ξs Quadratic stochastic operators : classification and limiting behaviour of their trajectories - Afifah Hanum binti Mohd Jamal (Kulliyyah of Science, International Islamic University Malaysia, 2012)
Quadratic dynamical systems have been proven to be a rich source of analysis for the investigation of dynamical properties and modeling in different domains. One of such operators is quadratic stochastic operator which naturally arises in modeling of a population dynamics. In this thesis we introduce a new class of quadratic stochastic operators called ξs-QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξs -QSO defined on 2D-simplex. We first describe ξs –QSO on 2D-simplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. We then investigate the behaviour of each operator from all six classes and prove convergence of trajectories of these classes and study their certain properties. This work also studies the limiting behaviour of one of ξs -QSO defined on two dimensional simplex. We investigate the behaviour of this operator and prove convergence of its trajectories and study its certain properties. Moreover, we have to stress that all the obtained operators are generalization and permutation of Volterra operators.