Global Continuation of Stable Periodic Orbits in Systems of Competing Predators

TL;DR

The article develops a rigorous, computer-assisted continuation framework to prove the global existence of a family of stable positive periodic orbits in a three-species predator–prey system with Holling type II functional response, connecting two boundary limit cycles via transcritical bifurcations.A zero-finding problem is formulated on a Fourier–Chebyshev coefficient space, and a Newton-like fixed-point operator is shown to be a contraction near a high-order numerical approximation, with interval-arithmetic verifications ensuring rigor. The main result demonstrates a stable branch parameterized by the carrying capacity $\kappa$ that spans from $\mathcal{C}_1$ to $\mathcal{C}_2$, with stability established along the entire branch and a detailed a posteriori framework for both existence and stability proofs. The methods are broadly applicable to other nonpolynomial vector fields and provide a concrete resolution of the stable connection problem proposed by Butler and Waltman.

Abstract

We develop a continuation technique to obtain global families of stable periodic orbits, delimited by transcritical bifurcations at both ends. To this end, we formulate a zero-finding problem whose zeros correspond to families of periodic orbits. We then define a Newton-like fixed-point operator and establish its contraction near a numerically computed approximation of the family. To verify the contraction, we derive sufficient conditions expressed as inequalities on the norms of the fixed-point operator, and involving the numerical approximation. These inequalities are then rigorously checked by the computer via interval arithmetic. To show the efficacy of our approach, we prove the existence of global families in an ecosystem with Holling's type II functional response, and thereby solve a stable connection problem proposed by Butler and Waltler in 1981. Our method does not rely on restricting the choice of parameters and is applicable to many other systems that numerically exhibit global families.

Figures (3)

• Figure 1: The parameter values are set to \ref{['parameter-set']}. (a) Fourier--Chebyshev approximation (with $K = 20$, $N = 30$, see Section \ref{['sec:contraction']}) of the global family of stable positive periodic orbits to the system \ref{['eq:original_model']} obtained in Theorem \ref{['thm:main']}. Within a distance $10^{-10}$ (in $C^0$-norm), there exists an exact family of stable periodic orbits. The true family is so close to the approximation that they are visually indistinguishable. The orange rings corresponds to the boundary limit circles $\mathcal{C}_1$ and $\mathcal{C}_2$ at $\hat{\kappa}_1 \approx 93.0545$ and $\hat{\kappa}_2 \approx 126.3145$, respectively. (b) Dependence of the minimal period on $\kappa \in [92, 129]$ along the global family. Here $\tau > 0$ is a time-rescaling parameter introduced in \ref{['eq:normalization']}.
• Figure 2: Real part of the non-trivial Floquet exponents $\mu_1$ and $\mu_2$ associated with the global family of periodic orbits detailed in Theorem \ref{['thm:main']}. The dashed red line $\{\mathrm{Re}(\mu) = 0\}$ is crossed twice -- at the intersection with the vertical dotted orange lines --, corresponding to the transcritical bifurcations at the invariant boundary planes $Q_1$ and $Q_2$ around $\hat{\kappa}_1 \approx 93.0545$ and $\hat{\kappa}_2 \approx 126.3145$, respectively. The real part of the Floquet exponents coincides when the multipliers $e^{2 \pi \mu_1}$ and $e^{2 \pi \mu_2}$ are complex conjugate. The line width is chosen sufficiently large to encompass the error.
• Figure 3: The parameter values are $a_2 = 41$, $d_1 = 0.8$, $d_2 = 0.5$, and $m_1 = m_2 = y_1 = y_2 = \gamma = 1$. (a) Numerical approximation of the global family for $a_1 = 6$. The orange rings corresponds to a numerical observation of a period-doubling bifurcation. (b) Numerical approximation of the Floquet multipliers associated to (a). The orange cross marks the crossing through $-1$, i.e., a potential period-doubling bifurcation. (c) Projection onto the $(X_1, X_2)$-plane of a numerical chaotic attractor in $\mathbb{R}^3_+$ for $a_1 = 4$.

Theorems & Definitions (23)

• Theorem 1.1: Global family of stable periodic orbits
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: Self-map
• proof
• Lemma 2.5: Contraction