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Gonosomal algebras and operators associated to genetic systems with a single male genotype

Yolanda Cabrera Casado, Manuel Ladra, Andrés Pérez-Rodríguez

TL;DR

The paper develops the theory of gonosomal algebras for genetic systems with a single male genotype and analyzes the dynamics of the associated normalised gonosomal operators. It establishes the mathematical framework via baric algebras and the duplication construction, defines the normalised operator $\widetilde{V}$ on the simplex $S^{n,1}$, and identifies invariant sets to study trajectory limits. The authors then apply these tools to diverse sex-determination scenarios—ZW with Wolbachia feminization, XY systems including wood lemming and Arctic lemming, and a combined XY–ZW model for African cichlids—deriving explicit fixed points and convergence results. These findings yield concrete biological interpretations about genotype persistence and extinction patterns, and demonstrate the modeling framework’s adaptability to complex, non-duplicate genetic systems.

Abstract

This article is devoted to studying gonosomal algebras and operators with a single male genotype. We compute the limit points of the trajectories of the corresponding normalised gonosomal operators, describing the development of specific populations and providing the corresponding biological interpretations.

Gonosomal algebras and operators associated to genetic systems with a single male genotype

TL;DR

The paper develops the theory of gonosomal algebras for genetic systems with a single male genotype and analyzes the dynamics of the associated normalised gonosomal operators. It establishes the mathematical framework via baric algebras and the duplication construction, defines the normalised operator on the simplex , and identifies invariant sets to study trajectory limits. The authors then apply these tools to diverse sex-determination scenarios—ZW with Wolbachia feminization, XY systems including wood lemming and Arctic lemming, and a combined XY–ZW model for African cichlids—deriving explicit fixed points and convergence results. These findings yield concrete biological interpretations about genotype persistence and extinction patterns, and demonstrate the modeling framework’s adaptability to complex, non-duplicate genetic systems.

Abstract

This article is devoted to studying gonosomal algebras and operators with a single male genotype. We compute the limit points of the trajectories of the corresponding normalised gonosomal operators, describing the development of specific populations and providing the corresponding biological interpretations.

Paper Structure

This paper contains 7 sections, 16 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on gonosomal algebras and operators
  3. ZW system with male feminization
  4. XY systems with fertile XY females
  5. Generalising the wood lemming sex-determination system
  6. The Arctic lemming sex-determination system
  7. A combination of XY systems and ZW systems

Key Result

Proposition 3.1

The operator $V_\eta$, with $\frac{1}{2} < \eta < 1$, has three non-zero fixed points: $(\frac{2}{2-\eta},\frac{2}{2-\eta},\frac{-2}{2-\eta},\frac{2}{\eta})$, $(0,2,0,2)$ and $(\frac{1}{1-\eta},0,0,\frac{1}{\eta})$. Moreover, $V_{\frac{1}{2}}$ and $V_{1}$ have infinitely many non-zero fixed points:

Theorems & Definitions (36)

  • Example 2.2: V_16
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 26 more