Spatial externalities and the stability of interacting populations near the center of a large area.

In this paper we analyze the spatial dynamics of a class of systems in which many agents continually reassess decisions of where to locate in a given region. Each agent gives rise to an externality whose impact diffuses to other agents. Thus every agent experiences a composite of externalities emitted by other agents. We term this composite a spatial externality. On the one hand, the level of the externality at any location depends on the spatial distribution of the population. On the other hand, the distribution of the externality induces changes in the distribution of the population. It proves of interest to analyze this joint unfolding of a population distribution and an externality distribution, particularly for the case of several classes of interacting agents. It is well established that the manner in which urban activities adjust to spatial externalities represents a problem at the core of current urban research priorities. Here we extend previous analyses of spatial equilibrium and adjustment dynamics in systems influenced by such externalities [see, for example, Miyao (1978)l. The main object of our concern is the relation between the structure of the externality diffusion operator and the form and stability of the equilibrium population distribution. The externality diffusion operator summarizes the processes by which an externality diffuses away from some agent in the space. Hence, the operator possesses a spatial structure. The stability of an equilibrium population distribution refers to a situation in which small perturbations to the distribution disappear over time. An implicit belief underlying our use of stability analysis is that unstable equilibrium distributions tend not to persist for any significant length of time; hence, conditions that lead to stability have an inherent interest. In particular, it is of interest to relate the conditions for stable population distributions and the spatial structure of the externality diffusion operator. Our plan is the following. We first display the model in full generality and briefly discuss the existence and uniqueness of the spatial equilibria. At this level of generality, however, it proves difficult to discuss necessary and sufficient conditions for the stability of the spatial equilibria, which is the main object of our