A mass-action system is a polynomial ordinary differential equation associated with a reaction
network, in which the flux of each reaction is proportional to the monomial
of its source complex (Boros and Hofbauer 2020). For positive
rate constants 
, its equation is
 |
where 
and 
is the 
th
concentration. The stoichiometric subspace 
contains this vector
field, and the positive compatibility class of 
is 
. For 
with forward rate 
and reverse rate 
, the equations are 
and 
.
A global positive trajectory is persistent when
 |
for every species 
.
The system is permanent on 
when one compact set 
eventually contains every positive
trajectory in the class. The entry time may depend on the initial state.
