Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 52 / 1 / a / 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:Solving the predator equation and inserting it into the prey equation gives the pre-breeding abundancesSuppose reproduction multiplies surviving prey by $R$ and adds $q{X}_n^{-}$ offspring per surviving predator. Retaining the surviving adults gives $X_{n+1}=R{X}_n^{-}$, $Y_{n+1}=Y_n^{-}(1+q{X}_n^{-})$. Define $r=R{e}^{-{\mu }_{X}T}$, $s=e^{-{\mu }_{Y}T}$, $\kappa =\beta (1-s)/{\mu }_Y$ and $b=q/R$. The resulting difference equations areThus 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 $Y_{n+1}=sb{Y}_n{X}_{n+1}$ instead. Both are consistent pulse models under their respective life-history assumptions.