The rainbow bridge: Hamiltonian limits and resonance in predator-prey dynamics

Area of Science:

• Ecology
• Mathematical Biology
• Dynamical Systems

Background:

• Predator-prey models are crucial for understanding ecological interactions.
• Realistic ecological models often lack Hamiltonian formulations, complicating dynamic analysis.
• Hamiltonian systems, though structurally unstable, offer insights into complex dynamics.

Purpose of the Study:

• To investigate the impact of non-integrable Hamiltonian predator-prey model topology on realistic, non-Hamiltonian systems.
• To demonstrate how Hamiltonian limits influence the dynamics and bifurcation structure of seasonally forced ecological models.
• To explain the emergence of multi-annual cycles in ecological systems through bifurcation theory.

Main Methods:

• Perturbing realistic models from a Hamiltonian limit (ℋ) containing diverse motion types.
• Analyzing the surface of Neimark-Sacker bifurcations (Gamma) emanating from ℋ.
• Mapping resonance horns connecting ℋ and Gamma, identifying saddle-node bifurcations and rotation numbers.

Main Results:

• Seasonal forcing destabilizes annual oscillations via Neimark-Sacker bifurcations on Gamma.
• Resonance horns preserve invariant motions from ℋ, with boundaries indicating saddle-node bifurcations.
• Dominant attractors within horns, determined by rotation numbers, often appear as multi-annual cycles in ecological data.
• Monotonically aligned horn tips on Gamma lead to a central tendency in observable periods under fluctuating parameters.

Conclusions:

• The intricate structure of Hamiltonian limits fundamentally shapes the dynamics of non-Hamiltonian ecological models.
• The bifurcation structure, particularly resonance horns, provides a theoretical basis for observed multi-annual cycles in nature.
• Hamiltonian limits play a pivotal role in ecological modeling, contrary to conventional views on their instability and applicability.