A discrete-time dynamical system and an evolution algebra of mosquito population

Abstract

Recently, continuous-time dynamical systems, based on systems of ordinary differential equations, for mosquito populations are studied. In this paper we consider discretetime dynamical system generated by an evolution quadratic operator of mosquito population and show that this system has two fixed points, which are saddle points (under some conditions on the parameters of the system). We construct an evolution algebra taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results.

Determining When an Algebra Is an Evolution Algebra

“…They were introduced in 2008 by Tian [1] to enlighten the study of non-Mendelian genetics. Since then, a large literature has flourished on this topic (see for instance [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) motivated by the fact that these algebras have connections with group theory, Markov processes, theory of knots, systems and graph theory. For instance, in [2], the theory of evolution algebras was related to that of pulse processes on weighted digraphs and applications were provided by reviewing and enlightening a report of the National Science Foundation about air pollution achieved by the Rand Corporation.…”

An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics

“…One of these types is that formed by evolution algebras of a chicken population. Another one is that formed by evolution algebras of a mosquito population (see [54,55], respectively, for further information).…”

Evolution algebras whose evolution operator is a homomorphism

“…On the one hand, evolution algebras, which were firstly introduced by Tian in his Ph.D. thesis 1 in 2004, have many connections with different mathematical theories, which makes them especially interesting for finding links between concepts from different branches of Mathematics, such as graph theory, group theory, dynamical systems, stochastic processes and mathematical physics 2‐6 . On the other hand, they have also many applications to different scientific disciplines, such as physics, engineering and, above all, biology 7‐10 . Both aspects explain the growing interest of researchers in them today.…”