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Rate effects on the growth of centres

Published online by Cambridge University Press:  07 July 2016

H. M. FRY  and
F. T. SMITH
H. M. FRY
Affiliation:
Centre for Advanced Spatial Analysis, University College London, Gower Street, London, United Kingdom WC1E 6BT email: hannah.fry@ucl.ac.uk
F. T. SMITH
Affiliation:
Department of Mathematics, University College London, Gower Street, London, United Kingdom WC1E 6BT email: f.smith@ucl.ac.uk
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Abstract

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Entropy maximising spatial interaction models have been widely exploited in a range of disciplines and applications: from trade and migration flows to the spread of riots and the understanding of spatial patterns in archaeological sites of interest. When embedded into a dynamic system and framed in the context of a retail model, the dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be addressed analytically. In this paper, we present some analysis of the continuous retail model and the corresponding discrete version, which yields insights into the effect of space on the evolving system, and an understanding of why certain retail centres are more successful than others. The slowly developing growths and the fast explosive growths that are of particular concern are explained in detail.

Keywords

Mathematical modelurban evolutionentropyasymptotics91D1091D2534E1034A3439A33

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 2 , April 2017 , pp. 221 - 242
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2016

References

Batty, M. (2010) Urban Modelling: Algorithms, Calibrations, Predictions. Cambridge University Press, Cambridge.Google Scholar
Batty, M. & Mackie, S. (1972) The calibration of gravity, entropy, and related models of spatial interaction. Environ. Plann. A 4 (2), 205–233.Google Scholar
Baudains, P., Braithwaite, A. & Johnson, S. D. (2013) Target choice during extreme events: A discrete spatial choice model of the 2011 london riots. Criminology 51 (2), 251–285.Google Scholar
Birkin, M., Clarke, G. & Clarke, M. (2010) Refining and operationalizing entropy-maximizing models for business applications. Geogr. Anal. 42 (4), 422–445.Google Scholar
Birkin, M., Clarke, G. & Clarke, M. (2003) Retail geography and intelligent network planning. International Journal of Geographical Information Science 17 (8), 815–816.Google Scholar
Clarke, G., Langley, R. & Cardwell, W. (1998) Empirical applications of dynamic spatial interaction models. Comput. Environ. Urban Syst. 22 (2), 157–184.Google Scholar
Clarke, M. & Wilson, A. G. (1983) The dynamics of urban spatial structure: Progress and problems. J. Reg. Sci. 23 (1), 1–18.CrossRefGoogle Scholar
Davies, T., Fry, H., Wilson, A., Palmisano, A., Altaweel, M. & Radner, K. (2014) Application of an entropy maximizing and dynamics model for understanding settlement structure: The khabur triangle in the middle bronze and iron ages. J. Archaeological Sci. 43 (0), 141–154.Google Scholar
Davies, T. P., Fry, H. M., Wilson, A. G. & Bishop, S. R. (2013) A mathematical model of the London riots and their policing. Sci. Rep. 3 (1303), 14–21.CrossRefGoogle ScholarPubMed
Dearden, J., Wilson, A. (2015) Explorations in Urban and Regional Dynamics: A Case Study in Complexity Science. Routledge. Vol. 7.Google Scholar
Dennett, A. & Wilson, A. (2013) A multilevel spatial interaction modelling framework for estimating interregional migration in europe. Environ. Planning A 45 (6), 1491–1507.CrossRefGoogle Scholar
Favaro, J.-M. & Pumain, D. (2011) Gibrat revisited: An urban growth model incorporating spatial interaction and innovation cycles. Geogr. Anal. 43 (3), 261–286.Google Scholar
Fotheringham, A. S. & Knudsen, D. C. (1986) Modeling discontinuous change in retailing systems: Extensions of the harris-wilson framework with results from a simulated urban retailing system. Geogr. Anal. 18 (4), 295–312.Google Scholar
Fry, H. (2012) A dynamic global trade model with four sectors: Food, natural resources, manufactured goods and labour. https://www.bartlett.ucl.ac.uk/casa/pdf/paper178.pdf. Casa working papers, 178.Google Scholar
Gould, P. (1972) Pedagogic review. Ann. Assoc. Am. Geogr. 62 (4), 689–700.Google Scholar
Guy, C. M. (1991) Spatial interaction modelling in retail planning practice: The need for robust statistical methods. Environ. Planning B: Planning Des. 18 (2), 191–203.Google Scholar
Haggett, P., Cliff, A. D. & Frey, A. E. (1977) Locational Analysis in Human Geography. Tijdschrift Voor Economische En Sociale Geografie 68.6, Wiley.Google Scholar
Harris, B. & Wilson, A. G. (1978) Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environ. Planning A 10 (4), 371–388.CrossRefGoogle Scholar
Huff, D. L. (1963) A probabilistic analysis of shopping center trade areas. Land Econ. 39 (1), 81–90.Google Scholar
Isard, W. (1975) A simple rationale for gravity model type behavior. Papers Reg. Sci. 35 (1), 25–30.Google Scholar
Johnston, R. J. & Thrift, N. J. (1993) Ringing the changes. Environ. Planning A 25.Google Scholar
Lakshmanan, J. R. & Hansen, W. G. (1965) A retail market potential model. J. Am. Inst. Planners 31 (2), 134–143.Google Scholar
May, R. (1976) Simple mathematical models with very complicated dynamics. Nature 261, 459–467.CrossRefGoogle ScholarPubMed
McFadden, D. (1980) Econometric models for probabilistic choice among products. J. Bus. 53 (3), S13–S29.Google Scholar
O'Kelly, M. E. (2010) Entropy-based spatial interaction models for trip distribution. Geogr. Anal. 42 (4), 472–487.CrossRefGoogle Scholar
Openshaw, S. (1976) An empirical study of some spatial interaction models. Environ. Planning A 8 (1), 23–41.Google Scholar
Rae, A. (2009) From spatial interaction data to spatial interaction information? Geovisualisation and spatial structures of migration from the 2001 UK census. Comput. Environ. Urban Syst. 33 (3), 161–178.Google Scholar
Senior, M. L. (1979) From gravity modelling to entropy maximizing a pedagogic guide. Progr. Human Geogr. 3 (2), 175–210.CrossRefGoogle Scholar
Singleton, A. D., Wilson, A. G. & OBrien, O. (2012) Geodemographics and spatial interaction: An integrated model for higher education. J. Geogr. Syst. 14 (2), 223–241.Google Scholar
Tang, J., Liu, F., Wang, Y. & Wang, H. (2015) Uncovering urban human mobility from large scale taxi GPS data. Phys. A: Stat. Mech. Appl. 438, 140–153.CrossRefGoogle Scholar
Wilson, A. G. & Wilson, (1971) A family of spatial interaction models, and associated developments. Environ. Planning A 3 (1), 1–32.CrossRefGoogle Scholar