nondimensionalisation for an ODE

Question

I need help to understand 'nondimensionalisation' for ODEs better. I stacked how to deal with a constant input in a prey-predator model. I reproduced the following ODE system $$\frac{dx}{dt}=X_{0}-ax$$ $$\frac{dy}{dt}=by-cxy-ey.$$ Denoting dimensionsless variables $T=t/t*,\bar{X}=x/x^*,\bar{Y}=y/y^*$ give us $$\frac{d\bar{X}}{dT}=\frac{t^*}{x^*}X_{0}-at^*\bar{X}$$ $$\frac{d\bar{Y}}{dT}=bt^*\bar{Y}-cx^*t^*\bar{X}\bar{Y}-et^*\bar{Y}.$$ I think choosing $t^*=\frac{1}{b},x^*=\frac{X_0}{b}$ would nondimensionalise the system but the term $cx^*t^*\bar{X}\bar{Y}$ in the second equation has still the dimension of the constant input $X_0$. What is the way to get rid of their dimension?