Table Of ContentPeter Nijkamp . Aura Reggiani
Interaction,
Evolution and Chaos
in Space
With 60 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. PETER NUKAMP
Faculty of Economics and Econometrics
Free University
De Boelelaan 1105
NL-lOS1HV Amsterdam, The Netherlands
Dr. AURA REGGIANI
Faculty of Economics
University of Bergamo
Via Salvecchio, 19
1-24100 Bergamo, Italy
ISBN-13: 978-3-642-77511-6 e-ISBN-13: 978-3-642-77509-3
001: 10.1007/978-3-642-77509-3
Library of Congress Cataloging-in-Publication Data. Nijkamp, Peter. Interaction, evolution, and
chaos in space / Peter Nijkamp, Aura Reggiani. p.cm. Includes bibliographical references and in
dex.
1. Spatial
analysis (Statistics) 2. Chaotic behavior in systems. 3. Econometrics. J. Reggiani, Aura. II. Title.
QA278.2.NS41992 330'.01'S118--dc20 92-1351S CIP.
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© Springer-Verlag Berlin· Heidelberg 1992
Softcover reprint of the hardcover I st edition 1992
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence
of a specific statement, that such names are exempt from the relevant protective laws and regula
tions and therefore free for general use.
214217130-S4321O -Printed on acid -free paper
This book is dedicated to those
whose spatial interaction is
governed by strange attractors
PREFACE
For many decades scholars from various disciplines have been intrigued by
the question whether there are unifying principles or models that have a validity in
different disciplines. The building of such analytical frameworks bridging the gaps
between scientific traditions is a very ambitious task and has not been very
successful up till now.
In the past - in a static context - several such principles have been defined
and advocated at the edge of the natural sciences on the one hand and social sciences
(in particular, economics and geography) on the other hand, mainly based on the
paradigm of 'social physics'. Some important contributions to the integration of the
spatial systems sciences and physics can be found in gravity theory and entropy
theory, which have formed the comer stones of interaction models in space.
This book is about spatial interaction models. It describes the origin, the
history and the correspondence of such models from a 'social physics' perspective. It
is emphasized that such models need a clear behavioural underpinning as a sine qua
non for a valid use in spatial systems analysis. This view also explains the use of
micro-based disaggregate choice models as a tool for analyzing spatial systems. This
is mainly analyzed in Part A of this book.
In recent years much attention has been devoted to qualitative (structural)
changes in dynamic systems, evolutionary theory, morphogenesis, bio-social sciences
and the like. Also here the question emerges as to the validity of such approaches in
social sciences in general and in spatial systems in particular. Is it, for instance,
possible to design models that describe indigenous behavioural shocks in spatial
systems models?
In this context, non-linear dynamics has many important lessons to offer to
the analysis of the dynamic behaviour of spatial systems. Especially modern chaos
theory, which has gained much popUlarity in recent years, presents fascinating new
analytical departures. At the same time the need for a behavioural explanation in
such qualitative structural change models has to be emphasized. Therefore, this book
contains in the second part (Part B) a critical overview of chaos types of models,
with a particular emphasis on the applicability in dynamic spatial systems.
We conclude in our book that spatial interaction models - interpreted in a
broad sense - may offer a general framework for many (static and dynamic)
phenomena in interconnected spatial systems. They appear to be compatible with
'social physics' and chaos principles. A major challenge is now to generate a more
solid empirical basis for such models.
On our 'chaotic' way to the fmal product in this book we have been guided
by many colleagues and friends. Furthermore, our thanks go to Dianne Biederberg,
Rita Hittema and Pilar van Breda who managed to create order out of the seemingly
chaotic pieces which formed the building blocks of the present study and had to be
typed in a logical and consistent order. We also wish to recognize the support offered
by the Netherlands Institute for Advanced Study in the Humanities and Social
Sciences (NIAS) in Wassenaar and the Italian Consiglio Nazionale delle Ricerche
(CNR) (no. 90.02147.CT11 and no. 91.02288.CTll), while completing this book.
Peter Nijkamp
Wassenaar, February 1992 Aura Reggiani
CONTENTS
PART A STATIC MODELS OF SPATIAL INTERACTION
1. SPATIAL INTERACTION MODELS AND GRAVITY THEORY
A CONCISE OVERVIEW
1.1 Introduction 3
1.2 Gravity Analysis and Spatial Interaction Models 3
1.3 Gravity Theory and the Social Sciences 8
1.4 Alternative Utility Foundations and Specifications of
Gravity Theory 10
1.4.1 Simple interaction theory 10
1.4.2 System-wide cost efficiency 12
1.4.3 Aggregate utility theory 13
1.5 The Scope of Gravity Models: Concluding Remarks 16
2. ENTROPY THEORY AND SPATIAL INTERACTION ANALYSIS
2.1 Prologue 17
2.2 Entropy Theory and Spatial Interaction 17
2.3 Alternative Specifications of the Entropy Model 25
2.4 Alternative Theoretical Backgrounds of the Entropy Model 30
2.4.1 An economic utility approach 30
2.4.2 A probabilistic utility approach 31
2.4.3 Statistical information theory 34
2.4.4 Bayesian statistics 35
2.4.5 Maximum likelihood approach 37
2.5 Concluding Remarks 38
3. ENTROPY AND GENERALIZED COST MINIMIZATION MODELS
AT THE MACRO LEVEL
3.1 Prologue 39
3.2 Entropy and Linear Programming 40
3.3 Entropy and Geometric Programming 42
3.4 Spatial Patterns of Entropy and Linear Programming Models 47
3.5 Entropy Revisited 52
3.6 Concluding Remarks 54
Annex 3A. Relationships Between Total Trip Costs and the Cost
Friction Coefficient 56
x
4. SPATIAL INTERACTION MODELS AND UTILITY MAXIMIZING
BEHA VIOUR AT THE MICRO LEVEL
4.1 Prologue 59
4.2 Spatial Interaction Behaviour and Individual Choice
Behaviour: Theory 59
4.2.1 Introduction 59
4.2.2 Spatial interaction models and deterministic
utility theory 62
4.2.3 Spatial interaction models and random
utility theory 63
4.2.3.1 Basic concepts of random utility theory 63
4.2.3.2 Analogies between spatial interaction
models and discrete choice models 68
4.2.4 Concluding remarks 71
4.3 Spatial Interaction Behaviour and Individual Choice
Theory: An Application 72
4.3.1 Introduction 72
4.3.2 The model 72
4.3.3 The data 75
4.3.4 Results and concluding remarks 75
4.4 Conclusions 82
Annex 4A. An Algorithm for Modal Split Choice with Congestion 84
PARTB DYNAMIC MODELS OF SPATIAL INTERACTION
5. DYNAMIC AND STOCHASTIC SPATIAL INTERACTION MODELS
5.1 Prologue 89
5.2 Spatial Interaction Models Analyzed by Means
of Optimal Control 90
5.2.1 Introduction 90
5.2.2 An optimal control approach 91
5.2.3 Concluding remarks 94
5.3 Spatial Interaction Models Analyzed by Means
of Stochastic Optimal Control 95
5.3.1 Introduction 95
5.3.2 A stochastic optimal control approach 97
5.3.3 Concluding remarks 102
5.4 Spatial Interaction Models with Catastrophe Behaviour
Analyzed in the Framework of Stochastic Optimal Control 102
5.4.1 The model 102
5.4.2 The stochastic optimal control version 104
5.5 Epilogue 107
Annex 5A. The Generalized Spatial Interaction Model as a
Solution to the Optimal Control Entropy Model 109
Annex 5B. A (Generalized) Stochastic Spatial Interaction Model
as a Solution to a Stochastic Optimal Control Problem 113
XI
Annex 5C. Stability and Bifurcations in a Phase Diagram
Analysis for a Stochastic Optimal Control Problem 116
6. SPATIAL MODELLING AND CHAOS THEORY
6.1 Prologue 119
6.2 Chaos Theory: A Brief Review 120
6.2.1 A general introduction to non-linear modelling 120
6.2.2 Key issues in the theory of chaos 125
6.3 Spatial Applications of Chaos Theory: A Brief Survey 133
6.3.1 Introduction 133
6.3.2 Dendrinos 135
6.3.3 Dendrinos and Sonis 137
6.3.4 Mosekilde, Aracil and Allen 137
6.3.5 Nijkamp 138
6.3.6 Reiner, Munz, Haag and Weidlich 139
6.3.7 White 139
6.3.8 Zhang 140
6.3.9 Concluding remarks 140
6.4 A Model of Chaos for Spatial Interaction and Urban Dynamics 140
6.4.1 Introduction 140
6.4.2 Results of simulation experiments 143
6.4.2.1 The onset of chaotic motion 143
6.4.2.2 Chaotic urban evolution 147
6.4.3 Concluding remarks 149
6.5 Epilogue 150
Annex 6A. Classification of Two-dimensional Critical Points 152
Annex 6B. Strange Attractors: A Brief Overview 155
Annex 6C. Steady State Solutions for a Generalized Lorenz System 161
7. SPATIAL INTERACTION MODELS AND CHAOS THEORY
7.1 Prologue 165
7.2 Chaos in Spatial Interaction Models 166
7.2.1 Introduction 166
7.2.2 Chaotic elements in dynamic logit model: theory 166
7.2.3 Simulation experiments for a dynamic 10git model 170
7.2.3.1 Dynamic processes in 10git models 171
7.2.3.2 Dynamic processes in spatial interaction
models 174
7.2.4 Concluding remarks 179
7.3 Delay Effects in Dynamic (Binary) Logit Models 180
7.3.1 Introduction 180
7.3.2 A logistic model with multiple delays 181
7.3.3 Concluding remarks 191
7.4 Conclusions 192
Annex 7 A. Stability Solutions for a Dynamic Logit Model 193
XII
Annex 7B. Stability Solutions for a Dynamic Spatial
Interaction Model 197
8. SPATIAL INTERACTION ANALYSIS AND ECOLOGICALLY
BASED MODELS
8.1 Prologue 199
8.2 Prey-Predator Models: Introduction 200
8.3 Synergetic Models of Spatial Interaction 203
8.4 An Optimal Control Model for a Spatial
Prey-Predator System 206
8.4.1 Introduction 206
8.4.2 Equilibrium analysis 207
8.4.3 Concluding remarks 209
8.5 Competition Models: Introduction 210
8.6 Impact of Chaotic Evolution in Spatial Competition 214
8.6.1 Introduction 214
8.6.2 The case of two competing regions 214
8.6.2.1 Equilibrium analysis 214
8.6.2.2 Simulation experiments 217
8.6.3 Concluding remarks 221
8.7 Epilogue 222
Annex 8A. Stability Solutions for an Optimal Control
Prey-Predator Problem 223
Annex 8B. Transformation of a Continuous System into
a Discrete System 228
Annex 8C. Stability Analysis for a Particular
Competing System 230
9. RETROSPECT AND PROSPECT
9.1 Retrospect 233
9.2 Typology of Dynamic Spatial Interaction Models 237
9.2.1 Introduction 237
9.2.2 Macro-dynamic approaches 238
9.2.3 Micro-meso dynamic approaches 240
9.3 Evolution of Spatial Interaction Models 241
9.4 New Research Areas 243
References 245
Index 275
PART A
STATIC MODELS OF SPATIAL INTERACTION
