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Quasi-toric differential inclusions
1. | Department of Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, USA |
2. | Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, USA |
Toric differential inclusions play a pivotal role in providing a rigorous interpretation of the connection between weak reversibility and the persistence of mass-action systems and polynomial dynamical systems. We introduce the notion of quasi-toric differential inclusions, which are strongly related to toric differential inclusions, but have a much simpler geometric structure. We show that every toric differential inclusion can be embedded into a quasi-toric differential inclusion and that every quasi-toric differential inclusion can be embedded into a toric differential inclusion. In particular, this implies that weakly reversible dynamical systems can be embedded into quasi-toric differential inclusions.
References:
[1] |
D. Anderson,
A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.
doi: 10.1137/11082631X. |
[2] |
D. Angeli, P. De Leenheer and E. Sontag,
A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks, Math. Biosci., 210 (2007), 598-618.
doi: 10.1016/j.mbs.2007.07.003. |
[3] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511804441. |
[4] |
J. Brunner and G. Craciun,
Robust persistence and permanence of polynomial and power law dynamical systems, SIAM J. Appl. Math, 78 (2018), 801-825.
doi: 10.1137/17M1133762. |
[5] |
G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, preprint, arXiv: 1501.02860.
doi: 1501.02860. |
[6] |
G. Craciun,
Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAGA, 3 (2019), 87-106.
doi: 10.1137/17M1129076. |
[7] |
G. Craciun and A. Deshpande, Endotactic networks and toric differential inclusions, preprint, arXiv: 1906.08384.
doi: 1906.08384. |
[8] |
G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels,
Toric dynamical systems, J. Symb. Comp., 44 (2009), 1551-1565.
doi: 10.1016/j.jsc.2008.08.006. |
[9] |
G. Craciun, F. Nazarov and C. Pantea,
Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.
doi: 10.1137/100812355. |
[10] |
M. Feinberg, Lectures on chemical reaction networks, Notes of Lectures Given at the Mathematics Research Center, University of Wisconsin, (1979), 49 pp. Google Scholar |
[11] |
M. Feinberg,
Chemical reaction network structure and the stability of complex isothermal reactors-I. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268.
doi: 10.1016/0009-2509(87)80099-4. |
[12] |
W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, The William
H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.
doi: 10.1515/9781400882526. |
[13] |
M. Gopalkrishnan, E. Miller and A. Shiu,
A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797.
doi: 10.1137/130928170. |
[14] |
C. M. Guldberg and P. Waage, Studies concerning affinity, J. Chem. Educ., 63 (1986), 1044.
doi: 10.1021/ed063p1044. |
[15] |
J. Gunawardena, Chemical reaction network theory for in-silico biologists, Notes available for download at http://vcp.med.harvard.edu/papers/crnt.pdf, (2003). Google Scholar |
[16] |
A. Kushnir and S. Liu, On linear transformations of intersections, ECON - Working Papers, 255 (2017), 17 pp. Google Scholar |
[17] |
C. Pantea,
On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.
doi: 10.1137/110840509. |
[18] |
R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. |
[19] |
E. Voit, H. Martens and S. Omholt, 150 years of the mass action law, PLOS Comput. Biol., 11 (2015), e1004012.
doi: 10.1371/journal.pcbi.1004012. |
[20] |
P. Yu and G. Craciun,
Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.
doi: 10.1002/ijch.201800003. |
[21] |
G. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4613-8431-1. |
show all references
References:
[1] |
D. Anderson,
A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.
doi: 10.1137/11082631X. |
[2] |
D. Angeli, P. De Leenheer and E. Sontag,
A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks, Math. Biosci., 210 (2007), 598-618.
doi: 10.1016/j.mbs.2007.07.003. |
[3] |
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
doi: 10.1017/CBO9780511804441. |
[4] |
J. Brunner and G. Craciun,
Robust persistence and permanence of polynomial and power law dynamical systems, SIAM J. Appl. Math, 78 (2018), 801-825.
doi: 10.1137/17M1133762. |
[5] |
G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, preprint, arXiv: 1501.02860.
doi: 1501.02860. |
[6] |
G. Craciun,
Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAGA, 3 (2019), 87-106.
doi: 10.1137/17M1129076. |
[7] |
G. Craciun and A. Deshpande, Endotactic networks and toric differential inclusions, preprint, arXiv: 1906.08384.
doi: 1906.08384. |
[8] |
G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels,
Toric dynamical systems, J. Symb. Comp., 44 (2009), 1551-1565.
doi: 10.1016/j.jsc.2008.08.006. |
[9] |
G. Craciun, F. Nazarov and C. Pantea,
Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.
doi: 10.1137/100812355. |
[10] |
M. Feinberg, Lectures on chemical reaction networks, Notes of Lectures Given at the Mathematics Research Center, University of Wisconsin, (1979), 49 pp. Google Scholar |
[11] |
M. Feinberg,
Chemical reaction network structure and the stability of complex isothermal reactors-I. The deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268.
doi: 10.1016/0009-2509(87)80099-4. |
[12] |
W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, 131, The William
H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993.
doi: 10.1515/9781400882526. |
[13] |
M. Gopalkrishnan, E. Miller and A. Shiu,
A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2014), 758-797.
doi: 10.1137/130928170. |
[14] |
C. M. Guldberg and P. Waage, Studies concerning affinity, J. Chem. Educ., 63 (1986), 1044.
doi: 10.1021/ed063p1044. |
[15] |
J. Gunawardena, Chemical reaction network theory for in-silico biologists, Notes available for download at http://vcp.med.harvard.edu/papers/crnt.pdf, (2003). Google Scholar |
[16] |
A. Kushnir and S. Liu, On linear transformations of intersections, ECON - Working Papers, 255 (2017), 17 pp. Google Scholar |
[17] |
C. Pantea,
On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.
doi: 10.1137/110840509. |
[18] |
R. T. Rockafellar, Convex analysis, Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970. |
[19] |
E. Voit, H. Martens and S. Omholt, 150 years of the mass action law, PLOS Comput. Biol., 11 (2015), e1004012.
doi: 10.1371/journal.pcbi.1004012. |
[20] |
P. Yu and G. Craciun,
Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.
doi: 10.1002/ijch.201800003. |
[21] |
G. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4613-8431-1. |








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