Aaron A. King, Ph.D.
Assistant Professor of Ecology & Evolutionary Biology and MathematicsUniversity of Michigan
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The Rainbow Bridge:
Hamiltonian Limits and Resonance in Predator-Prey Dynamics
Journal of Mathematical Biology, 39: 439-69, 1999.
Abstract
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, 
, in which are contained infinite numbers of
periodic, quasiperiodic and chaotic motions. From , there
emanates a surface, 
, of Nejmark-Sacker bifurcations whereby the
annual oscillations induced by seasonality are destabilized.
Connecting and 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 , 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
and 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.
For reprints of this paper, contact me at aaron.king@umich.edu. An electronic reprint (PDF) is also available.