Extinction in Reaction Network Models

TL;DR

This paper investigates extinction phenomena in reaction network models, introducing notions of weak and strong extinction, and relating them to network structure using Lyapunov functions, with results on deficiency-zero and linear networks.

Contribution

It introduces new concepts of weak and strong extinction, and provides geometric and analytical tools to analyze extinction in reaction networks, especially for deficiency-zero and linear cases.

Findings

Weak extinction occurs in bounded invariant subspaces of deficiency-zero, non-weakly reversible networks.

Species outside terminal strongly connected components undergo strong extinction in linear, non-weakly reversible networks.

An example network demonstrates weak but not strong extinction, highlighting their differences.

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.