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Extinction in Reaction Network Models

Pranav Agarwal, Gheorghe Craciun, Abhishek Deshpande, Jiaxin Jin

TL;DR

This work introduces two extinction notions—weak and strong—for mass-action reaction networks and ties them to Lyapunov functions via LaSalle’s invariance principle. It proves that deficiency-zero, non–weakly reversible networks admit a linear Lyapunov function, which implies weak extinction within bounded stoichiometric classes, while linear non–weakly reversible networks exhibit strong extinction outside terminal components. The authors provide a concrete construction of these Lyapunov functions (via Stiemke’s theorem) and illustrate the distinctions between weak and strong extinction with an Ivanova-based example. The results advance understanding of long-term behavior in reaction networks and offer a framework for predicting which species may vanish under various structural constraints. Overall, the paper bridges network structure, Lyapunov analysis, and extinction outcomes with potential applications in biology and epidemiology.

Abstract

In this paper, we study extinction in dynamical systems generated by reaction networks. We introduce two notions: weak extinction and strong extinction, and relate them to the structure of the underlying network through Lyapunov functions and LaSalle's invariance principle. In particular, for all deficiency-zero networks that are not weakly reversible, we provide a geometric construction of linear Lyapunov functions. Using these functions, we establish that if these networks have bounded invariant subspaces, then they must exhibit weak extinction within every such subspace. Also, for linear networks that are not weakly reversible, we show that every species outside a terminal strongly connected component undergoes strong extinction. Moreover, in order to further emphasize the difference between weak and strong extinction, we construct an example of a reaction system (based on the Ivanova network) that exhibits weak extinction for all the species, but does not exhibit strong extinction in any species.

Extinction in Reaction Network Models

TL;DR

This work introduces two extinction notions—weak and strong—for mass-action reaction networks and ties them to Lyapunov functions via LaSalle’s invariance principle. It proves that deficiency-zero, non–weakly reversible networks admit a linear Lyapunov function, which implies weak extinction within bounded stoichiometric classes, while linear non–weakly reversible networks exhibit strong extinction outside terminal components. The authors provide a concrete construction of these Lyapunov functions (via Stiemke’s theorem) and illustrate the distinctions between weak and strong extinction with an Ivanova-based example. The results advance understanding of long-term behavior in reaction networks and offer a framework for predicting which species may vanish under various structural constraints. Overall, the paper bridges network structure, Lyapunov analysis, and extinction outcomes with potential applications in biology and epidemiology.

Abstract

In this paper, we study extinction in dynamical systems generated by reaction networks. We introduce two notions: weak extinction and strong extinction, and relate them to the structure of the underlying network through Lyapunov functions and LaSalle's invariance principle. In particular, for all deficiency-zero networks that are not weakly reversible, we provide a geometric construction of linear Lyapunov functions. Using these functions, we establish that if these networks have bounded invariant subspaces, then they must exhibit weak extinction within every such subspace. Also, for linear networks that are not weakly reversible, we show that every species outside a terminal strongly connected component undergoes strong extinction. Moreover, in order to further emphasize the difference between weak and strong extinction, we construct an example of a reaction system (based on the Ivanova network) that exhibits weak extinction for all the species, but does not exhibit strong extinction in any species.
Paper Structure (7 sections, 12 theorems, 72 equations, 3 figures)

This paper contains 7 sections, 12 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Reaction Networks
  3. Extinction
  4. Lyapunov Functions for Reaction Networks
  5. Lyapunov Functions for Deficiency-Zero, Non-Weakly Reversible Networks
  6. Extinction for Two Classes of Reaction Networks
  7. Discussion

Key Result

Proposition 1.1

Let $G = (V, E)$ be a deficiency-zero reaction network that is not weakly reversible. For any choice of rate constants $\boldsymbol{k}$, the mass-action system $(G, \boldsymbol{k})$ admits a linear Lyapunov function $\mathcal{V}$ such that

Figures (3)

  • Figure 1: Examples of reaction networks. (a) a reaction network with two linkage classes. (b) a reaction network with one linkage class and two terminal strongly connected components (circled in light green). (c) A weakly reversible reaction network with a single, strongly connected linkage class.
  • Figure 2: The reaction network considered in Example \ref{['ex:stoichiometric_compatibility_class']}.
  • Figure 3: (a) Phase space portrait of the modified Ivanova network. (b) Concentration of species $X$, $Y$, and $Z$ as a function of time (for the first few instances of time). (c) Concentration of species $X$, $Y$, and $Z$ as a function of time (for later instances of time).

Theorems & Definitions (34)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: craciun2015toriccraciun2019polynomialcraciun2020endotactic
  • Definition 2.2: feinberg1979lecturesgunawardena2003chemicalyu2018mathematical
  • Definition 2.3: feinberg1979lecturesgunawardena2003chemicalvoit2015150yu2018mathematical
  • Example 2.4
  • Definition 2.5: feinberg1979lecturesgunawardena2003chemical
  • Definition 2.6: feinberg1979lecturesgunawardena2003chemical
  • Lemma 2.7: craciun2019realizationsfeinberg2019foundations
  • ...and 24 more