Gravity-Type Interactive Markov Models—Part II: Lyapunov Stability of Steady States

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.     

Environmental limitations on prehistoric pastoralism in Africa

In several areas of Africa, pastoralism that existed prehistorically no longer survives today. Two such areas, the Southwestern Cape of South Africa and the Sahara, are discussed here, outlining methodologies for reconstruction of adaptive strategies on the basis of environmental parameters that constrained the pastoral economies.

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Résumé En plusieurs régions d'Afrique, le pastoralisme qui existait en temps préhistorique a maintenant disparu. L'article discute de deux de ces régions, le cap sud-ouest de l'Afrique du Sud et le Sahara, donnant un aperçu des méthodologies qui visent la reconstruction des stratégies adaptatrices en se fondant sur les paramètres écologiques qui ont contraint les économies pastorales.