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Sustained oscillations for density dependent Markov processes

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Abstract

Simulations of models of epidemics, biochemical systems, and other bio-systems show that when deterministic models yield damped oscillations, stochastic counterparts show sustained oscillations at an amplitude well above the expected noise level. A characterization of damped oscillations in terms of the local linear structure of the associated dynamics is well known, but in general there remains the problem of identifying the stochastic process which is observed in stochastic simulations. Here we show that in a general limiting sense the stochastic path describes a circular motion modulated by a slowly varying Ornstein–Uhlenbeck process. Numerical examples are shown for the Volterra predator–prey model, Sel’kov’s model for glycolysis, and a damped linear oscillator.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, USA

    Peter H. Baxendale

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

    Priscilla E. Greenwood

Authors
  1. Peter H. Baxendale
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  2. Priscilla E. Greenwood
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Correspondence to Peter H. Baxendale.

Additional information

P. Baxendale supported in part by NSF Grant DMS-05-04853. P. Greenwood supported by the Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, N.C., and the Mathematical, Computational and Modeling Sciences Center at Arizona State University.

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Baxendale, P.H., Greenwood, P.E. Sustained oscillations for density dependent Markov processes. J. Math. Biol. 63, 433–457 (2011). https://doi.org/10.1007/s00285-010-0376-2

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  • DOI: https://doi.org/10.1007/s00285-010-0376-2

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