The Sel'kov model is a planar dynamical system for self-sustained oscillations in glycolysis (Sel'kov 1968). A reduced form with a basal reaction term is
 |  |  |
(1)
|
 |  |  |
(2)
|
where primes denote derivatives of 
and 
with respect to time 
,
and 
and 
are positive parameters (Das and Acharyya 2021). At 
, the substitutions 
, 
, and 
recover Sel'kov's limiting system for reaction exponent
2.
For 
, the unique equilibrium
point is 
. The Jacobian
matrix there has determinant 
and trace 
. A negative trace
gives an attracting equilibrium point. If the
trace is positive, the equilibrium
point repels, and there is a periodic orbit
in the positive quadrant. Indeed, the vector field
points inward on the boundary away from the origin, 
bounds 
from above, and 
then bounds 
. The Poincaré-Bendixson
theorem gives a periodic orbit from the bounded
trajectories away from the repeller.
