# Persistence and the Global Attractor Conjecture

Persistence is a concept from population dynamics. In words, a model of species interactions (e.g. predator-prey) is persistent if, given an initial condition where all species have non-zero population, none of the species can become extinct. The meaning in chemical reaction network theory is analogous: a CRN model is called persistent if, given an initial condition where all chemical species have non-zero concentration, none of the species can be completely "used up", i.e. none of their concentrations can tend to zero. In mathematical terms, a CRN model is persistent if

$\omega(x) \cap \partial \mathbb{R}_{\geq 0}^n = \empty\ \forall x \in \mathrm{int}(\mathbb{R}_{\geq 0}^n)$

where $\omega(x)$ denotes the omega limit set of $x$.

## The persistence conjecture

Feinberg - 1987
Given a chemical reaction network system, assume:

1. All reactions follow mass action kinetics,
2. The network is weakly reversible and
3. Each trajectory remains bounded.

Then the system is persistent.

## Counterexample for power law kinetics

The assumption that all reactions obey the law of mass action is an essential part of the conjecture. Consider the following simple example network:

$2A \begin{array}{c} k_1 \\ \rightleftharpoons \\ k_2 \end{array} A + B$

Let $a, b$ represent the concentrations of $A$ and $B$ respectively. It is easy to demonstrate that $b$ cannot tend to zero under standard kinetic assumptions. Note also that $a + b = c$, where $c$ is a constant, and that all trajectories are bounded as a result. Under mass action kinetics,

$\dot{a} = -k_1 a^2 + k_2 a b = -(k_1 + k_2) a^2 + k_2 a c$

For $a \ll 1$, the second term $k_2 a c$ dominates, and so $\dot{a} > 0$, implying that the system is persistent. Now consider the same system under more general kinetics, where the right-to-left reaction rate has been replaced by more general power law kinetics:

$\dot{a} = -k_1 a^2 + k_2 a^3 b = -k_1 a^2 - k_2 a^4 + k_2 a^3 c$

In this case, for $a \ll 1$ the term $-k_1 a^2$ dominates, and so $\dot{a} < 0$, meaning that the system is not persistent.