= Solution
Suppose $r>1$ and set $q=\log r/(r-1)$. Substitution of the positive <equilibrium> into the <Jacobian matrix> gives
$$
J_*=
\begin{pmatrix}
1&-r\log r/[b(r-1)]\\
b(r-1)/r&q
\end{pmatrix},\quad
\operatorname{tr}J_*=1+q,\quad
\det J_*=q+\log r=\frac{r\log r}{r-1}.
$$
Part (b) showed $f(x)<0$ whenever $x\ne1$. In particular,
$$
f(1/r)=\frac{1-r+\log r}{r}<0
$$
gives $\log r<r-1$, whereas
$$
f(r)=r-1-r\log r<0
$$
gives $\det J_*>1$. The latter already violates the <Jury stability criterion> condition $1-\det J_*>0$. More directly, the product of the two <eigenvalues> is greater than one, so at least one has <modulus> greater than one. Therefore the positive <equilibrium> is \b[unstable]. This is the <instability of the Nicholson-Bailey equilibrium>; parasitoid feedback in this simple model does not stabilize coexistence.
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