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ESMTB Summer School on

"Spatial Structures in Biology and Ecology: Models and Methods"

MARTINA FRANCA   (TA) -  ITALY, September 2000

PROGRAMMES

  Metapopulation dynamics (10 units)
 
Coordinator: M.Gyllenberg, University of Turku, Finland
Instructor: M.Gyllenberg
Unit 1
The notion of a metapopulation. The importance of the  metapopulation concept for different ecological questions.  Historical background.

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.
 
 

  Morphogenesis (10 units)

Coordinator: P.Maini, Oxford University, England
Instructors: P.K. Maini, N.A.M. Monk (University of Sheffield, UK),
                         E. Plahte (Norway), L. Davidson (University of Virginia, USA)
Abstract.
In the past decade there have been huge advances in biotechnology leading to the discovery of new genes and an explosion of information on their interactions. Despite these advances, there is still relatively little known about the mechanisms underlying the spatiotemporal cues which control gene dynamics and how the mechanochemical properties encoded by genes determine form and structure. This course examines the mathematical theories underlying these issues beginning with the classical continuum modelling approach, then investigating the cellular approach for fine-grained patterning and finally modelling how cell-cell interactions can lead to tissue morphogenesis.
 
 

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
 

Taxis and diffusion (6 units)

Coordinator: A.Stevens, Max-Planck Institute, Leipzig, Germany
Instructors: A.Stevens
 
Units 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)

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, Scotland
Instructor: M.Chaplain
 
Abstract.

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, Italy
Course instructors: Vincenzo Capasso and Shay Gueron
1. Elements of stochastic analysis (V.Capasso)
  1.1. The Brownian Motion
  1.2. Ito Stochastic Differential Equations
  1.3. Ito Formula
  1.4. Systems of SDE's modelling interacting particles(Lagrange)
  1.5. Evolution equation of the collective stochastic system (empirical measure)
  1.6. Convergence to continuum deterministic spatial systems (Euler)
  1.7. Examples for collective behaviour of biological populations

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 1  (A. Deutsch)
 Cellular Automata: principles and key-concepts
 
 Unit 2  (A. Deutsch)
 Cellular Automata: an example of application in the field of ecology
 
 Unit 3  (A. Lomnicki)
 Individual-based models: a historical perspective, with examples in ecology
 
 Unit 4  (A. Lomnicki; Ch. Le Page)
 * Definition of a very simple IBM case-study. This model should be seen as a toy situation, that is, a situation which bas been stripped  of all rough aspects of reality, but which, because of its austerity, provide ideal conditions for a good understanding of the mechanis  brought into action.
 * Presentation of some tools to describe and formalize the specifications of a model. Application to the case-study model.

 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, Spain
Instructors: P.Auger, R.Bravo de la Parra, J.-C. Poggiale, E.Sanchez
Abstract.
Aggregation methods try to approximate a large scale dynamical system, the general system, involving many coupled variables by a reduced system, the aggregated system, that describes the dynamics of a few global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the general system. Aggregation methods have been developed for systems of ordinary differential equations as well as for discrete models. The course develops applications of population dynamics in patchy environments. The models include two processes acting at different time scales: a slow demography and a fast migration. As examples, prey-predator and Leslie type models are presented.

 
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, Italy
Instructors: J.-N. Bacro, INAPG, Paris, France; A. De Gaetano
The 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.
 

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, Spain
Instructors: L.M. Abia; Dr. Rodolfo Bermejo, University Complutense, Madrid, SPAIN
Unit 1: ( L.M. Abia)
Introduction to Numerical Methods for Models of Structured Populations

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.
 

 
 
 
 



 
 
 

Last revision: May 22, 2000-  Daniela Morale  by