# American Institute of Mathematical Sciences

September  2009, 4(3): 501-526. doi: 10.3934/nhm.2009.4.501

## Boltzmann maps for networks of chemical reactions and the multi-stability problem

 1 Institute for Cancer Research and Treatment (IRCC), Str Prov 142, Km 3.95, 10060 Candiolo (Torino), Italy 2 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received  March 2008 Revised  April 2009 Published  July 2009

Boltzmann Maps are a class of discrete dynamical systems that may be used in the study of complex chemical reaction processes. In this paper they are generalized to open systems allowing the description of non-stoichiometrically balanced reactions with unequal reaction rates. We show that they can be widely used to describe the relevant dynamics, leading to interesting insights on the multi-stability problem in networks of chemical reactions. Necessary conditions for multistability are thus identified. Our findings indicate that the dynamics produced by laws like the mass action law, can hardly produce multistable phenomena. In particular, we prove that they cannot do it in a wide range of chemical reactions.
Citation: Andrea Picco, Lamberto Rondoni. Boltzmann maps for networks of chemical reactions and the multi-stability problem. Networks & Heterogeneous Media, 2009, 4 (3) : 501-526. doi: 10.3934/nhm.2009.4.501
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