= Solution
The origin has Jacobian $\begin{pmatrix}r&0\\1-b&0\end{pmatrix}$ and is unstable because $r>1$. At a positive equilibrium, $re^{-p_c}=1$, giving
$$
\boxed{p_c=\log r,\qquad n_c=\frac{r\log r}{r-b}.}
$$
The linearization is
$$
\binom{n'_{t+1}}{p'_{t+1}}=\begin{pmatrix}1&-n_c\\1-b/r&bn_c/r\end{pmatrix}\binom{n'_t}{p'_t}.
$$
Its <trace> is $\tau=1+bn_c/r$ and <determinant> $n_c$. Elimination gives $n'_{t+2}-\tau n'_{t+1}+n_cn'_t=0$. For distinct roots,
$$
\boxed{n'_t=A\lambda_1^t+B\lambda_2^t,\quad p'_t=\frac{A(1-\lambda_1)\lambda_1^t+B(1-\lambda_2)\lambda_2^t}{n_c},\quad\lambda_1+\lambda_2=\tau,\quad\lambda_1\lambda_2=n_c.}
$$
At a repeated root replace the first expression by $(A+Bt)\lambda^t$ and use $p'_t=(n'_t-n'_{t+1})/n_c$.
As $r\downarrow1$, $n_c\to0$, so the roots approach one and zero. The <characteristic polynomial> at one is $n_c(1-b/r)>0$, putting the larger real root just below one and the other <positive root> below it. No real root can ever equal one. Moreover
$$
\frac{dn_c}{dr}=\frac{r-b-b\log r}{(r-b)^2}>0,
$$
since its numerator exceeds $r-1-\log r>0$. At $n_c=1$, $\tau=1+b/r<2$, so the roots are conjugate and nonreal. Just above this point their modulus is $\sqrt{n_c}>1$.
The unit-disc conditions are $1-n_c>0$, $1-\tau+n_c>0$, and $1+\tau+n_c>0$. The last two hold automatically here, so stability is equivalent to $n_c<1$. The unique crossing and its <eigenvalue> argument satisfy
$$
\boxed{r_*\log r_*=r_*-b,\qquad 1<r_*<e,\qquad 2\cos\theta_*=1+b/r_*.}
$$
Indeed $\log r_*=1-b/r_*<1$. This is the <stability threshold of a discrete exponential predator-prey model>. The sketch uses $b=1/2$: the real moduli merge and then coincide, increasing through one at the threshold. Linearization decides stability below and instability above; it does not determine nonlinear stability exactly at the crossing.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-2-predator-eigenvalues.png]
{title=Eigenvalue moduli and the stability threshold of the discrete predator-prey model}
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