For the 3D system of ODEs:

$$\begin{eqnarray}\dot{x} &=& -\beta x y \\ \dot{y} &=& \beta x y + \hat{\beta} z y - \delta y \\ \dot{z} &=& -\hat{\beta} z y + \delta y, \end{eqnarray}$$

the quantities $Q_1 = x+y+z$ (assume $Q_1=1$) and $Q_2 = (\delta/\hat{\beta}-z)(x)^{\hat{\beta}/\beta}$ are conserved quantities (i.e., $\dot{Q}_1 = \dot{Q}_2=0$).

$Q_2$ can be found by first reducing the 3D system to 2D by using $Q_1$ (e.g. $z = 1-x-y$) and then computing $dy/dx$, which can be solved analytically.

My question is about the following generalization of the previous system to the following 6D system of ODEs:

$$\begin{eqnarray}\dot{x}_1 &=& -\beta x_1 y_2 \\ \dot{y}_1 &=& \beta x_1 y_2 + \hat{\beta} z_1 y_2 - \delta y_1 \\ \dot{z}_1 &=& -\hat{\beta} z_1 y_2 + \delta y_1, \end{eqnarray}$$

$$\begin{eqnarray}\dot{x}_2 &=& -\beta x_2 y_1 \\ \dot{y}_2 &=& \beta x_2 y_1 + \hat{\beta} z_2 y_1 - \delta y_2 \\ \dot{z}_2 &=& -\hat{\beta} z_2 y_1 + \delta y_2. \end{eqnarray}$$

It is easy to see that $Q_{11} = x_1 + y_1 + z_1$ and $Q_{12} = x_2 + y_2 + z_2$ are conserved quantities for the new system.

Are there any other conserved quantities in the new system? If there are, what are they?

Please assume that $Q_{11} = Q_{12} = 1$.

I have played with many analogous forms to $Q_2$ without success and I would like some help from the community to help me find the other conserved quantity(ies).


Even without assuming that $Q_{11} = Q_{12} = 1$, you can use $y_2 = -\frac{1}{\beta} \frac{\dot{x}_1}{x_1}$ (and the equivalent for $y_1$) to obtain the conserved quantity

$$ Q_2 = \beta(y_1-y_2) + (\beta-\hat{\beta})(x_1-x_2) - (\delta + \hat{\beta} Q_{12}) \log x_2 + (\delta + \hat{\beta}Q_{11}) \log x_1 . $$

Indeed, we see that

\begin{align} \dot{Q}_2 =& \beta (\dot{y}_1 - \dot{y}_2) + (\beta-\hat{\beta})(\dot{x}_1 - \dot{x}_2) - (\delta + \hat{\beta} Q_{12}) \frac{\dot{x}_2}{x_2} + (\delta + \hat{\beta} Q_{11}) \frac{\dot{x}_1}{x_1}\\ =& \beta\left(\beta x_1 y_2 + \hat{\beta}(Q_{11} - x_1 - y_1)y_2 - \delta y_1 - \beta x_2 y_1 - \hat{\beta}(Q_{12} - x_2 - y_2)y_1 + \delta y_2\right) \\ & + (\beta-\hat{\beta})(-\beta x_1 y_2 + \beta x_2 y_1) - (\delta + \hat{\beta} Q_{12})(-\beta y_1) + (\delta + \hat{\beta}Q_{11})(-\beta y_2)\\ =& 0. \end{align}

Unfortunately, I haven't been able to find another one, for now.

  • $\begingroup$ Thanks! I however cannot see that this is a conserved quantity. Could you please edit your answer to show that this is indeed a conserved quantity? $\endgroup$ – rpa Oct 5 '18 at 12:43
  • 3
    $\begingroup$ Sure, give me a minute. $\endgroup$ – Frits Veerman Oct 5 '18 at 13:05

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