Lotka-Volterra coordinates transformation

Question

I would like to ask the following:

Given a Lotka-Volterra predator-prey system, \begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} , with all the parameters ${\alpha}, {\beta}, {\gamma}, {\delta}$ to be positive integers, I would like to apply a transformation in order to reach to the following form: \begin{align} & \frac{dx}{dt}=-{\mu_1}x(1-y) \\ & \frac{dy}{dt}={\mu_2}y(1-x) \end{align} where ${\mu_1},{\mu_2} \in \mathbb{Z}^{+}$. Any ideas? Thank you all!

Answer 1

\begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} Let $x=kY$ and $y=hX$ \begin{align} & \frac{dY}{dt}={\alpha}Y-{h\beta}XY=\alpha Y (1-\frac{h\beta}{\alpha}X) \\ & \frac{dX}{dt}=-{\gamma}X+{k\delta}XY=-\gamma X (1-\frac{k\delta}{\gamma}Y) \end{align} With $\mu_2=\alpha \\ \mu_1=\gamma \\ h=\frac{\alpha}{\beta} \\ k=\frac{\gamma}{\delta}$

the wanted form is obtained : \begin{align} & \frac{dX}{dt}=-{\mu_1}X(1-Y) \\ & \frac{dY}{dt}={\mu_2}Y(1-X) \end{align}