Abstract
We analyze the relationship between the macroscopic and microscopic descriptions of two-state systems, in particular the regime in which the microscopic one shows sustained ‘stochastic oscillations’ while the macroscopic tends to a fixed point. We propose a quantification of the oscillatory appearance of the fluctuating populations, and show that good stochastic oscillations are present if a parameter of the macroscopic model is small, and that no microscopic model will show oscillations if that parameter is large. The transition between these two regimes is smooth. In other words, given a macroscopic deterministic model, one can know whether any microscopic stochastic model that has it as a limit, will display good sustained stochastic oscillations.
Similar content being viewed by others
References
G Abramson et al, Bull. Math. Biol. 65, 519 (2003)
I D Peixoto and G Abramson, Ecology 87, 873 (2006)
L Giuggioli et al, Bull. Math. Biol. 67, 1135 (2005)
G Abramson, Mathematical modelling of Hantavirus: From the mean field to the individual level, in: Progress in mathematical biology research edited by James T Kelly (Nova Science Publishers, 2008), pp. 219–245
S Risau-Gusmán and G Abramson, Eur. Phys. J. B60, 515 (2007)
E B Wilson and O M Lombard, Pathology 31, 367 (1945)
M S Bartlett, J. R. Stat. Soc. A120, 48 (1957)
J Güémez and M A Matías, Phys. Rev. E48, R2351 (1993)
M A Matías and J Güémez, J. Chem. Phys. 103, 1597 (1995)
H Wang and Q Li, J. Chem. Phys. 108, 7555 (1998)
A J McKane and T J Newman, Phys. Rev. E70, 041902 (2004)
N G van Kampen, Stochastic processes in physics and chemistry (Elsevier Science B.V., Amsterdam, 2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abramson, G., Risau-Gusman, S. Assessing the quality of stochastic oscillations. Pramana - J Phys 70, 1047–1053 (2008). https://doi.org/10.1007/s12043-008-0109-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-008-0109-x