Analysis of mutual inhibition with autocatalysis, and leak

If $\alpha\ge 0$, $c\ge1$, and one of these inequalities is strict, the function $f(x)=x^{1-c}-\alpha x^{-c}$ can take the same value for at most 2 positive values of $x$. Thus, there are only two values a variable can take at a given steady state (0 cannot be a steady state value). If two different equilibrium values are taken by some variables, one of these values is higher than $\alpha \frac{c} {c-1}$, and the other lower.

If $\alpha>0$ and $c=1$, the system only has one equilibrium, with all variables equal.