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<p>The work is devoted to the study of the dynamic properties of discrete Lotka-Volterra mappings with degenerate skew-symmetric matrices operating in a four-dimensional simplex. Mappings of this kind have not been considered before. We have studied specific representatives of this class, since it is not possible to generalize them. For the considered cases, based on the fact that these mappings are homeomorphisms of the simplex, using the construction of the Lyapunov function and the analysis of the Jacobian spectrum, sets of limit points and positive and negative trajectories are found. The paper also proposes a new methodology for constructing these sets, splitting the simplex into polyhedra. The considered specific representatives of these mappings are relevant in that they act as discrete compartmental models for epidemiological and environmental tasks.</p>

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DISCRETE DYNAMIC SYSTEMS OF LOTKA-VOLTERRA WITH DEGENERATE SKEW-SYMMETRIC MATRICES ACTING ON SIMPLEX \({{S}^{4}}\)

  • 期刊论文

  • 发表日期：

    2025年11月13日

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  • Rasul Ganikhodzhaev,
  • Dilfuza Eshmamatova,
  • Ulugbek Muminov

摘要

The work is devoted to the study of the dynamic properties of discrete Lotka-Volterra mappings with degenerate skew-symmetric matrices operating in a four-dimensional simplex. Mappings of this kind have not been considered before. We have studied specific representatives of this class, since it is not possible to generalize them. For the considered cases, based on the fact that these mappings are homeomorphisms of the simplex, using the construction of the Lyapunov function and the analysis of the Jacobian spectrum, sets of limit points and positive and negative trajectories are found. The paper also proposes a new methodology for constructing these sets, splitting the simplex into polyhedra. The considered specific representatives of these mappings are relevant in that they act as discrete compartmental models for epidemiological and environmental tasks.