Summary of $\alpha$ threshold values

For the system to be able to sustain $k$ switch elements "on" at the same time, the condition $\alpha<1/k^2$ must be met (for $\sigma \gg 1$, this condition is also sufficient). Thus, for $\alpha>1/4$, only 1 switch element can be on at a time. The corresponding equilibrium value is a decreasing function of $\alpha$. For $\alpha>\frac{\sigma^{2}}{4\left(n\sigma +1 \right)}$, there is an "extinction domain" around the diagonal $x_1=..=x_n$: matching sufficiently closely the concentrations of the switch elements, at whatever value, makes the system switch off all switch elements. For large $\sigma$, the extent of this domain is small, except in a very narrow range of $\alpha$ values. Finally, for $\alpha>\frac{\sigma^2}{4\left( \sigma +1 \right)}$, a condition which becomes $\alpha > \sigma/4$ for large $\sigma$, there are no non-0 steady states.