Local stability analysis

With a leak $\alpha$, equation A.3 becomes

$$\bar{x}^c < \frac{\sigma}{c} \frac{\bar{x}^{c+1}}{\left( \bar{x}-\alpha\right)^2} -1 -\left(k -1 \right) \bar{x}^c$$

$$1+k \bar{x}^c<\frac{\sigma}{c} \frac{\bar{x}^{c+1}}{\left(\bar{x}-\alpha \right)^2}$$

$$\frac{\sigma \bar{x}^c}{\bar{x}-\alpha} < \frac{\sigma}{c} \frac{\bar{x}^{c+1}}{\left(\bar{x}-\alpha \right)^2}$$

$$\bar{x}-\alpha < \frac{1}{c}\bar{x}$$

Thus the stability condition A.3 is met iff $\bar{x}<\alpha \frac{c}{c-1}$ (in that case, since non-diagonal terms of the Jacobian are obviously negative, diagonal terms are also negative, and the equilibrium is stable). Since solutions to equation 8 can be made arbitrarily high by increasing $\sigma$, increasing $\sigma$ past a threshold value (other parameters being equal) will prevent the existence of a stable equilibrium with all variables equal.