Extinction in Reaction Network Models

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.