ (basic knowledge about these softwares is required and their use is mandatory)

The examination takes place in two phases : the first phase is in the form of a written exam about the contents of the lectures (books or personal notes are not allowed), its duration is of 2 hours and its evaluation is weighted by 2/3 for the final score; the second phase consists in showing basic abilities in using the numerical programs developed for the simulations, its duration is of about 15 minutes and its evaluation is weighted by 1/3 for the final score (an example of "numerical examination" is to choose a model from the last section - which is not directly considered for "theoretical examination" - to be simulated using the codes developed during the laboratory sessions)