Interactive Markovian models of ‘progressive’ trends. (English) Zbl 0687.92020
Summary: Non-linear Markovian models are investigated for a developing society stratified in terms of literacy or some other sociocultural attribute and characterized by two opposing features: a) progressive trends which forbid upper to lower state intergeneration transitions, and b) fertility differential in favour of lower states. The progressive trends in the models arise due to “attraction” of the higher states.
Two different limits are considered: one in which all upward transitions are permitted (Maximal Attractor Model) and the other in which only one- step transitions are allowed (Adjacent Attractor Model). The nature of the critical points of the two models is examined analytically and/or numerically as appropriate to the case in question. In the first model, the society converges to the highest state in due course despite the fertility differential. The three-state and four-state cases of the models are considered in some detail. The main interesting result concerns the Adjacent Attractor Model. For the three-state case, numerical computation reveals this model society locked in an endless cyclic pattern of evolution despite the existence of a progressive trend.
Two different limits are considered: one in which all upward transitions are permitted (Maximal Attractor Model) and the other in which only one- step transitions are allowed (Adjacent Attractor Model). The nature of the critical points of the two models is examined analytically and/or numerically as appropriate to the case in question. In the first model, the society converges to the highest state in due course despite the fertility differential. The three-state and four-state cases of the models are considered in some detail. The main interesting result concerns the Adjacent Attractor Model. For the three-state case, numerical computation reveals this model society locked in an endless cyclic pattern of evolution despite the existence of a progressive trend.
MSC:
| 91D99 | Mathematical sociology (including anthropology) |
| 60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |
Keywords:
cyclic evolution; Non-linear Markovian models; developing society; literacy; sociocultural attribute; progressive trends; fertility differential; Maximal Attractor Model; Adjacent Attractor Model; critical points; four-state cases; three-state caseReferences:
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