Existence of equilibria

Let us consider a ss of the system in which the variable $x_i$ has a given value, and all other variables share the same value $\epsilon$. We have

$$\begin{array}{ll} \nonumber x_i=\frac{\sigma}{1+(n-1)\epsilo... ...\epsilon=\frac{\sigma}{1+x_{i}^{c}+(n-2)\epsilon^c} \end{array}$$

Rewriting the last equation as

$$\epsilon(1+(\frac{\sigma}{1+(n-1)\epsilon^{c}})^c+(n-2)\epsilon^c)=\sigma,$$

and considering limits when $\epsilon$ tends to 0 and to infinity, one sees that the system has at least one solution.