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In this paper we consider a class of strictly non-Volterra quadratic stochastic operators defined on the two-dimensional simplex. We show that such operators have a unique fixed point and the set of limit points is either a single point or an infinite set. Algebra, Analysis and Quantum Probabilitydoi:10.1088/1742-6596/697/1/012013 fatcat:rlt35jdtdbdlnjhpc34is54tym