A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study ξ (s) -QSO defined on 2D simplex. We first classify ξ (s) -QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.Mathematics Subject Classification 2010: 37E99; 37N25; 39B82, 47H60, 92D25. Key words: Quadratic stochastic operator; ℓ−Volterra quadratic stochastic operator; ξ (s) −quadratic stochastic operator; permuted ℓ−Volterra quadratic stochastic operator; dynamics10 = {V 11 , V 12 }, K 11 = {V 19 , V 31 }, K 12 = {V 20 , V 33 }, K 13 = {V 21 , V 32 }, K 14 = {V 22 , V 34 }, K 15 = {V 23 , V 36 }, K 16 = {V 24 , V 35 }, K 17 = {V 25 }, K 18 = {V 26 , V 27 }, K 19 = {V 28 }, K 20 = {V 29 , V 30 }.

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with dim E 2 = dim E − 1. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.Mathematics Subject Classification: 46S10, 82B26, 12J12, 39A70, 47H10, 60K35.

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Abstract. The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e. when the algebra is not associative, we provide a sufficient condition to get trivial derivation on generic Volterra algebras. Furthermore, we describe all derivations of three dimensional generic Volterra algebras, which allowed us to prove that any local derivation is a derivation of the algebra.

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