The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex

U. U. Zhamilov   ; U. A. Rozikov

Sbornik. Mathematics, Tome 200 (2009) no. 9, pp. 1339-1351

Résumé

An arbitrary strictly non-Volterra quadratic operator on the 2-simplex is shown to have a unique fixed point, which is established as being nonattracting. A description of the $\omega$-limit set of the trajectory of some subclasses of these operators is obtained. Strictly non-Volterra operators, as distinct from the Volterra operators, are shown to have cyclic trajectories. For two particular operators, we show that there exists a cyclic trajectory with period 3. Each trajectory which starts at the boundary of the simplex converges to this cyclic trajectory, whereas trajectories which begin at an interior point of the simplex (not at the fixed point) must diverge. Furthermore, the $\omega$-limit set of such a trajectory is infinite, and lies at the boundary of the simplex. Also, we study subclasses of strictly non-Volterra operators whose trajectories tend to a cyclic trajectory with period 2. Bibliography: 18 titles.

Détail

Keywords: quadratic stochastic operators; simplex; trajectory.

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