Structural stability in the model of spatial economy

Abstract

Urban problems have become very complicated as a result of technological progress and change in the behavior of human beings. The urban systems of our epoch are characterized by increasing spatial and temporal variation. Many urban models have been suggested to explain and forecast urban pattern formation in urban economics, regional science and geography. Within the framework of the spatial economy model, equilibrium states of commodity flows and labor, which are described by two-dimensional vector fields, are considered. These configurations are obtained on the assumption of the validity of divergent and gradient laws, which characterize the influence on the commodity flows and labor by external factors, such as the distance from the places of habitation of labor to industrial objects, the local costs of transportation and so on. The structural stability of such a system for various types of configuration perturbations is investigated. It is very difficult to obtain general characteristics of structurally stable systems, but in the two-dimensional case it turns out to be possible to draw precise picture of structurally stable flows and spatial organization of the economy corresponding to such flows. In the paper possible flow structures for a hyperbolic umbilical for a cubic potential function that depends on three parameters are constructed numerically in accordance with the concept of catastrophe theory. Bifurcation varieties are found in the parameter space, flux fields for various parameter combinations are constructed. A qualitative change in the character of the flow structure in the neighbourhood of a bifurcation manifold is demonstrated.