So what I'm messing around with are these two coupled ODES:
$$\frac{dx}{dt}=\gamma x\left(1 - \frac{\alpha x+\beta y}{N}\right)$$
$$\frac{dy}{dt}=\theta y\left(1 - \frac{\alpha x+\beta y}{N}\right)$$
where $\gamma , \alpha , \beta , \theta \space and \space N$ are constants
As an exercise I try to nondimensionalize them by setting $$x =c_1X, \space y=c_2Y \space and \space t = c_3\tau$$
I get
$$\frac{dX}{d\tau}= \gamma c_3X - \frac{\gamma c_3c_1\alpha X^2+\beta c_3c_2\gamma XY}{N}$$
$$\frac{dY}{d\tau} = \theta c_3Y - \frac{\theta c_3c_2\beta Y^2 + \alpha c_2c_3\theta XY}{N}) $$
At this particular critical point in time, I set $$c_3 = 1/\gamma,\space\gamma c_3c_1\alpha = 1, \space\beta c_3c_2\gamma = 1 $$
where I will finally get $$\frac{dX}{d\tau}=X\left(1 - \frac{X+Y}{N}\right)$$
$$\frac{dY}{d\tau}=\kappa Y\left(1 - \frac{X+Y}{N}\right)$$
Now I'm just curious, if I'm doing it right, and if there are other methods to nondimensionalize it.