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An annual pulse-breeding predator-prey model combines continuous predation and mortality between short reproduction episodes with an annual difference equation. If surviving predators produce offspring in proportion to prey surviving the year, retained adults give the displayed map. Here $r$ combines prey reproduction and natural survival, $s$ is predator survival, and $\kappa$ integrates predation over declining predator abundance. The positive annual equilibrium is $X_{\ast }=(1-s)/(sb)$, $Y_{\ast }=\log r/\kappa$ for $r>1$ and $0<s<1$.

A difference equation relates values of a sequence at different indices, or its finite differences. A first-order autonomous system has the displayed form, with a scalar or vector state. Writing $z_{n+1}-z_n=F(z_n)-z_n$ gives its equivalent finite-difference form. Its fixed point stability is determined by the Jacobian matrix of the update and the unit circle, whereas a continuous-time differential equation uses the real parts of its linearized eigenvalues.

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.