= Solution
Use an <annual pulse-breeding predator-prey model>. Let $X_n,Y_n$ be the abundances just after reproduction, and let the year have length $T$. Between breeding pulses, assume mass-action <predation>, natural mortality rates $\mu_X,\mu_Y>0$, and no reproduction:
$$
\dot X=-\mu_XX-\beta XY,\qquad \dot Y=-\mu_YY.
$$
Solving the <predator> equation and inserting it into the <prey> equation gives the pre-breeding abundances
$$
Y^-_n=e^{-\mu_YT}Y_n,
\qquad
X^-_n=X_n\exp\!\left[-\mu_XT-
\frac{\beta(1-e^{-\mu_YT})}{\mu_Y}Y_n\right].
$$
Suppose reproduction multiplies surviving <prey> by $R$ and adds $qX^-_n$ offspring per surviving <predator>. Retaining the surviving adults gives $X_{n+1}=RX^-_n$, $Y_{n+1}=Y^-_n(1+qX^-_n)$. Define $r=Re^{-\mu_XT}$, $s=e^{-\mu_YT}$, $\kappa=\beta(1-s)/\mu_Y$ and $b=q/R$. The resulting <difference equations> are
$$
\boxed{X_{n+1}=rX_ne^{-\kappa Y_n},\qquad
Y_{n+1}=sY_n(1+bX_{n+1}).}
$$
Thus offspring depend on <prey> available at the end of the year, not on the number killed during the year. The adult-survival convention is part of the model; if breeding replaces all adults, use $Y_{n+1}=sbY_nX_{n+1}$ instead. Both are consistent pulse models under their respective life-history assumptions.
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