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<p>In this work, we examine the dynamical behavior of discrete Lotka–Volterra operators associated with a given signature. The primary objectives are as follows: determine the polyhedra corresponding to the signature structure of the operator; establish the order of their arrangement based on the trajectory operator; identify invariant polyhedra under the action of the operator; calculate the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-limit and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-limit sets of the trajectories; determine the transition routes governed by the transformation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(T_{\sigma _1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <msub> <mi>σ</mi> <mn>1</mn> </msub> </msub> </math></EquationSource> </InlineEquation>. The transition of points between polyhedra is analyzed for understanding the asymptotic behavior of the system.</p>

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DEGENERATE CASES IN LOTKA–VOLTERRA SYSTEMS ACTING IN THE SIMPLEX \(S^3\)

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • R. N. Ganikhodzhaev,
  • S. I. Masharipov,
  • J. X. Bekturdiyeva

摘要

In this work, we examine the dynamical behavior of discrete Lotka–Volterra operators associated with a given signature. The primary objectives are as follows: determine the polyhedra corresponding to the signature structure of the operator; establish the order of their arrangement based on the trajectory operator; identify invariant polyhedra under the action of the operator; calculate the \(\alpha \) α -limit and \(\omega \) ω -limit sets of the trajectories; determine the transition routes governed by the transformation \(T_{\sigma _1}\) T σ 1 . The transition of points between polyhedra is analyzed for understanding the asymptotic behavior of the system.