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Limit Cycles under Concave Per-Capita Growth

A mix of human-written and AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.CA — Classical Analysis and ODEs

Contributed by Zhenyu Sun ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We study predator--prey systems with a fixed globally linear functional response and a strictly concave prey per-capita growth rate. For every N≥1N\ge1 and K≥2K\ge2, we construct a growth polynomial of degree 2N+2⌈K/2⌉+22N+2\lceil K/2\rceil+2 with exactly NN positive periodic orbits, all hyperbolic. The construction retains a chosen center KK-jet and the low-density growth rate and carrying capacity of its limiting profile, subject to an explicit relation between the center height and curvature. A limiting Lyapunov function and return-map estimates at three amplitude scales exclude additional cycles. A dissipative extension gives exact global counts with prescribed limiting center energies. We also prove that profiles with at least NN hyperbolic cycles are C2C^2-dense in the admissible class, while preserving the entire germ at the maximum and both ecological endpoint values. The first nonzero canonical odd term determines the sharp differentiability threshold. Further results give parity restrictions for fixed focus jets, degree lower bounds, quantitative polynomial approximation under concavity and interpolation constraints, and amplitude and multiplicity interpolation by Abelian integrals. Finally, a single smooth strictly concave profile has infinitely many nested hyperbolic cycles converging to coexistence.

Author comments: 45 pages, 3 figures. Symbolic and numerical verification scripts and data are included in anc/.

Provenance statement

The text combines human-written and AI-generated material. OpenAI Codex assisted with manuscript revision, review of mathematical arguments and references, computational cross-checks, and figure preparation. Python with SymPy, NumPy, SciPy, and Matplotlib was used for symbolic checks, numerical experiments, and figures. Some parts, but not all, have been checked and understood by a human. The computational checks are finite tests and do not constitute a formal verification of the general theorems.

References

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Version history

  1. v1 · 45 pagesSubmitted by Zhenyu Sun45 pages, 3 figures. Symbolic and numerical verification scripts and data are included in anc/.CurrentNov 12, 2026