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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?

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  • $\begingroup$ Hamiltonian form doesn't exist for three-dimensional systems: the number of equations should be even in order to be presented as $\dot{q} = \frac{\partial H}{\partial p}, \; \dot{p} = -\frac{\partial H}{\partial q}$ for some function $H(q, p)$. There is some kind of generalization called Hamiltonian-Poisson realization that can be applied to odd-dimensional systems, but I'm not an expert in this stuff, just heard this notion. ... $\endgroup$
    – Evgeny
    Commented Mar 4, 2019 at 16:48
  • $\begingroup$ @Evgeny sure, but we could add more variables, something like $\dot q s$ or any other.. $\endgroup$
    – Nikita Kalinin
    Commented Mar 4, 2019 at 17:43
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    $\begingroup$ Hm, such extension of phase space is used sometimes, but the ways of extending that I know won't help here. I'm not 100% sure, but even if these equations were "embeddable" into some two-degree-of-freedom Hamiltonian system, the usual way of studying the dynamics (from what I've read and encountered) would use restriction onto level sets of Hamiltonian and figuring out its internal dynamics. So in the end you go back to the same or equivalent three-dimensional system. $\endgroup$
    – Evgeny
    Commented Mar 4, 2019 at 18:51
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    $\begingroup$ @NikitaKalinin A complete global analysis of the Droop equation is carried out in detail by Lange and Oyarzun in pdfs.semanticscholar.org/7ed9/…. The authors show that trajectories are globally stable, and in particular, they converge to one of two fixed points (5) or (6) depending on the positivity of $a_2 a_3-a_2 -a_3-a_1 a_2$. Authors use comparison arguments, a Lyapunov function, and the Poincaré-Bendixson Theorem to conclude this. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Mar 7, 2019 at 12:05

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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.

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  • $\begingroup$ Yes, thanks, I have seen it. Unfortunately, it does not answer my question but seems to be the best knowledge at this moment... $\endgroup$
    – Nikita Kalinin
    Commented Mar 11, 2019 at 15:09
  • $\begingroup$ The OP asks two questions: "How to find another constant of motion? Do you have any idea how to present this system in the Hamiltonian or Lagrangian form?" This answer: (1) puts these questions in proper context; (2) shows that the planar vector field associated with the 2-dimensional Droop system has neither Hamiltonian nor gradient structure, which directly addresses the second question, and in the absence of structure, there are no general techniques for finding such time-dependent first integrals, which addresses the first question. Stress that a complete global analysis is available. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Mar 11, 2019 at 17:50

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