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


Institute for Cancer Research and Treatment (IRCC), Str Prov 142, Km 3.95, 10060 Candiolo (Torino), Italy


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

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017


Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i


Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091


Jacky Cresson, Bénédicte Puig, Stefanie Sonner. Stochastic models in biology and the invariance problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2145-2168. doi: 10.3934/dcdsb.2016041


Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250


Monique Chyba, Benedetto Piccoli. Special issue on mathematical methods in systems biology. Networks and Heterogeneous Media, 2019, 14 (1) : i-ii. doi: 10.3934/nhm.20191i


Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215


Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009


Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016


Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485


José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401


Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165


Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056


Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011


Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008


Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845


Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028


Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250


Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028


Yuan Gao, Guangzhen Jin, Jian-Guo Liu. Inbetweening auto-animation via Fokker-Planck dynamics and thresholding. Inverse Problems and Imaging, 2021, 15 (5) : 843-864. doi: 10.3934/ipi.2021016

2020 Impact Factor: 1.213


  • PDF downloads (71)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]