In a stable steady state, " $\kappa \ne \kappa'$"

We now show that there is no stable steady state with $r_n=\max_i r_i$. Suppose that $Q_n>0$, $Q_{n-1}<0$, and $r_n=\max_i r_i$. Then

$$P(0)=\Pi_{j\ne n}Q_j/P_j + Q_n/P_n \Sigma_{i=1}^{n-1} \Pi_{j=1,j\ne i}^{n-1}Q_j/P_j -\Pi_{j=1}^{n}Q_j/P_j$$

$$P(0)=\left( \Pi_{j\ne n}Q_j/P_j + Q_n/P_n \Pi_{j=1}^{n-2} Q_j/P_j... ...Q_n/P_n \Sigma_{i=1}^{n-2} \Pi_{j=1,j\ne i}^{n-1}Q_j/P_j -\Pi_{j=1}^{n}Q_j/P_j$$

$$P(0)=\left( Q_{n-1}/P_{n-1} + Q_n/P_n\right) \Pi_{j=1}^{n-2}Q_j/P... ...Q_n/P_n \Sigma_{i=1}^{n-2} \Pi_{j=1,j\ne i}^{n-1}Q_j/P_j -\Pi_{j=1}^{n}Q_j/P_j$$

The last two terms in the sum both have the same sign as $(-1)^n$.

Now consider

$$\frac{Q_n}{P_n}\frac{P_{n-1}}{Q_{n-1}} = \frac{1-2\beta_n x_n}{1-2\beta_{n-1} x_{n-1}} \frac{\beta_{n-1}}{\beta_n}$$

$$\frac{Q_n}{P_n}\frac{P_{n-1}}{Q_{n-1}} = - \frac{\sqrt{1-4 \alpha... ... \beta_n^2}} {\sqrt{1-4 \alpha D^2 \beta_{n-1}^2}} \frac{\beta_{n-1}}{\beta_n}$$

By hypothesis, $\frac{\beta_{n-1}}{\beta_n}\ge1$, and thus

$$Q_n/P_n\ge \left\vert {Q_{n-1}/P_{n-1}} \right\vert$$

Therefore, $P(0)$ has the same sign as $(-1)^n$. Since $P(Q_n)$ has the same sign as $(-1)^{n-1}$, P has a positive root, and the steady state is unstable.