If you could find physical meaning for a, b, x0/ or a, b, that'd be the best approach.
If you have a hard time coming up with a chemically/physically valid formula, then you can compare some special values of the functions. We should assume x to be non-negative real numbers given the physical meaning of the project. Let's also assume a, b, x0 to be positive, since f is finite and positive. If not, some of the conclusions will be different, but the methodology is the same.
1st, x =0:
when x=0, f =0 only for Hyperbola.
this seems to be physically valid. So sigmoidal doesn't seem to work when x is
close to 0.
2nd, x = infinity, f should approach an asympotic value, since the quantity of B is limited. This is achieved for both functions.
3rd, min/max values for the x in experimental range:
sigmoidal has a minimum value a/2 when x=x0, is that expected from the experiment? Is that x0 in the range of x you are interested?
Hyperbola is monotonic, with min at x=0, and max at x=infinity.
This would be how I would start comparing the two formulae.
