Special case: no cooperativity ($c=1$)

We assume that $\sigma>1$. The set of steady states for the system defined by equations 1 is 0 and the attracting hyperplane $\{x \vert 1 + \Sigma_{i=1}^{n} x_i=\sigma\}$. Let $s=\Sigma_{i=1}^{n}x_i$. Then $s$ never crosses the value $\sigma-1$, and since $\dot{x_i}=x_i \left( \frac{\left(\sigma -1\right)-s}{1+s} \right)$, $\dot{x_i}$ is of constant sign, and each $x_i$ convergent.

Simulations show that there is a great number of stable steady states.

For $c>1$, the convergence of the dynamical system (defined by equations 1) to an equilibrium, from any initial condition, will be derived in a more general context, in section B.1. In the rest of the appendix we assume $c>1$.