Local stability analysis

It is useful, for the Jacobian term computations to follow in the rest of the appendix, to note that if $g(x)=\frac{x^m}{\alpha+x^m}$, $g'(x)=\frac{\alpha m x^{m-1}}{\left( \alpha + x^m \right) ^2}$.

If $x_j$ is at a non-zero steady-state and $\forall i\ne j, x_i=0$, and if $c>1$, the stability at that steady state depends only on the sign of the $(j,j)$ coefficient of the Jacobian matrix (this coefficient will be called $J_{j,j}$ in the remainder of the appendix).

$$J_{j,j}= -1 + \sigma c \left( 1+ \Sigma_{i \ne j} x_{i}^{c} \right) \frac{x_{j}^{c-1}}{\left( 1+\Sigma_{i=1}^{n}x_{i}^\mathrm{c} \right)^2}$$ (5)

$$J_{j,j}= -1 + \sigma c \frac{x_{j}^{c-1}}{\left( 1+x_{j}^\mathrm{c} \right)^2},$$

with

$$\sigma \bar{x}_{j}^{c-1}=1+\bar{x}_j^{c},$$ (6)

at equilibrium

$$J_{j,j}= -1 + c \frac{1}{\sigma \bar{x}_{j}^{c-1}},$$

the equilibrium is stable iff

$$\bar{x}_{j}^{c-1}>\frac{c}{\sigma}, \mathrm{ie} 1+\bar{x}_{j}^{c}>c$$

It is possible to give a sufficient condition for the equilibrium with the greatest solution to equation 6 to be stable. Let $f(x)=x^{c} - \sigma x^{c-1}$. If $f\left( \left( \frac{c}{\sigma} \right) ^ {\frac{1}{c-1}} \right) < -1$, then the greatest root of equation 6 will be greater than $\left( \frac{c}{\sigma} \right) ^ {\frac{1}{c-1}}$, and the corresponding equilibrium will be stable. A sufficient stability condition is thus

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

Numerical investigation shows that this condition is met for $\sigma \ge 2$.