I have the following coupled ODEs, which describe a system of inter-specific interactions within an ecosystem:
$$\frac{dx}{dt} = rx\left(1 - \frac{x}{K}\right) -bxy$$
$$\frac{dx}{dt}= -cy + dxy$$
with $ b,c,d,r$ and $K$ are positive constants. I have already provided a nondimensionalized system with a change in variables in the following fashion:
$$ u =\frac{x}{K}, v =\frac{b}{r}y, \tau=rt $$
And a change of parameters like the following:
$$ \alpha = \frac{c}{r}, \beta = \frac{dK}{r} $$
so that I end up with the following nondimensionalized system:
$$ \frac{du}{d\tau}= u(1 - u - v) $$
$$ \frac{dv}{d\tau}= v(- \alpha + \beta u) $$
My problem is that now I want to nondimendionalize the followint system, which is the same except for the inclusion of an extra parameter, $a > 0$, in the first ODE:
$$\frac{dx}{dt} = rx(x - a)\left(1 - \frac{x}{K}\right) -bxy$$
$$\frac{dx}{dt}= -cy + dxy$$
I'm getting really stuck when I try to provide a nondimensionalization of the system. ¿Any hints on how to handle that $(x-a)$ ?