On mixing in the class of quadratic stochastic operators

We study different types of limit behavior of quadratic stochastic operators acting on ℓ1ℓ1 (or ℓd1) spaces in both strong and uniform topologies. The main motif of the paper is to express the uniform and strong asymptotic stability of the quadratic stochastic operator in terms of convergence of the associated (linear) nonhomogeneous Markov chain. We also examine which type of uniform convergence of iterates of the quadratic stochastic operator is prevalent.