# 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

where denotes the omega limit set of .

## The persistence conjecture

**Feinberg - 1987**

Given a chemical reaction network system, assume:

- All reactions follow mass action kinetics,
- The network is weakly reversible
*and* - 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:

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

For , the second term dominates, and so , 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 Now consider the same system under more general kinetics, where the right-to-left reaction rate has been replaced by more general power law kinetics:

In this case, for the term dominates, and so , meaning that the system is not persistent.