TITLE: Constants of motion for Droop equation
QUESTION [8 upvotes]: There is an important ODE system in biochemistry, Droop's equations:
$$s'=1-s-\frac{sx}{a_1+s}$$
$$x'=a_2\big(1-\frac{1}{q}\big)x-x$$
$$q'=\frac{a_3s}{a_1+s}-a_2(q-1)$$
Relatively easy one finds a constant of motion $e^t(\frac{xq}{a_3}+s-1)$.
How to find another constant of motion? Do you have any idea how to present this system in the Hamiltonian or Lagrangian form?

REPLY [5 votes]: A complete global analysis of the Droop equations is carried out in

Lange, Kenneth;
  Oyarzun, Francisco J., The attractiveness of the Droop
  equations, Math.
  Biosci. 111, No. 2, 261-278 (1992).
  ZBL0762.92021.

The authors show that solutions are globally stable, in the sense that they converge to one of two fixed points (5) or (6) in their paper, depending on the positivity of $a_2 a_3 - a_2 - a_3 - a_1 a_2$; see Theorems 1 and 2. Authors use comparison arguments, a Lyapunov function, the Poincaré-Bendixson theorem, and the Bendixson-Dulack theorem to conclude this. Moreover, the vector field associated with the limiting 2-dimensional Droop system on the set $\{ a_3^{-1} x q + s - 1 = 0 \}$ is neither Hamiltonian nor is it conservative.