I have the following system of 3 ODEs of 3 variables $(\ell,M,h)$ and 11 parameters $(\sigma_{\ell},\mu_{\ell},d_{\ell},\sigma_M,\alpha_1,\beta,\alpha_2,\nu_M,\sigma_h,\nu_h,d_h)$:
$\frac{d\ell}{dt}=\sigma_{\ell} - \mu_{\ell}\frac{M \ell}{1+\ell} - d_{\ell} \ell$,
$\frac{dM}{dt}=\sigma_M\left(\alpha_1\frac{\ell}{\beta+\ell} +\alpha_2\frac{M \ell}{1+\ell} \right) - \nu_M \frac{Mh}{1+h} $,
$\frac{dh}{dt}=\sigma_h-\nu_h \frac{Mh}{1+h} - d_h h$.
To find any possible equilibria I clearly need to look for the intersection of the nullclines of the system. These nullclines graphed are 3 surfaces in $(\ell,M,h)$ space. The issue is that I have 11 unknown parameters and the only restrictions on them (and the variables) are that they be real and either zero or positive.
I considered that it may be possible to write a matlab or mathematica script where I could vary the parameters and get it to output excatly when an intersection occurs (ie when all of the above equations are zero on the right-hand-side).
This question is certainly vague, and I would welcome any range of mathematical/computational techniques that could be useful for this problem. Cheers!