Previous result

It was shown by Cinquin (2005) that, in the case where $\forall i, d_i=1, \sigma_i=\sigma$ and $\sigma » 1$, there are stable steady states with $k$ elements "on" (i.e. non-0) if and only if

$$\alpha<1/k^2$$

(when the condition $\sigma » 1$ is not met, the above condition is necessary but not sufficient).

Since $\alpha$ is a measure of the harshness of the competition in the system (as it depends on the quantity of the common class A activators, and the heterodimer concentration giving half-maximal transcription), this shows that the harsher the competition in the system, the lower the number of elements which can co-exist.