$k$ variables on, others off

With identical parameters, there can be no equilibrium with 2 variables having different, non-zero values.

At any equilibrium, variables can be renumbered so that, in the Jacobian matrix, variables at 0 form an independent block. This block is stable, and the stability of the whole system depends only on the block formed by non-0 variables. Thus, in the following we suppose that no steady-state variable has 0 for a value.

For $i \ne j$,

$$J_{i,j}(\bar{x})=-\sigma c \frac{\bar{x}^{2 c -1}}{\left( 1 + k \bar{x} ^{c} \right)^2}$$

With the same kind of analysis as in Cinquin and Demongeot (2002), the equilibrium is stable only if

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

$$\sigma c \left( 1 + \left( k - 1 \right) \bar{x}^{c} \right) \frac{\bar{x}^{c-1}}{\left( 1 + k \bar{x}^{c} \right) ^2}$$ (7)

With the definition of the equilibrium,

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

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

$$\frac{\sigma}{c} x^{c-1} > k x^c +1$$

Again with the definition of the equilibrium,

$$\frac{1}{c}x^{c-1}>x^{c-1},$$

ie $c<1$, in which case no interesting equilibria exist.