Persistence

A chemical reaction network endowed with some kinetic scheme is said to be persistent if initial present species are not allowed to approach extinction, either directly or asymptotically (i.e. as a limit of subsequences). That is to say, a network is persistent if $c_{i}(0)>0$ for $i=1,\ldots ,m$ implies

$\liminf _{t>0}c_{i}(t)>0$

for $i=1,\ldots ,m$. For systems which are known to have bounded trajectories, this is equivalent to the condition $\omega ({\vec {c}}(0))\cap \partial \mathbb {R} _{\geq 0}^{n}=\emptyset$ where $\omega ({\vec {c}}(0))$ is the omega-limit set of the trajectory with initial condition ${\vec {c}}(0)$.

See also Persistence and the Global Attractor Conjecture.