A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems

Gheorghe Craciun; Stefan Muller; Casian Pantea; Polly Y. Yu; Gheorghe Craciun; Stefan Muller; Casian Pantea; Polly Y. Yu

doi:10.3934/mbe.2019417

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 8243-8267. doi: 10.3934/mbe.2019417

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A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems

1.
Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Dr, Madison, WI 53706, USA;
2.
Faculty of Mathematics, University of Vienna, 1010 Vienna, Austria;
3.
Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA;
4.
Department of Biomolecular Chemistry, University of Wisconsin-Madison, 420 Henry Mall, WI 53706, USA

Received: 03 May 2019 Accepted: 26 August 2019 Published: 16 September 2019

Mass-action kinetics and its generalizations appear in mathematical models of (bio)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.
- reaction network,
- generalized Birch's theorem,
- generalized mass-action,
- vertex-balanced steady states
Citation: Gheorghe Craciun, Stefan Muller, Casian Pantea, Polly Y. Yu. A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 8243-8267. doi: 10.3934/mbe.2019417

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Abstract

Mass-action kinetics and its generalizations appear in mathematical models of (bio)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.

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