In this paper we will show that for a suitable choice of the parameters in an urban retail model as developed by Huff and by Lakshmanen and Hansen one finds a cusp catastrophe in the surface of equilibrium points. We give some economic consequences of the fact that we have a cusp catastrophe. Furthermore, we show that for other choices of the parameters the equilibrium points of the model depend smoothly on the parameters.
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References
1.
BeaumontJ RClarkeMWilsonA G, 1981a, “Changing energy parameters and the evolution of urban spatial structure”Regional Science and Urban Economics11287–316
2.
BeaumontJ RClarkeMWilsonA G, 1981b, “The dynamics of urban spatial structures: Some exploratory results using difference equations and bifurcation theory”Environment and Planning A131473–1483
3.
BellmanR, 1970An Introduction to Matrix Analysis (McGraw-Hill, New York)
4.
ChowS NHaleJ K, 1982Methods of Bifurcation Theory (Springer, New York)
5.
ChudzyńskaISłodkowskiZ, 1984, “Equilibria of a gravity demand model”Environment and Planning A16185–200
6.
HarrisBChoukrounJ-MWilsonA G, 1982, “Economies of scale and the existence of supply-side equilibria in a production-constrained spatial-interaction model”Environment and Planning A14823–837
7.
HarrisBWilsonA G, 1978, “Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models”Environment and Planning A10371–388
8.
HirschM WSmaleS, 1974Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, New York)
9.
HuffD, 1964, “Defining and estimating a trading area”Journal of Marketing2834–38
10.
LakshmanenT RHansenW G, 1965, “A retail market potential model”Journal of the American Institute of Planners31134–143
11.
PuuT, 1979, “Regional modelling and structural stability”Environment and Planning A111431–1438
12.
PuuT, 1981, “Catastrophe structural change in a continuous regional model”Regional Science and Urban Economics11317–334
13.
RijkF J AVorstA C F, 1983a, “Equilibrium points in an urban retail model and their connection with dynamical systems”Regional Science and Urban Economics13383–399
14.
RijkF J AVorstA C F, 1983b, “On the uniqueness and existence of equilibrium points in an urban retail model”Environment and Planning A15475–482
15.
Taussky-ToddO, 1961, “A generalization of a theorem of Lyapunov”SIAM Journal of Applied Mathematics9640–643
16.
van der WaerdenB L, 1966Algebra. Volume 1 (Springer, New York)
17.
WilsonA G, 1981Catastrophe Theory and Bifurcation: Applications to Urban and Regional Systems (Croom Helm, Beckenham, Kent)
