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On a family of non-Volterra quadratic operators acting on a simplex

Uygun Jamilov, Manuel Ladra

TL;DR

This work analyzes a family of quadratic stochastic operators acting on the simplex, formed as a convex combination of non-Volterra structures and parameterized by $α$. It delivers a multi-part investigation: (i) a regular non-Volterra QSO with global attraction to the center via a Lyapunov function; (ii) a quasi strictly non-Volterra QSO in higher dimensions with invariant sets, Lyapunov structure, and finite periodic limits; (iii) the convex combination $V_α$, whose dynamics are globally attracted to a fixed interior point through a logistic-map conjugacy. The results provide explicit fixed points, Lyapunov functions, and a complete description of ω-limit sets across regimes, enhancing understanding of non-Volterra QSOs and the impact of convex combinations on population-dynamics trajectories.

Abstract

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter $α$ and study their trajectory behaviors. We showed that for any $α\in [0,1)$ the trajectories of such operator converge to a fixed point. For $α=1$ any trajectory of the operator converges to a periodic trajectory.

On a family of non-Volterra quadratic operators acting on a simplex

TL;DR

This work analyzes a family of quadratic stochastic operators acting on the simplex, formed as a convex combination of non-Volterra structures and parameterized by . It delivers a multi-part investigation: (i) a regular non-Volterra QSO with global attraction to the center via a Lyapunov function; (ii) a quasi strictly non-Volterra QSO in higher dimensions with invariant sets, Lyapunov structure, and finite periodic limits; (iii) the convex combination , whose dynamics are globally attracted to a fixed interior point through a logistic-map conjugacy. The results provide explicit fixed points, Lyapunov functions, and a complete description of ω-limit sets across regimes, enhancing understanding of non-Volterra QSOs and the impact of convex combinations on population-dynamics trajectories.

Abstract

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter and study their trajectory behaviors. We showed that for any the trajectories of such operator converge to a fixed point. For any trajectory of the operator converges to a periodic trajectory.
Paper Structure (5 sections, 17 theorems, 77 equations)

This paper contains 5 sections, 17 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and known results
  3. A regular non-Volterra quadratic stochastic operator
  4. Quasi strictly non-Volterra quadratic stochastic operator
  5. Convex combination of operators

Key Result

Theorem 2.1

For the operator $V_\theta$ the following statements are true:

Theorems & Definitions (28)

  • Theorem 2.1: Vlen
  • Theorem 2.2: Gdan
  • Theorem 2.3: Vsp
  • Theorem 2.4: GSN
  • Theorem 2.5: Jjph
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1: Khukr
  • ...and 18 more