Evolution algebra of a bisexual population

TL;DR

This paper introduces an algebraic framework for bisexual population inheritance, analyzing its properties, structure, and dynamics, including fixed points and special cases, to better understand population evolution.

Contribution

It develops the concept of an evolution algebra for bisexual populations, studying its properties, structure, and dynamics, and provides detailed analysis of special cases and fixed points.

Findings

The algebra is commutative, not associative, and not power associative.

It is a dibaric algebra with a baric square, and forms a Banach algebra.

Necessary conditions for fixed points and limit sets are established.

Abstract

We introduce an (evolution) algebra identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. The basic properties of the algebra are studied. We prove that this algebra is commutative (and hence flexible), not associative and not necessarily power associative. We show that the evolution algebra of the bisexual population is not a baric algebra, but a dibaric algebra and hence its square is baric. Moreover, we show that the algebra is a Banach algebra. The set of all derivations of the evolution algebra is described. We find necessary conditions for a state of the population to be a fixed point or a zero point of the evolution operator which corresponds to the evolution algebra. We also establish upper estimate of the limit points set for trajectories of the evolution operator. Using the necessary conditions we give a detailed analysis of a special case of the evolution algebra (bisexual population of which has a preference on type "1" of females and males). For such a special case we describe the full set of idempotent elements and the full set of absolute nilpotent elements.