Part 1. Modeling the Relationships between Societies and Nature... 1

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1 Contents Introduction xi Part 1. Modeling the Relationships between Societies and Nature Chapter 1. The Theoretical Context of Classical Geography Environmentalism a theory that is still rejected The theoretical double paradox of classical geography Majority refusal of the theoretical approach Theories in physical geography viewed by sub-disciplines Two paradigms borrowed from social sciences: structuralism and Marxism Evolutionary theories at the core of classical geography The general theory of systems and the theories derived therefrom A formal theory to unite geography Theories stemming from the general theory of systems Conclusion Appendix 2: Importing data within Mathematica Chapter 2. Statistical and Probability Models for Given Relationships Between Societies and the Natural Environment Acknowledging the probability model for recorded data Four approaches to determine the probability law for a given series of data Case study: GNP/inhabitant in Africa Modeling the relationships between two or several variables Responding to issues raised by a data table Avoiding the pitfalls of linear and monotonic correlation

2 vi Geographical Models with Mathematica Regression models for all data types Interpolation to bridge the missing data gap Moving from classification to regionalization Factorial analyses to jointly analyze both the structure of variables and the structure of geographic objects Temporalities and time series models The geographer faced with seven lines of questioning Understanding and testing the non-stationarity of given evolution Determining persistence: the correlogram and the variogram Breaking down a series into trends, cycles and random behavior Pinpointing exceptional events, their intensity and their repetition The difficult choice of a probability model Modeling bivariate and multivariate time series Conclusion Appendix 3: chronological program processing Chapter 3. Models of Ordinary Dynamic Systems Four lines of questioning to understand the behavior of a dynamic system Initiation in the modeling of dynamic systems The stages of modeling through a system of ODEs Discrete time and continuous time: difference equations and differential equations Three methods, which are analytical, numerical and qualitative as a means to study a dynamic system The qualitative approach of dynamic systems Assets and restrictions of ODE models The contributions to differential equation models in understanding the evolution of geographical systems A constraint overcome: the reductionism of these models More realistic models of geographical systems Exploitation of stock: fisheries, non-renewable resources and sustainable development Time delays and introducing exceptional events into a dynamic model

3 Contents vii Patterns of interaction between two populations Multi-stock system behavioral models Conclusion Appendix 4: crowd behavior in catastrophic situations Part 2. Modeling Geographic Locations Chapter 4. Theories of Geographical Locations Introduction to spatial economic theories Von Thünen s theory: the location of agricultural produce Weber s theory of industrial location Two categories of theories to explain service activity location A new urban economy and a new economic geography New locations in contemporary space Alonso: von Thünen s theory applied to the city Issues with the new economic geography Works behind numerous theoretical models Conclusion Chapter 5. Theoretical Geolocation Models Von Thünen and d Alonso s monocentric and polycentric models Steiner s model generalizes Weber s Central place models in the making Conclusion Part 3. Spatial Structures and Territorial Dynamics Chapter 6. Theories Used to Understand Territorial Structures and Dynamics From terrestrial to geographical space Space as a multidisciplinary concept A reality: terrestrial space. A concept: geographical space Territory: an ecological notion borrowed by geography Some theories drawn from various fields and used to explain simple territorial forms Some linear-gradient spatial structures imposed by nature or produced by societies Several theories used to explain concentrated territorial organizations Three discontinuity theories, including a geographical one

4 viii Geographical Models with Mathematica 6.3. From morphology to morphogenesis The basic mechanisms of morphogenesis: growth, movement, emergence and symmetry breaking An overview of morphogenetic theories Schelling s theory and social interaction theories The reaction diffusion theory and its generalization The theory of self-organized critical systems Bejan s constructal law and its ramifications Nottale s theory of scale relativity Conclusion Appendix 5: globalization at the root of a paradox: homogenization and global fracturing Chapter 7. Models of Basic Structures: Points and Fields Modeling the point structures of a geographical space Four categories of questions Observing and summarizing a distribution of points: density, mean and median centers Testing the random, regular or concentrated distribution of points Spatial dependence: autocorrelation and variography Spatial interpolation: creating fields from data points Modeling geographical fields Some contributions of related fields Global study of geographical fields Local study of geographical fields Conclusion Appendix 6: Introduction to the morphometric analysis of the Grenoble Alps Chapter 8. Models of Basic Structures: Networks The two aspects of a network: graphs and matrices From the graphic representation of a network To its matrix representations Modeling the structure of a spatial network Centrality, hierarchy, power and prominence in a network Cohesion and communities of a network Three models of network structures Qualitative geographical models and graph theory Modeling affiliation relationships Elaborating conceptual models Modeling online social networks

5 Contents ix 8.4. Modeling network dynamics Optimization in fixed networks Temporal evolution of networks Models of movements channeled by networks Conclusion Appendix 7: A geometric approach to the network of French metropolises Chapter 9. Geographical Space as a Mixture of Basic Spatial Structures Testing links between two elementary spatial structures Finding correspondences between two elementary structures of the same type Relationships between two different basic structures Modeling complex spatial structures: machine learning and choremes Two examples of complex spatial structures A chorematic approach updated by machine learning Modeling multiscale spatial structures Wavelet decomposition models of a spatial structure Models of information entropy decomposition for mesh data Multifractal models Comparing two multiscale spatial structures Conclusion Chapter 10. Morphogenetic Macro- and Micro-models Time series typical of morphogenetic theories Testing the chaotic behavior of a time series Testing the fractality of data to verify the theory of self-organized critical systems Log-periodic behaviors represent an indicator of the scale relativity theory Modeling the dynamics of territorial systems: from ODEs to PDEs The components of a spatialized dynamic system Some instructive models Cellular automata, Brownian motions and multi-agent systems The definition of a cellular automaton Brownian motions for the simulation of movements The contribution of multi-agent systems A kind of universal modeling

6 x Geographical Models with Mathematica Building a simulation micro-model Critical observations on the use of CA and MASs Conclusion Appendix 8: simulating urban growth along the French Riviera with a cellular automata model Conclusion Bibliography Index