Is the Share of a Central Place in the Population of the Area, Served by This Central Place, a Constant for All Levels of the Christaller’s Hierarchy?
https://doi.org/10.31857/S2587-556620191128-135
Abstract
One of the conditions of the central place theory is the assumption of a constant k parameter - a share of a central place in the population of the area served by this central place - for all levels of the Christaller’s hierarchy. Nevertheless, we did not find a rigorous proof of this assertion (underlying the Beckmann-Parr equation) in the bibliography on the central place theory. If this condition is assumed true, it also remains unclear - whether for all or only for strictly defined k -values. We have established that if the chosen K-value of the Christaller’s hierarchy is constant at every lattice level, the Beckmann-Parr equation holds for all meaningful values of k. At the same time we found that the range of k -values for an ideal Christaller’s lattice is bounded above by not an asymptote at k = 1, but an exact almost twice smaller value equal to \(K - \sqrt{K^{2}-K}\). Since the latter changes very slightly during a radical rearrangement of the lattice from K = 3 to K = 7, we can state that we have discovered the new nonstrict invariant in the central place theory - the maximum value of k.
Keywords
Supporting agencies: Scientific Research Plan of the Institute of Geography of the Russian Academy of Sciences, Institute for African Studies of the Russian Academy of Sciences
About the Author
R. V. Dmitriev
Institute of Geography, Russian Academy of Sciences; Russia Institute for African Studies, Russian Academy of Sciences
Russian Federation
Russian Federation
Moscow
References
1. Armand A.D. Samoorganizatsiya isamoregulirovanie geograficheskikh system [Self-Organization and SelfRegulation in Geographic Systems], Targul’yan V.O., Ed. Moscow: Nauka Publ., 1988. 264 p.
2. Bogdanov V.S., Bogdanov S.V. Net invariants and tensor invariants. Izv. Volgogradskogo Gos. Tekhn. Univ., 2013, vol. 18, no. 22 (125), pp. 21-25. (In Russ.).
3. Vazhenin A.A. Invariants of the urbanization processes and hierarchy of population systems. In Geografiya mirovogo razvitiya [Geography of World Development], vol. 1, Sintserov L.M., Ed. Moscow: Inst. Geogr. RAN, 2009, pp. 220-232. (in Russ.).
4. Vazhenin A.A. The causality directions of settlement processes in self-organizing systems. In Geografiya mirovogo razvitiya [Geography ofWorld Development], vol. 2, Sintserov L.M., Ed. Moscow: Tov-vo Nauchn. Izdanii KMK Publ. 2010, pp. 195-206. (In Russ.).
5. Hilbert D. Osnovaniya geometrii [The Foundations of Geometry], Vasil'ev A.V., Ed. Leningrad: Seyatel' Publ., 1923. 152 p.
6. Gorokhov S.A., Dmitriev R.V. The paradoxes of urbanization in modern India. Geografiya v Shkole, 2009, no. 2, pp. 17-23; no. 3, pp. 24-29. (In Russ.).
7. Dmitriev R.V. Application of gravity models to spatial analysis of settlement systems. Narodonaselenie, 2012, no. 2, pp. 41-47. (In Russ.).
8. Harvey D. Explanation in Geography. London: Edward Arnold, 1969. 521 p.
9. Haggett P. Geography: A Modern Synthesis. London: Harper and Row, 1972. 483 p.
10. Shuper V.A. Central places relativistic theory and settlement in the post-industrial era. In Geografiya mirovogo razvitiya [Geography of World Development], vol. 2, Sintserov L.M., Ed. Moscow: Tov-vo Nauchn. Izdanii KMK Publ., 2010, pp. 177-194. (In Russ.).
11. Shuper V.A. Samoorganizatsiya gorodskogo rasseleniya [Self-organization of Urban Settlement]. Moscow: Rossiiskii Otkrytyi Univ., 1995. 168 p.
12. Shuper V.A. The epoch and its creator: the 90th anniversary of Yu.V. Medvedkov. Izv. Ross. Akad. Nauk, Ser. Geogr., 2018, no. 1, pp. 139-146. (In Russ.).
13. Shuper V.A., Em P.P. Expansion of Moscow city: an alternative from the central place theory point of view. Regional’nye Issledovaniya, 2012, no. 4 (38), pp. 97-107. (In Russ.).
14. Em P.P. Fuzziness and relativity: the comparison of central place theory approaches (in case of the Republic of Korea). In Geografiya mirovogo razvitiya. [Geography of World Development], vol. 3, Sintserov L.M., Ed. Moscow: Tov-vo Nauchn. Izdanii KMK Publ., 2016, pp. 267-283. (In Russ.).
15. Abu-Hijleh Y.M. Central place model: a case study on southwest Iowa. Master of Science Thesis. Ames, 1989. 66 p.
16. Beckmann M.J. City hierarchies and the distribution of city size. Econ. Dev. Cult. Change, 1958, vol. 6, no. 3, pp. 243-248.
17. Beguin H., Peeters D. Urbanization in some hierarchical urban models. Reg. Sci. Urban Econ., 1981, no. 11, pp. 19-37.
18. Berry B.J.L., Garrison W. Recent developments of central place theory. Reg. Sci. Assoc., Pap. Proc., 1958, vol. 4, pp. 107-120.
19. Christaller W. Central Places in Southern Germany. Englewood Cliffs: Prentice-Hall, 1966. 230 p.
20. Dacey M.F. A probability model for central place location. Ann. Assoc. Am. Geogr., 1966, vol. 56, no. 3, pp. 550-568.
21. Dennis C., Marsland D., and Cockett T. Central place practice: shopping centre attractiveness measures hinterland boundaries and the UK retail hierarchy. Journal of Retailing and Consumer Servic., 2002, no. 9, pp. 185-199.
22. Fujita M., Krugman P., and Mori T. On the evolution of hierarchical urban systems. Eur. Econ. Rev., 1999, no. 43, pp. 209-251.
23. Parr J.B. City hierarchies and the distribution of city size: a reconsideration of Beckmann's contribution. J. Reg. Sci., 1969, vol. 9, no. 2, pp. 239-253.
24. Sonis M. Central Place Theory after Christaller and Losch: Some further exploration. Presentation at 45th Congress of the Regional Science Association, 23-27 August 2005. Amsterdam: Vrije Univ. Amsterdam, 2005. 30 p.
25. Tabuchi T., Thisse J-F. A new economic geography model of central places. J. Urban Econ., 2011, no. 69, pp. 240-252.
26. Ullman E.L. A theory of location for cities. Am. J. Sociol., 1941, vol. 46, no. 6, pp. 853-864.
27. Wang F. Modeling a central place system with interurban transport costs and complex rural hinterlands. Reg. Sci. Urban Econ., 1999, no. 29, pp. 381-409.
- The Beckmann–Parr equation holds for all levels of the Christaller’s hierarchy for any value of k.
- The maximum value of k (a share of a central place in the population of the service area) is \(K - \sqrt{K^{2}-K}\).
- The maximum value of k in a numerical expression is a non-strict invariant for changes in the K value.
Review
For citations:
Dmitriev R.V. Is the Share of a Central Place in the Population of the Area, Served by This Central Place, a Constant for All Levels of the Christaller’s Hierarchy? Izvestiya Rossiiskoi Akademii Nauk. Seriya Geograficheskaya. 2019;(1):128-135. (In Russ.) https://doi.org/10.31857/S2587-556620191128-135
Views:
598
Email this article 
