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 
is the time derivative of the concentration vector
, 
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 
. The Schnakenberg
system has a cubic monomial from a reaction with
two molecules of one species and one molecule of the other in its source complex.
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.
