A generalization of Birch's theorem and vertexbalanced steady states for generalized massaction systems
Abstract
Massaction 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 nonlinear 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 vertexbalanced steady state for a generalized massaction 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 vertexbalanced 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 vertexbalanced steady states.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.06919
 Bibcode:
 2018arXiv180206919C
 Keywords:

 Mathematics  Dynamical Systems;
 37N25;
 92C42;
 80A30;
 92D25;
 92C45
 EPrint:
 21 pages