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Global stability of Wright-type equations with negative Schwarzian

Mauro Díaz, Karel Hasík, Jana Kopfová, Sergei Trofimchuk

TL;DR

This work extends the Wright-type global stability framework to scalar delay equations with decreasing nonlinearities $f$ of negative Schwarzian, under negative feedback and boundedness assumptions. It introduces two key bounding constructs, $L_a(M,m)$ and $ abla_a(M,m)$ (via $ ext{Sigma}_a$), and biases the analysis with a billiard-type separation method together with rigorous interval arithmetic to certify nonexistence of slowly oscillating periodic solutions for $f'(0)60[-37/24,0]$, thereby proving global attractivity of the trivial equilibrium in this extended range. The main result tightens the global stability threshold from Wright’s original bound to $f'(0)60[-37/24,0]$ (and improves via local analysis near the origin), with a detailed computational framework (Appendices A–F) supporting the rigorous numerics. These contributions advance the understanding of global dynamics for Wright-type delay equations and related scalar models, offering a robust approach applicable to a broader class of nonlinearities with negative Schwarzian.

Abstract

Simplicity of the $37/24$-global stability criterion announced by E.M. Wright in 1955 and rigorously proved by B. Bánhelyi et al in 2014 for the delayed logistic equation raised the question of its possible extension for other population models. In our study, we answer this question by extending the $37/24$-stability condition for the Wright-type equations with decreasing smooth nonlinearity $f$ which has a negative Schwarzian and satisfies the standard negative feedback and boundedness assumptions. The proof contains the construction and careful analysis of qualitative properties of certain bounding relations. To validate our conclusions, these relations are evaluated at finite sets of points; for this purpose, we systematically use interval analysis.

Global stability of Wright-type equations with negative Schwarzian

TL;DR

This work extends the Wright-type global stability framework to scalar delay equations with decreasing nonlinearities of negative Schwarzian, under negative feedback and boundedness assumptions. It introduces two key bounding constructs, and (via ), and biases the analysis with a billiard-type separation method together with rigorous interval arithmetic to certify nonexistence of slowly oscillating periodic solutions for , thereby proving global attractivity of the trivial equilibrium in this extended range. The main result tightens the global stability threshold from Wright’s original bound to (and improves via local analysis near the origin), with a detailed computational framework (Appendices A–F) supporting the rigorous numerics. These contributions advance the understanding of global dynamics for Wright-type delay equations and related scalar models, offering a robust approach applicable to a broader class of nonlinearities with negative Schwarzian.

Abstract

Simplicity of the -global stability criterion announced by E.M. Wright in 1955 and rigorously proved by B. Bánhelyi et al in 2014 for the delayed logistic equation raised the question of its possible extension for other population models. In our study, we answer this question by extending the -stability condition for the Wright-type equations with decreasing smooth nonlinearity which has a negative Schwarzian and satisfies the standard negative feedback and boundedness assumptions. The proof contains the construction and careful analysis of qualitative properties of certain bounding relations. To validate our conclusions, these relations are evaluated at finite sets of points; for this purpose, we systematically use interval analysis.

Paper Structure

This paper contains 13 sections, 39 theorems, 135 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Billiards and verified proofs of the inequalities
  3. A bound for the minimum of a slowly oscillating periodic solution
  4. A bound for the maximum of a slowly oscillating periodic solution
  5. Proof of Theorem \ref{['1.4']}
  6. Appendix 1: For $a=-37/24$ the curves $m= L_a(M,m)$ and $M= \Sigma _a(M,m)$ are tangent at the origin
  7. Local analysis of $m= L_{-37/24}(M,m)$
  8. Local analysis of $M = \Sigma_{a}(M,m)$
  9. Appendix A: Scripts for $L_a(M,m)$ and Lemma \ref{['L2.15']} (a), (c), (d)
  10. Appendix B: Lemma 3.16, computation of $\partial L_{-37/24}(M,m)/\partial m$ and its estimates
  11. Appendix D: MATLAB scripts for Lemma \ref{['4.8']}
  12. Appendix E: MATLAB scripts for Lemma \ref{['L4.18']}
  13. Appendix F: MATLAB scripts for Theorem \ref{['1.4']}

Key Result

Proposition 1.2

The zero equilibrium is a global attractor of 11 if and only if equation 11 does not have any slowly oscillating periodic solutions.

Figures (11)

  • Figure 1: Separation of the regions $\mathcal{U}(-{37}/{24})$, $\mathcal{L}(-{37}/{24})$.
  • Figure 1: On the left: the ball running across the billiard table and rebounding off its edges; $j_0=5$. On the right: real computation algorithm uses lower approximations of $t_j$ and $q(t_j)$; $J_0=8$.
  • Figure 1: Slowly oscillating periodic solution $x(t)$, $x(t_{max})= \mathbf M,$ of \ref{['4']} and its upper bounds.
  • Figure 1: Graphs of solution $x(t)$ (in black) and its bounding functions $x=z_{+}(t)$, $x=z_{-}(t)$, $x=\tilde{z}(t)$ (in red). Parabolic parts of the bounding graphs are shown as dashed curves, their linear parts are represented by continuous lines.
  • Figure 1: Schematic diagram with the graphs of the functions $m_k(M)$ (blue) and $\theta_n(m)$ (red), and the path of the 'billiard ball'.
  • ...and 6 more figures

Theorems & Definitions (74)

  • Definition 1.1
  • Proposition 1.2: Theorem 3.1 in BCKN
  • Proposition 1.3: Theorem 1.3 in LPRTT
  • Conjecture 1: Conjecture 1.2 in LPRTT
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6: Theorem 4.1 in BCKN
  • Corollary 1.7
  • Corollary 1.8
  • Lemma 2.1
  • ...and 64 more