On stabilization of an automaton model of migration processes

A dynamic system of cities with migrants is considered. The wage function is each city depends on the number of migrants in the city. The system is modeled by an automaton whose state is the vector consisting of the numbers of migrants in the cities. The transition function of the automaton reflects the conditions for transfers of migrants between cities. The system stabilizes if the moves are stopped at some point. We find conditions for stabilization of such system depending on the restrictions on the wage function and the automaton transition function. It is shown that if the functions of wages are strictly decreasing, if their ranges are disjoint, and if the transition function is defined so that a migrant moves to another city if and only if its salary increases, then the system necessarily stabilizes and its final state depends only on the total number of migrants and does not depend on their initial distribution over the cities. However, if the transition function is changed so that a migrant moves also if its salary is preserved, but the total wages in all cities are increased, then a monotonous decrease in the wage functions is sufficient for stabilization of the system.