Journal of Scientific Computing

, Volume 34, Issue 2, pp 127–151

A Hierarchy of Approximations of the Master Equation Scaled by a Size Parameter


DOI: 10.1007/s10915-007-9179-z

Cite this article as:
Ferm, L., Lötstedt, P. & Hellander, A. J Sci Comput (2008) 34: 127. doi:10.1007/s10915-007-9179-z


Solutions of the master equation are approximated using a hierarchy of models based on the solution of ordinary differential equations: the macroscopic equations, the linear noise approximation and the moment equations. The advantage with the approximations is that the computational work with deterministic algorithms grows as a polynomial in the number of species instead of an exponential growth with conventional methods for the master equation. The relation between the approximations is investigated theoretically and in numerical examples. The solutions converge to the macroscopic equations when a parameter measuring the size of the system grows. A computational criterion is suggested for estimating the accuracy of the approximations. The numerical examples are models for the migration of people, in population dynamics and in molecular biology.


Master equation Reaction rate equations Linear noise approximation Moment equations 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Division of Scientific Computing, Department of Information TechnologyUppsala UniversityUppsalaSweden

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