Coordinator: M.Gyllenberg, University of Turku, FinlandUnit 1
The Levins model and the mainland-island model. The restricting assumptions on which these unstructured models are built. The solutions of the models.
Predictions of the simple models of Lecture 2. The use of these predictions in different contexts
How can the restricting assumptions of the simple unstructured models be relaxed? Structured metapopulation models
Modelling philosophy of structured metapopulations
Derivation of a general structured metapopulation model.
Asymptotic behaviour, steady states, extinction, bifurcation theory. Differences with the prediction of the unstructured models.
Adaptive metapopulation dynamics. How should one define fitness in a metapopulation?
Evolution of dispersal in a metapopulation
Current issues in metapopulation dynamics. Future prospects.
Morphogenesis (10 units)
Coordinator: P.Maini, Oxford University, EnglandAbstract.
Instructors: P.K. Maini, N.A.M. Monk (University of Sheffield, UK),
E. Plahte (Norway), L. Davidson (University of Virginia, USA)
Introduction and motivation
Units 2 and 3 (PK Maini)
Mathematical Models for Pattern Formation
Unit 4 (PK Maini)
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
Taxis and diffusion (6 units)
Coordinator: A.Stevens, Max-Planck Institute, Leipzig, GermanyUnits 1 and 2
- 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)
- The stationary Keller-Segel model for chemotaxis:
- Reduction to one equation, bifurcation results
- Selfsimilar solutions and traveling waves
- Blow-up results for the Keller-Segel model
- Overview, scaling arguments, explicit examples
- Hyperbolic models for chemotaxis in 1D
- Comparison with the Keller-Segel model (formal limit and simulations)
- Transport equations
- Mathematical setting
- Parabolic limit
- Applications to chemotaxis
Modelling cell migration (6 units)
Coordinator: M.Chaplain, University of Dundee, ScotlandAbstract.
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.
Chemotactic cell movement in Dyctiostelium discoideum
Lymphocyte (T-cell) invasion of solid tumours
Endothelial cell migration in angiogenesis
Pattern formation in solid tumours
Cancer cell migration and invasion
Trophoblast migration and invasion in embryo implantation;
Interacting particles and stochastic differential equations (6 units)
Coordinator: V.Capasso, University of Milan, Italy1. Elements of stochastic analysis (V.Capasso)
Course 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, France
Course 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
Coordinator: R.Bravo de la Parra, Universidad de Alcala de Henares, SpainAbstract.
Instructors: 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
Estimation (6 units)
Coordinator: A.De Gaetano, CNR, Roma, ItalyThe general objective of the course is to introduce parameter estimation for (nonlinear) spatially distributed models and spatial prediction. The two main examples will be nonlinear PDE and kriging for geographical data.
Instructors: 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, ...).
Numerical methods (6 units)
Coordinator: Luis Abia, Universidad de Valladolid, SpainUnit 1: ( L.M. Abia)
Instructors: 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.