Convergence

Let $y_{i}= \sqrt{x_i}$, and

$$P=\frac{1}{4}\Sigma_{i=1}^{n} y_{i}^2 - \frac{\sigma}{4c}\log{\left( 1+\Sigma_{i=1}^{n}y_{i}^{2c} \right)} -$$

$$\frac{1}{2}\log{\Pi_{i=1}^{n}y_{i}^{ \alpha}}$$

$$\dot{y_i}=\frac{\dot{x_i}}{2\sqrt{x_i}}$$

$$2\dot{y_i}=- y_i + \sigma \frac{y_{i}^{2c-1}} {1+\Sigma_{i=1}^{n} y_{i}^{2c}} + \frac{ \alpha}{y_i}=2\frac{\partial P}{\partial y_i}$$

Thus, $P$ is a potential for the system.

If its equilibria are isolated, a gradient system converges to a steady-state regardless of the initial conditions. It is shown below that the number of solutions of the system is finite when the cooperativity $c$ is an integer, and the system thus always converges to a steady state (we expect this result to also hold for non-integer values of $c$). The model without leak corresponds to $\alpha=0$, and this convergence result thus also applies to it, for $c>1$.