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

*Journal of Mathematical Biology*

**39**: 439–469, 1999.

In the presence of seasonal forcing, the intricate topology of non-integrable Hamiltonian predator-prey models is shown to exercise profound effects on the dynamics and bifurcation structure of more realistic schemes which do not admit a Hamiltonian formulation. The demonstration of this fact is accomplished by writing the more general models as perturbations of a Hamiltonian limit, ℋ, in which are contained infinite numbers of periodic, quasiperiodic and chaotic motions, From ℋ, there emanates a surface, Γ, of Nejmark-Sacker bifurcations whereby the annual oscillations induced by seasonality are destabilized. Connecting Γ and ℋ is a bridge of resonance horns within which invariant motions of the Hamiltonian case persist. The boundaries of the resonance horns are curves of tangent (saddle-node) bifurcations corresponding to subharmonics of the yearly cycle. Associated with each horn is a rotation number which determines the dominant frequency, or “color”, of attractors within the horn. When viewed through the necessarily coarse filter of ecological data acquisition and regardless of their detailed topology, these attractors are often indistinguishable from multi-annual cycles. Because the tips of the horns line up monotonically along Γ, it further follows that the distribution of observable periods in systems subject to fluctuating parameter values induced, for example, by year-to-year variations in the climate, will often exhibit a discernible central tendency. In short, the bifurcation structure is consistent with the observation of multi-annual cycles in Nature. Fundamentally, this is a consequence of the fact that the bridge between ℋ and Γ is a rainbow bridge. While the present analysis is principally concerned with the two-species case (one predator and one prey), Hamiltonian limits are also observed in other ecological contexts: 2n-species (n predators, n prey) systems and periodically-forced three level food chain models. Hamiltonian limits may thus be common in models involving the destruction of one species by another. Given the oft-commented upon structural instability of Hamiltonian systems and the corresponding lack of regard in which they are held as useful caricatures of ecological interactions, the pivotal role assigned here to Hamiltonian limits constitutes a qualitative break with the conventional wisdom.