= Solution
For the adult-survival <annual pulse-breeding predator-prey model>, a positive annual <fixed point> exists when $r>1$ and $0<s<1$:
$$
X_* =\frac{1-s}{sb},\qquad Y_* =\frac{\log r}{\kappa}.
$$
At this point the <Jacobian matrix> is
$$
J_*=
\begin{pmatrix}
1&-\kappa X_*\\sbY_*&1-(1-s)\log r
\end{pmatrix},
\qquad
\boxed{\det J_*=1,\quad \operatorname{tr}J_*=2-(1-s)\log r.}
$$
If $0<(1-s)\log r<4$, the two <eigenvalues> are a <complex conjugate> pair of <moduli> one. The annual samples show linearly neutral seasonal oscillations. If $(1-s)\log r>4$, they form a reciprocal negative real pair and the <fixed point> is a <saddle fixed point of a map>. Equality at four is a degenerate boundary needing nonlinear analysis. For replacement rather than surviving adults the same calculation gives <trace> $2-\log r$ and <determinant> one.
The absence of an attracting annual cycle is stronger than a linear observation. Put $u=\log X$, $v=\log Y$. The update becomes
$$
u'=u+\log r-\kappa e^v,
\qquad v'=v+\log s+\log(1+be^{u'}).
$$
It is the composition of two <nonlinear shear maps>, each of <determinant> one: an <area-preserving seasonal predator-prey map>. No isolated positive <periodic orbit> has an open attracting basin. Neutral linear oscillations alone also do not prove nonlinear stability at every resonance. In particular, this density-independent model does not select an <asymptotically stable> oscillation amplitude.
The <Nicholson-Bailey model> instead counts <parasitoid> offspring from attacked <hosts>:
$$
X'=rXe^{-\kappa Y},\qquad Y'=cX(1-e^{-\kappa Y}).
$$
Its positive <equilibrium> has $Y_*=\log r/\kappa$ and $X_*=Y_*/[c(1-1/r)]$. Its <Jacobian matrix> has
$$
\det J_* =\frac{r\log r}{r-1}>1,
\qquad \operatorname{tr}J_*=1+\frac{\log r}{r-1}<2.
$$
Since the <determinant> exceeds one while the positive <trace> is below two, the <eigenvalues> are complex with <moduli> greater than one: <instability of the Nicholson-Bailey equilibrium> produces outward oscillations near it. The two maps share an exponential prey-survival factor but have different reproduction laws and stability properties.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-52-seasonal-orbits.png]
{title=Neutral seasonal predator-prey oscillations compared with the unstable Nicholson-Bailey <equilibrium>}
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