In this paper, we investigate the dynamics of the Lebesque quadratic stochastic operator on the set of all Lebesque measures of the set X = [0,1]. We consider the family of functions such that for any fixed x, y a probability measure P(x, y,.) is absolutely continuous with respect to usual Lebesque measure on X with simple Radon-Nikodym derivative. We construct the family of strictly non-Volterra quadratic stochastic and show that their dynamic behavior coincides with dynamic of strictly non-Volterra quadratic stochastic operator on 3-dimensional simplex.

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