Dynamics of Lebesgue Quadratic Stochastic Operator with Nonnegative Integers Parameters Generated by 2-Partition
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Keywords:
quadratic stochastic operator; nonlinear operator; Lebesgue measure; infinite state space; trajectory behaviorAbstract
The theory of quadratic stochastic operator (QSO) has been significantly developed since it was introduced in 1920s by Bernstein on population genetics. Over the century, many researchers have studied the behavior of such nonlinear operators by considering different classes of QSO on finite and infinite state spaces. However, all these studies do not comprehensively represent the core problem of QSO; i.e., the trajectory behavior. Recently, a class of QSO called Lebesgue QSO has been introduced and studied. Such an operator got its name based on Lebesgue measure which serves as a probability measure of the QSO. The conditions of the Lebesgue QSO have allowed us to consider the possibility of introducing a new measure for such QSO. This research presents a new class
of Lebesgue QSO with nonnegative integers parameters generated by a measurable 2-partition on the continual state space X = [ , 0 1]. This research aims to study the trajectory behavior of the QSO by reducing its infinite variables into a mapping of one-dimensional simplex. The behavior of such operators will be investigated computationally and analytically, where the computational results
conform to the analytical results. We will apply measure and probability theory as well as functional analysis to describe the limit behavior of such QSO. We show that for the new class of Lebesgue QSO generated by a 2-partition, one could analyze the behavior of such an operator by describing the existence of fixed points and periodic points of period-2. These results demonstrate that such
Lebesgue QSO generated by a measurable 2-partition can be a regular or nonregular transformation dependence of fixed parameters
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