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In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras, allows to investigated the set $\exp(Der(E))$. Moreover, it is proved that $\exp(Der(E))$ is a normal subgroup of $Aut(E)$, and its corresponding index is calculated., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1805.10285

Lotka-Volterra (LV) algebras are generally applied in solving biological problems and in examining the interactions among neighboring individuals. With reference to the methods applied by Gutierrez-Fernandez and Garcia in \cite{17}. this study examines the derivations of five-dimensional LV algebras. Before explicitly describing its derivation space, anti-symmetric matrices are used to define the LV algebra A.

In this paper, we introduce Volterra evolution algebras which are evolution algebras whose structural matrices are described by skew symmetric matrices. A main result of the present paper gives a connection between such kind of algebras with ergodicities of Volterra quadratic stochastic operators. Furthermore, some of properties of the considered algebras such as nilpotency, derivations have been studied as well., Comment: 19 pages

Many systems are presented using theory of nonlinear operators. A quadratic stochastic operator (QSO) is perceived as a nonlinear operator. It has a wide range of applications in various disciplines, such as mathematics, biology, and other sciences. The central problem that surrounds this nonlinear operator lies in the requirement that behavior should be studied. Nonlinear operators, even QSO (i.e., the simplest nonlinear operator), have not been thoroughly investigated. This study aims to present a new class of $\xi^{(as)}$-QSO defined on 2D simplex and to classify it into 18 non-conjugate (isomorphic) classes based on their conjugacy and the remuneration of coordinates. In addition, the limiting points of the behavior of trajectories for four classes defined on 2D simplex are examined.

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with $dimE^2 = dimE-1$. 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., Comment: 23 pages

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., Comment: 15 pages, Communications in Algebra (accepted)

In the present paper we study some algebraic properties of evolution algebras. Moreover, we reduce the study of evolution algebras of permutations to two special types of evolution algebras, idempotents and absolute nilpotent elements of the algebra. We study three-dimensional evolution algebras whose each element of evolution basis has infinite period. In addition, for an evolution algebra with some properties we describe its associative enveloping algebra., Comment: 11 pages

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 $\xi^{(s)}$--QSO defined on 2D simplex. We first classify $\xi^{(s)}$--QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators., Comment: 20 pages