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A quadratic stochastic operator (in short QSO) is usually used to present thetime evolution of differing species in biology. Some quadratic stochasticoperators have been studied by Lotka and Volterra. In the present paper, wefirst give a simple characterization of Volterra QSO in terms of absolutelycontinuity of discrete measures. Moreover, we provide its generalization incontinuous setting. Further, we introduce a notion of orthogonal preservingQSO, and describe such kind of operators defined on two dimensional simplex. Itturns out that orthogonal preserving QSOs are permutations of Volterra QSO. Theassociativity of genetic algebras generated by orthogonal preserving QSO isstudied too.

On Volterra and orthoganality preserving quadratic stochastic operators