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Stochastically modeled weakly reversible reaction networks with a single linkage class

Published online by Cambridge University Press:  04 September 2020

David F. Anderson*
Affiliation:
University of Wisconsin-Madison
Daniele Cappelletti*
Affiliation:
ETH Zurich
Jinsu Kim*
Affiliation:
University of California at Irvine
*
*Postal address: Department of Mathematics, University of Wisconsin-Madison. Email address: anderson@math.wisc.edu
**Postal address: Department of Biosystems Science and Engineering, ETH-Zurich.
***Postal address: Department of Mathematics, University of California, Irvine.

Abstract

It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary, and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the n-step embedded discrete-time Markov chain.

Type
Research Papers
Copyright
© Applied Probability Trust 2020

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