An important class of interactive Markov migration models is characterized by gruuity-type transition kernels, in which migration flows in each time period are postulated to vary inversely with some symmetric measure of migration costs and directly with some population-dependent measure of attractiveness. This two-part study analyzes the uniqueness and stability properties of steady states for such processes. In this first part, it is shown that a flow version of the steady-state problem can be given a programming formulation which permits global analysis of steady-state behavior. Within this programming framework, it is shown that when attractiveness is diminished by increased population congestion, the steady states for such processes are unique. The second part of the study will employ these results to analyze the stability properties of such steady states.

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In Part I of this paper (Smith and Hsieh, 1997) a programming formulation of steady states was developed for gravity-type interactive Markov chains in terms of their associated spatial-flow chains. These results are here applied to analyze the stability properties of interactive Markov chains. In particular, the objective function for this programming formulation is shown to constitute a Lyapunov function for an appropriately defined continuous-time version of spatial-flow chains. The Lyapunov stability properties of these spatial flows are then shown to yield corresponding stability properties for the continuous-time versions of interactive Markov chains. In particular, these processes always exhibit global convergence to steady states. Finally, it is shown that when steady states are unique, these convergence results are inherited by those interactive Markov chains that are 'sufficiently close' to their continuous-time versions.

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