Results

Let $r_i=\sigma_i/d_i$. Then at any stationary state,

$$\forall i \mathrm{st} x_i\ne 0, x_{i}^2-r_i x_i + \alpha D^2=0,$$

and

$$\forall i \mathrm{st} x_i\ne 0, x_{i}=\frac{r_i \pm \sqrt{r_{i}^2-4\alpha D^2}}{2}$$ (2)

$x_i$s at 0 can be discarded from the rest of the analysis. It will be shown below that if the stationary state is stable, at most one $x_i$ can be at the lower solution of equation 2 (inequality 5). Suppose that there is such an $x_\kappa$ (if there is not, a stronger inequality is derived, see Appendix B), and let $\kappa'$ be such that $r_{\kappa'}=\max_{i}{r_i}$. It will be shown below that any steady state where $\kappa=\kappa'$ is unstable, and we can therefore suppose $\kappa \ne \kappa'$. Then

$$2\left(D-1\right)=\Sigma_{i \ne \kappa}\left( r_i + \sqrt{r_{i}^2-4\alpha D^2}\right) + r_\kappa - \sqrt{r_{\kappa}^2-4\alpha D^2}$$

$$\Sigma_{i}r_i+2=2D-\Sigma_{i \ne \kappa}\sqrt{r_{i}^2-4\alpha D^2} + \sqrt{r_{\kappa}^2-4\alpha D^2}$$

$$\Sigma_{i}r_i+2 \le 2D - \Sigma_{i\ne \kappa, i\ne \kappa'}\sqrt{r_{i}^2-4\alpha D^2}$$

Consider the right-hand side of the above inequality as a function of $D$. It is an increasing function, and for the above inequality to hold, it must also hold for the maximum of that function (which is for $D=r_s/2\sqrt{\alpha}$), where $r_s=\min_i r_i$, ie

$$\frac{r_{s}}{\sqrt{\alpha}} \ge 2+ \Sigma_{i}r_i +\Sigma_{i\ne \kappa, i\ne \kappa'}\sqrt{r_{i}^2-r_{s}^2}$$ (3)

This implies in particular $\alpha\le 1/k^2$, where $k$ is the number of non-zero $x_i$s, generalizing the result obtained by Cinquin (2005).