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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.