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29 March 2024
 
  » arxiv » 1305.5303

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A geometric approach to the Global Attractor Conjecture
Manoj Gopalkrishnan ; Ezra Miller ; Anne Shiu ;
Date 23 May 2013
AbstractThis paper introduces the class of "strongly endotactic networks", a subclass of the endotactic networks introduced by G. Craciun, F. Nazarov, and C. Pantea. The main result states that the global attractor conjecture holds for complex-balanced systems that are strongly endotactic: every trajectory with positive initial condition converges to the unique positive equilibrium allowed by conservation laws. This extends a recent result by D. F. Anderson for systems where the reaction diagram has only one linkage class (connected component). The results here are proved using differential inclusions, a setting that includes power-law systems. The key ideas include a perspective on reaction kinetics in terms of combinatorial geometry of reaction diagrams, a projection argument that enables analysis of a given system in terms of systems with lower dimension, and an extension of Birch’s theorem, a well-known result about intersections of affine subspaces with manifolds parameterized by monomials.
Source arXiv, 1305.5303
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