Separable Quadratic Stochastic Operators

Abstract

We consider quadratic stochastic operators, which are separable as a product of two linear operators. Depending on properties of these linear operators we classify the set of the separable quadratic stochastic operators: first class of constant operators, second class of linear and third class of nonlinear (separable) quadratic stochastic operators. Since the properties of operators from the first and second classes are well-known, we mainly study properties of the operators of the third class. We describe some Lyapunov functions of the operators and apply them to study $\omega$-limit sets of the trajectories generated by the operators. Also we compare our results with known results of the theory of quadratic operators and give some open problems.