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Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective
Institution:1. Kotel''nikov''s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov 410019, Russia;2. Saratov State University, Astrakhanskaya, 83 410012, Russia;3. Yuri Gagarin State Technical University of Saratov, Politehnicheskaya 77,Saratov 410054, Russia;4. St.Petersburg State University, Universitetskiy proezd, 28, Peterhof 198504, Russia
Abstract:We conjecture that the well-known oscillations (3- to 5-yr and 10-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 maxima 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.
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