## Introducing Vacuous Persistence

The previous blog post “Persistence via reachability vs. via Petri nets” mentioned vacuous persistence. This post is devoted to this concept.

$\omega$-limit point (Definition)
Consider a trajectory $c$ in a closed subset $P \subseteq {\mathbb{R}}^{n}$, i.e. a map $c : {\mathbb{R}}_{\geqslant{0}} \rightarrow P$. An $\omega$-limit point of $c$ is any point in $P$ that $c$ approaches in discrete time. More precisely, an $\omega$-limit point of $c$ is any point $x \in P$ such that $x = \lim_{k \rightarrow \infty}c( t_{k})$ for some sequence ${(t_{k})}_{k \geqslant 0}$ in $\mathbb{R}_{\geqslant{0}}$ such that $\lim_{k \rightarrow \infty} t_{k} = \infty$.

Persistent trajectory (Definition)
A trajectory in ${\mathbb{R}}_{\geqslant{0}}^{n}$ is persistent if it has no omega-limit points on the boundary $\partial\mathbb{R}_{\geqslant{0}}^{n} = \mathbb{R}_{\geqslant{0}}^{n}\!\setminus\!\mathbb{R}_{>0}^{n}$.

If $c$ is a trajectory in ${\mathbb{R}}_{\geqslant{0}}^{n}$ whose components $c_{1}, \ldots, c_{n}$ quantify $n$ particular entities, to say that $c$ is persistent is to say that these entities do not experience recurring, worse-trending episodes of near extinction, and in particular do not approach extinction.

Positive dynamical system (Definition)
A dynamical system ${\dot{x}} = f(x)$, where $f$ is a vector field ${\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$, is a positive dynamical system if the nonnegative orthant ${\mathbb{R}}_{\geqslant{0}}^{n}$ is forward-invariant, i.e. if all solution trajectories originating in $\mathbb{R}_{\geqslant{0}}^{n}$ range in $\mathbb{R}_{\geqslant{0}}^{n}$.

Examples of positive dynamical systems include the dynamical systems that describe the kinetics of mass-action reaction networks.

Persistent dynamical system (Definition)
A positive dynamical system in ${\mathbb{R}}^{n}$ is persistent if all solution trajectories originating in ${\mathbb{R}}_{>0}^{n}$ are persistent.

I will state an equivalent formulation of persistence which applies to reaction networks and provides a convenient transition to vacuous persistence. I will use the language of Chemical Reaction Network Theory. For readers unfamiliar with this area, two classic references are the lecture notes of Martin Feinberg at http://www.che.eng.ohio-state.edu/~Feinberg/LecturesOnReactionNetworks/ and the tutorial of Jeremy Gunawardena at http://www.jeremy-gunawardena.com/papers/crnt.pdf

If $P$ is a (nonnegative) stoichiometric compatibility class of a reaction network, I shall denote $P_{>0}$ and $P_{\ngtr{0}}$ the (relative) interior and the (relative) boundary of $P$ respectively. These notations are chosen to highlight the fact that the interior $P_{>0}$ consists of the points of $P$ whose components are all positive, and the boundary $P_{\ngtr{0}}$ consists of the points of $P$ whose components are not all positive. We have $P = P_{>0} \sqcup P_{\ngtr{0}}$. I shall say that the class $P$ is degenerate if $P = P_{\ngtr{0}}$, or equivalently if $P_{>0} = \varnothing$. Also, a set $\mathcal{Z}$ of species is $P$-admissible if there exists in $P$  a point of which $\mathcal{Z}$ is the support.

Persistence for mass-action reaction networks (Reformulated definition)
For every nondegenerate stoichiometric compatibility class $P$, there are no omega-limit points on the boundary $P_{\ngtr{0}}$ for trajectories in $P$ originating in the interior $P_{>0}$.

Vacuous persistence for mass-action reaction networks (Definition)
For every nondegenerate stoichiometric compatibility class $P$, there are no omega-limit points on the boundary $P_{\ngtr{0}}$ for trajectories in $P$.

Ordinary persistence allows opportunities for non-persistence, in the sense that there can be boundary points that are omega-limit points, as long as they are so for trajectories originating on (and confined to) the boundary. The attribute “vacuous” is to indicate that such opportunities for non-persistence do not exist in vacuous persistence. One can see that vacuous persistence is equivalent to persistence together with the non-existence of trajectories entirely contained in the boundary of nondegenerate stoichiometric compatibility classes. It has been suggested to me (on 5 January 2010) that vacuous persistence might be equivalent to persistence together with the non-existence of equilibria on the boundary of nondegenerate stoichiometric compatibility classes. This condition is obviously necessary, but it is not yet fully established that it is sufficient.

Persistence has long been an area of interest in mathematical modeling, in particular in population modeling. To my knowledge, mathematical models of population dynamics assume that all living species (plants, animals, etc) are present at the chosen initial time. This may explain why vacuous persistence has as far as I can tell not been considered before. But vacuous persistence is relevant in chemistry because chemical species can appear after initial time as a result of chemical reactions, e.g. association (i.e. binding), dissociations (i.e. unbinding), isomerization, etc. The box below shows an if-and-only-if condition for the vacuous persistence of mass-action networks with bounded trajectories. The result is from my paper Reachability, Persistence, and Constructive Chemical Reaction Networks.

 Characterization of vacuous persistence (Theorem) Consider a mass-action reaction network for which all concentration trajectories are bounded. The following are equivalent: The reaction network is vacuously persistent.   Among the subsets of the set of all species, only the full set is both reach-closed and $P$-admissible for every nondegenerate stoichiometric compatibility class $P$.

What it means to be reach-closed is discussed in the paper. I might in the future have a blog post dedicated to reachability. The fact that there is this nice structural (i.e. independent of rate constants) characterization (i.e. necessary and sufficient condition) makes me think that vacuous persistence is a natural concept of non-extinction for mass-action networks with bounded trajectories. This is of course a very subjective view, as evidenced by disputable attributes such as “nice” and “natural”.