Coordinator: M.Gyllenberg, University of Turku, FinlandInstructor: M.Gyllenberg

**Unit 2**
**The Levins model and the mainland-island
model. The restricting assumptions on which these unstructured models
are built. The solutions of the models.**

**Unit 3**
**Predictions of the simple models
of Lecture 2. The use of these predictions in different contexts**

**Unit 4**
**How can the restricting assumptions
of the simple unstructured models be relaxed? Structured metapopulation
models**

**Unit 5**
**Modelling philosophy of structured
metapopulations**

**Unit 6**
**Derivation of a general structured
metapopulation model.**

**Unit 7**
**Asymptotic behaviour, steady
states, extinction, bifurcation theory. Differences with the prediction
of the unstructured models.**

**Unit 8**
**Adaptive metapopulation dynamics.
How should one define fitness in a metapopulation?**

**Unit 9**
**Evolution of dispersal in a
metapopulation**

**Unit 10**
**Current issues in metapopulation
dynamics. Future prospects.**

Coordinator: P.Maini, Oxford University, EnglandInstructors: P.K. Maini, N.A.M. Monk (University of Sheffield, UK),E. Plahte (Norway), L. Davidson (University of Virginia, USA)

**Unit 1 **
**(PK Maini)**
**Introduction and motivation**

**Units 2 and 3 (PK Maini)**
**Mathematical Models for Pattern
Formation**

**Unit 4 (PK Maini)**
**Applications**

**Units 5 and 6 (N. Monk)**
**Pattern Formation in Discrete
Cellular lattices**

**Units 7 and 8 (E. Plahte)**
**Mathematical theory for pattern
formation in Discrete Cellular lattices**

**Units 9 (L. Davidson)**
**Biomechanical models of tissue folding**

**Units 10 (L. Davidson)**
**Biomechanical models of cell rearrangement and tissue
extension**

Coordinator: A.Stevens, Max-Planck Institute, Leipzig, GermanyInstructors: A.Stevens

** - Diffusion vs transport**
**(random walks and velocity jump processes)**
** - Taxis and kinesis in biology - motivation**
** - Random walk models for chemotaxis**
** - Derivation of the classical Keller-Segel model
for chemotaxis**
** - Many particle models and simulations** **
(short outlook to session about many particle systems)**

**Unit 3**
** - The stationary Keller-Segel model for chemotaxis:**
** - Reduction to one equation, bifurcation results**
** - Selfsimilar solutions and traveling waves**

**Unit 4**
** - Blow-up results for the Keller-Segel model**
** - Overview, scaling arguments, explicit examples**

**Unit 5**
** - Hyperbolic models for chemotaxis in 1D**
** - Comparison with the Keller-Segel model** **
(formal limit and simulations)**

**Unit 6**
** - Transport equations**
** - Mathematical setting**
** - Parabolic limit**
** - Applications to chemotaxis**

** **
**
Modelling cell migration
(6 units)**

Coordinator: M.Chaplain, University of Dundee, ScotlandInstructor: M.Chaplain

**This course will present an overview
account of the mathematical modelling of cell migration using a number
of important examples drawn from developmental biology. For example, the
migratory response of Dictyostelium discoideum amoebae to cyclic AMP (cAMP),
the migratory response of immune cells to a solid tumour, the migratory
response of endothelial cells to angiogenic cytokines and matrix macromolecules,
the response of tumour cells to growth factors and growth inhibitors, the
migratory response of invading cancer cells, and the migratory response
of trophoblast cells in embryo implantation. The modelling will utilise
two different techniques - continuum models (partial differential
equations) and discrete models (biased random walks) - to examine how cells
respond to external stimuli (such as cytokines) and also interact with
their substratum (the extracellular matrix, for example). Mechanisms of
cell migration such as chemokinesis, chemotaxis, haptotaxis will be investigated.**

**Unit 1**
**Chemotactic cell movement in
Dyctiostelium discoideum**

**Unit 2**
**Lymphocyte (T-cell) invasion
of solid tumours**

**Unit 3**
**Endothelial cell migration in
angiogenesis**

**Unit 4**
**Pattern formation in solid tumours**

**Unit 5**
**Cancer cell migration and invasion**

**Unit 6**
**Trophoblast migration and invasion
in embryo implantation;**

**
Interacting particles
and stochastic differential equations
(6 units)**

Coordinator: V.Capasso, University of Milan, ItalyCourse instructors: Vincenzo Capasso and Shay Gueron

**2. From Stochastic to Deterministic
models: actual case studies (S. Gueron)**
** 2.1. Birth and death processes
and mass action laws in biology and chemistry: an interacting particle
systems approach.**
** 2.2. Coagulation-fragmentation
processes and their applications to population biology: from stochastic
to deterministic models**

**
Computer simulation of populations**
**(10 un.)**

Coordinator: C. LePage, CIRAD, Montpellier, FranceCourse instructors: A. Deutsch, A. Lomnicki, C. LePage

** Unit 5 (Ch. Le Page)**
** i-state configuration
model and agent-based simulation: principles and key concepts**
** * Object-Oriented Programming**
** * Multi-Agent Systems
(MAS)**
** * Defining the Environment
of a MAS as a virtual landscape**
** **
** Unit 6 (Ch. Le Page)**
** Generic simulation platforms
: Cormas**
** * Defining spatial entities
at multiple scales**
** * Coupling MAS and Geographical
Information Systems (GIS)**
** **
** Unit 7 (H. Lorek)**
** Generic simulation platforms
: Ecosim**
** * General overview of
the toolkit**
** * Example of IBM built
with Ecosim**

** Unit 8 (M. Hare)**
** Generic simulation platforms
: Swarm**
** * General overview of
the toolkit**
** * Example of IBM built
with Swarm: Weaver**

** Unit 9 (Ch. Le Page;
H. Lorek; M. Hare)**
** Comparisons and discussion
about the implementations with the three generic platforms (Cormas,
EcoSim, Swarm) of the** ** case-study
model proposed during unit 4**

** Unit 10 ()**
** Use and validation of
computer simulation models**
** **

**
Time scales and space
(6 units)**

Coordinator: R.Bravo de la Parra, Universidad de Alcala de Henares, SpainInstructors: P.Auger, R.Bravo de la Parra, J.-C. Poggiale, E.Sanchez

**Unit 1 (P. Auger)**
**Introduction to aggregation
methods for o.d.e.s.**

**Unit 2 (P.Auger)**
**Applications to population dynamics
in heterogeneous environments**

**Unit 3 (J.-C. Poggiale)**
**Center manifold and aggregation**

**Unit 4 (J.-C. Poggiale)**
**Center manifold and aggregation:
applications to spatial dynamics**

**Unit 5 (E.Sanchez)**
**Aggregation on linear discrete
models**

**Unit 6 (E.Sanchez)**
**Aggregation on non-linear discrete
models**

Coordinator: A.De Gaetano, CNR, Roma, ItalyInstructors: J.-N. Bacro, INAPG, Paris, France; A. De Gaetano

**Unit 1 (A. De Gaetano)**
**General presentation of the concepts and goals of
the process of estimating model structural parameters from observations:
from probability densities to maximum likelihood to optimization in parameter
space.**

**Unit 2 (A. De GAetano)**
**Inference on nonlinear model parameters, shape and
type of (asymptotic) confidence regions, local curvature.**

**Unit 3 (A. De Gaetano)**
**Architecture of a parameter estimation software and
application of estimation of parameters for a PDE model.**

**Unit 4 (J.N. Bacro)**
**Presentation of the goals and fundamental techniques
of geostatistical analysis, with emphasis on the hypothesis related to
the scale of variation (large scale/small scale), the stationarity assumption
and the construction of a variogram estimator.**

**Unit 5 (J.N. Bacro)**
**Introduction to spatial prediction : kriging procedures
and related models**
** - ordinary kriging
and robust kriging;**
** - universal kriging
and median-polish kriging.**

**Unit 6 (J.N. Bacro)**
**An application of kriging to geographically distributed
data. Hints to further topics in the geostatistical analysis of spatial
data (trans-gaussian kriging, co-kriging, non-linear geostatistics, ...).**

Coordinator: Luis Abia, Universidad de Valladolid, SpainInstructors: L.M. Abia; Dr. Rodolfo Bermejo, University Complutense, Madrid, SPAIN

**The Lecture is an introduction to the basic of the discretization
of partial differential equations by the finite difference method:
difference equations, the usual concepts of consistency, stability
and convergence. This introduction is motivated on very simplified
structured population models.**
** **
**Unit2: ( L.M. Abia)**
**Numerical Methods for Age-Structured Population Models**
** **
**The Lecture reviews some standard finite difference methods and
finite-difference methods along the characteristics proposed in the literature
for nonlinear age-structured population models. The models are prototype
of models appearing in epidemiologie, in demographie, in ecology,
etc.**
** **
**Unit 3: ( L.M. Abia)**
**Numerical Methods for Size-Dependent Population Models **
** **
**The Lecture reviews some numerical methods for size-dependent models
of structured populations with emphasis in the algorithms for changing
adaptativily the mesh along the characteristics. Here we cover also models
with two or more variables structuring the population (age, size, ...)
and methods based on discretizations of the Volterra integral equations
in what the can be reformulated.**
** **
**Units 4,5: (R. Bermejo)**
**Numerical Methods of Diffusive and Ecological Models**
** **
**The first Lecture introduce the basic finite difference schemes
used in the** **discretization of mathematical models in biology
and ecology whit diffusion.** **Also the interaction of diffusion
and transport will be considered.**
** **
**The second Lecture will review the discretization of more far reaching
diffusive** ** models in biology and** ** ecology: reaction-diffusion
systems, models with memory, etc..., featuring the** **main numerical
problems to be solved.**
** **