Aaron A. King, Ph.D.

Professor of Ecology & Evolutionary Biology and Mathematics
University of Michigan

Subharmonic resonance and multi-annual oscillations in northern mammals: a nonlinear dynamical systems perspective

W. M. Schaffer, B. S. Pederson, B. K. Moore, O. Skarpaas, A. A. King, and T. V. Bronnikova
Chaos, Solitons, and Fractals 12: 251–264, 2001.
Seasonal predator-prey dynamics and complex dynamics

We conjecture that the well-known oscillations (3- to 5-yr and I0-yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predator-prey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a bare-bones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxims in the corresponding time series should be more nearly constant than their amplitude - a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.

The official version of the paper is here.   Please contact Prof. King if you'd like a reprint.

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