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I already showed that the Lotka Volterra Equations preserve the weighted area $ dx \wedge dy/xy$ see here. Now I need that a modification of the Forward Euler Method preserves the same weighted area. Here is the modification:

$$ \frac{x_{n+1}-x_n}{\Delta t} = x_n -x_ny_n $$ $$\frac{y_{n+1}-y_n}{\Delta t} = -y_n -x_{n+1}y_n $$

side-question: why is this Euler scheme still explicite?

Proof that this modification preserves the weighted area $dx \wedge dy/xy$:

to simplify notitation :

$$ X = \Delta tx +x - \Delta txy$$ and $$ Y = -\Delta ty+ y + \Delta tXy $$ where we set $X:=x_{n+1}, Y:=y_{n+1}$, $x:= x_n$ and $ y := y_n$ solved for the unknown $X$ and $Y$.

now I showed $dX \wedge dX = 0$, $dY \wedge dY = 0$ and compute $dX \wedge dY$ and compute $XY$.

I found out that:

$$ \frac{1}{XY}{dX \wedge dY} = \frac{1}{xy}{dx \wedge dy} $$

Question: why is this enough to say that the modification preserves the weighted area above?

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