Forward-Invariant Peeling in Chemical Dynamics: a Simple Case Study

Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

^⋆ Corresponding author. E-mail: ag153@le.ac.uk

Abstract

Forward-invariant peeling aims to produce forward-invariant subset from a given set in phase space. The structure of chemical kinetic equations allows us to describe the general operations of the forward-invariant peeling. For example, we study a simple reaction network with three components A₁,A₂,A₃ and reactions A₁ → A₂ → A₃ → A₁, 2A₁ ⇌ 3A₂ (without any stoichiometric conservation law). We assume that kinetics obey the classical mass action law and reaction rate constants are positive intervals 0 <k_{i min} ≤ k_i ≤ k_{i max}< ∞. Kinetics of this system is described by a system of differential inclusions. We produce forward-invariant sets for these kinetic inclusions from the sets { c | c_i ≥ 0, ∑ c_i ≥ ε } by the forward-invariant peeling (for sufficiently small ε> 0). In particular, this construction proves persistence of this kinetic system (a positive solution cannot approach the origin even asymptotically, as t → ∞).