I have the following ODE system $$ \dfrac{d \ell}{dt}=\sigma_{\ell} - \mu_{\ell} \dfrac{M \ell}{1+\ell}-d_{\ell}\ell \\ \dfrac{dM}{dt}=\dfrac{\ell}{\beta+\ell}+\sigma_M \dfrac{M \ell}{1+\ell}-M $$ where both the variables $\ell$ and $M$ and parameters $\sigma_{\ell}$, $\mu_{\ell}$, $d_{\ell}$, $\sigma_M$ and $\beta$ are all positive reals.
Is there someway to show that the nullclines of this system (i.e. when the left hand side of the above 2 equations equals zero) intersect exactly once for all $\ell$, $M$ $\in \mathbb{R}^+$ and for all $\sigma_{\ell}$, $\mu_{\ell}$, $d_{\ell}$, $\sigma_M$, $\beta$ $\in \mathbb{R}^+$.
ps I don't mind if you set $\beta=1$. It isn't really that important to my model.