41 research outputs found
Weakly reversible mass-action systems with infinitely many positive steady states
We show that weakly reversible mass-action systems can have a continuum of
positive steady states, coming from the zeroes of a multivariate polynomial.
Moreover, the same is true of systems whose underlying reaction network is
reversible and has a single connected component. In our construction, we relate
operations on the reaction network to the multivariate polynomial occurring as
a common factor in the system of differential equations
Some dynamical properties of delayed weakly reversible mass-action systems
This paper focuses on the dynamical properties of delayed complex balanced
systems. We first study the relationship between the stoichiometric
compatibility classes of delayed and non-delayed systems. Using this relation
we give another way to derive the existence of positive equilibrium in each
stoichiometric compatibility class for delayed complex balanced systems. And if
time delays are constant, the result can be generalized to weakly reversible
networks. Also, by utilizing the Lyapunov-Krasovskii functional, we can obtain
a long-time dynamical property about -limit set of the complex balanced
system with constant time delays. An example is also provided to support our
results
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics
Motivated by questions in mass-action kinetics, we introduce the notion of
vertexical family of differential inclusions. Defined on open hypercubes, these
families are characterized by particular good behavior under projection maps.
The motivating examples are certain families of reaction networks -- including
reversible, weakly reversible, endotactic, and strongly endotactic reaction
networks -- that give rise to vertexical families of mass-action differential
inclusions. We prove that vertexical families are amenable to structural
induction. Consequently, a trajectory of a vertexical family approaches the
boundary if and only if either the trajectory approaches a vertex of the
hypercube, or a trajectory in a lower-dimensional member of the family
approaches the boundary. With this technology, we make progress on the global
attractor conjecture, a central open problem concerning mass-action kinetics
systems. Additionally, we phrase mass-action kinetics as a functor on reaction
networks with variable rates.Comment: v5: published version; v3 and v4: minor additional edits; v2:
contains more general version of main theorem on vertexical families,
including its accompanying corollaries -- some of them new; final section
contains new results relating to prior and future research on persistence of
mass-action systems; improved exposition throughou