Fixed-quota harvesting of Lotka-Volterra populations ($\det DH=X^{-}/(X^{-}-h)>1$)

In logarithmic prey/predator coordinates, the Lotka-Volterra predator-prey model has divergence zero. Removing a fixed prey quota $h$ changes $\log X$ to $\log (X-h)$ and expands logarithmic area by $X/(X-h)$. Thus a periodic annually harvested orbit has an expanding return map determinant and cannot be asymptotically stable. A continuous-quota approximation gives a coexistence equilibrium with positive Jacobian matrix trace $h/X_{\ast }$, unlike proportional harvesting, which only changes the prey growth parameter in this idealized model.

The predator-prey first integral $H=cX-d\log X+bY-a\log Y$ changes by the displayed amount when $h$ prey are removed. At a fixed prey threshold this increment is constant; sufficiently high thresholds reduce oscillation energy until the threshold is no longer crossed. At a fixed predator threshold, the increment depends on the prey abundance and on the crossing direction. A rising-predator balanced-quota cycle has negative return multiplier of magnitude greater than one; its leading small-oscillation approximation is marginal rather than a robust attractor.