Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 52 / 1

Let $X,Y$ be prey and predator abundances. With constant positive demographic and predation coefficients, the Lotka-Volterra equations areTheir positive equilibrium is $X_{\ast }=\gamma /\delta$, $Y_{\ast }=\alpha /\beta$. Its Jacobian matrix and eigenvalues areThe linearized populations oscillate with period $2\pi /\sqrt{\alpha \gamma }$, with the predator peak following the prey peak. This is a center equilibrium, not an attracting equilibrium. The conclusion is supported beyond linearization by the Logarithmic first integral of the Lotka-Volterra equations: in these units one may useDirect differentiation gives $\dot{V}=0$. Its Hessian matrix is a positive-definite matrix in the positive quadrant, and its minimum is at $(X_{\ast },Y_{\ast })$. Nearby level curves are closed periodic orbits, so there is no damping toward a preferred oscillation amplitude.

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.

For the adult-survival annual pulse-breeding predator-prey model, a positive annual fixed point exists when $r>1$ and $0<s<1$:At this point the Jacobian matrix isIf $0<(1-s)\log r<4$, 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 becomesIt 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:Its positive equilibrium has $Y_{\ast }=\log r/\kappa$ and $X_{\ast }=Y_{\ast }/[c(1-1/r)]$. Its Jacobian matrix hasSince 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.