In this paper it is shown that if the consumers' scale economies parameter α in the urban retail model is less than one then the unique positive equilibrium is globally asymptotically stable. This is one of the reasons one might add a stochastic term to the model. This will be done in the paper.
Get full access to this article
View all access options for this article.
References
1.
ArnoldL, 1974Stochastic Differential Equations, Theory and Applications (John Wiley, New York)
2.
BeaumontJ RClarkeMWilsonA G, 1981a, “The dynamics of urban spatial structure: Some exploratory results using difference equations and bifurcation theory”Environment and Planning A131473–1483
3.
BeaumontJ RClarkeMWilsonA G, 1981b, “Changing energy parameters and the evolution of urban spatial structure”Regional Science and Urban Economics11287–316
4.
ChudzyriskaISîodkowskiZ, 1984, “Equilibria of a gravity demand model”Environment and Planning A16185–200
5.
FreidlinM IWentzellA O, 1983Random Perturbations of Dynamical Systems (Springer, New York)
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.
KaashoekJ FVorstA C F, 1984, “The cusp catastrophe in the urban retail model”Environment and Planning A16851–862
11.
LakshmanenT RHansenW G, 1965, “A retail market potential model”Journal of the American Institute of Planners31134–143
12.
MalliarisA GBrockW A, 1982Stochastic Methods in Economics and Finance (North-Holland, Amsterdam)
13.
RijkF J AVorstA C F, 1983, “Equilibrium points in an urban retail model and their connection with dynamical systems”Regional Science and Urban Economics13383–399
14.
SchussZ, 1980Theory and Applications of Stochastic Differential Equations (John Wiley, New York)
15.
SikdarP KKarmeshu, 1982, “On population growth of cities in a region: A stochastic nonlinear model”Environment and Planning A14585–590
16.
WilsonAG, 1981Catastrophe Theory and Bifurcation: Applications to Urban and Regional Systems (Croom Helm, Beckenham, Kent)
