Finding complex balanced and detailed balanced realizations of chemical reaction networks

Reversibility, weak reversibility and deficiency, detailed and complex balancing are generally not "encoded" in the kinetic differential equations but they are realization properties that may imply local or even global asymptotic stability of the underlying reaction kinetic system when further conditions are also fulfilled. In this paper, efficient numerical procedures are given for finding complex balanced or detailed balanced realizations of mass action type chemical reaction networks or kinetic dynamical systems in the framework of linear programming. The procedures are illustrated on numerical examples.Comment: submitted to J. Math. Che

Finding weakly reversible realizations of chemical reaction networks using optimization

An algorithm is given in this paper for the computation of dynamically equivalent weakly reversible realizations with the maximal number of reactions, for chemical reaction networks (CRNs) with mass action kinetics. The original problem statement can be traced back at least 30 years ago. The algorithm uses standard linear and mixed integer linear programming, and it is based on elementary graph theory and important former results on the dense realizations of CRNs. The proposed method is also capable of determining if no dynamically equivalent weakly reversible structure exists for a given reaction network with a previously fixed complex set.Comment: 18 pages, 9 figure

Kinetic discretization of one-dimensional nonlocal flow models

We show that one-dimensional nonlocal flow models in PDE form with Lighthill-Whitham-Richards flux supplemented with appropriate in- and out-flow terms can be spatially discretized with a finite volume scheme to obtain formally kinetic models with physically meaningful reaction graph structure. This allows the utilization of the theory of chemical reaction networks, as demonstrated here via the stability analysis of a flow model with circular topology. We further propose an explicit time discretization and a Courant-Friedrichs-Lewy condition ensuring many advantageous properties of the scheme. Additional characteristics, including monotonicity and the total variation diminishing property are also discussed. Copyright (C) 2022 The Authors

Traffic Reaction Model

In this paper a novel non-negative finite volume discretization scheme is proposed for certain first order nonlinear partial differential equations describing conservation laws arising in traffic flow modelling. The spatially discretized model is shown to preserve several fundamentally important analytical properties of the conservation law (e.g., conservativeness, capacity) giving rise to a set of (second order) polynomial ODEs. Furthermore, it is shown that the discretized traffic flow model is formally kinetic and that it can be interpreted in a compartmental context. As a consequence, traffic networks can be represented as reaction graphs. It is shown that the model can be equipped with on- and off- ramps in a physically meaningful way, still preserving the advantageous properties of the discretization. Numerical case studies include empirical convergence tests, and the stability analysis presented in the paper paves the way to scalable observer and controller design