math.DS — Dynamical Systems

Limit Cycles under Concave Per-Capita Growth

Contributed by Zhenyu Sun

We study predator--prey systems with a fixed globally linear functional response and a strictly concave prey per-capita growth rate. For every $N\ge1$ and $K\ge2$, we construct a growth polynomial of degree $2N+2\lceil K/2\rceil+2$ with exactly $N$ positive periodic orbits, all hyperbolic. The construction retains a chosen center $K$-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 $N$ hyperbolic cycles are $C^2$-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/.