На окружности для гладкого однопараметрического семейства пар управляемых систем и плотностей выгоды изучены типичные переходы между оптимальными вращением и стационарной стратегией в задаче максимизации средней временной выгоды на бесконечном горизонте. Показано, что имеется только два вида таких переходов, найдены соответствующие им особенности средней выгоды как функции параметра семейства и доказана устойчивость этих особенностей к малому шевелению типичного семейства. Завершена классификация особенностей максимальной средней выгоды для типичных семейств. Библиография: 16 названий.

Abstract-Understanding the emergence of sustainable behavior in dynamic models of resource consumption is essential for control of coupled human and natural systems. In this paper we analyze a mathematical model of resource exploitation recently reported by the authors. The model incorporates the cognitive decision-making process of consumers and has previously been studied in a game-theoretic context as a static two-player game. In this paper we extend the analysis by allowing the agents to adapt their psychological characteristics according to simple bestresponse learning dynamics. We show that, under the selected learning scheme, the Nash Equilibrium is reachable provided certain conditions on the psychological attributes of the consumers are fulfilled. Moreover, the Equilibrium solution obtained is found to be sustainable in the sense that no players exhibit free-riding behavior, a phenomenon which occurs in the original open-loop system. In the process, via a Lyapunov-function based approach, we also provide a proof for the asymptotic global stability of the original system which was previously known to be only locally stable.

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We consider a model of exploitation of a size-structured population when the birth, growth and mortality rates depend on the individual size and interspecies competition, while the exploitation intensity is a function of the size only. For a given exploitation intensity and under natural assumptions on the rates, we establish existence and uniqueness of a nontrivial stationary state of the population. In addition, we prove existence of an exploitation intensity which maximizes a selected profit functional of exploitation.