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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.

This Lotka-Volterra predator-prey model includes self-limitation of the prey through the term $-b{x}^2$, with positive constants $a,b,c,d,e$. A positive coexistence equilibrium point exists when $a>be/d$, at $x_{\ast }=e/d$, $y_{\ast }=(a-b{x}_{\ast })/c$. The Lyapunov functionsatisfies $\dot{V}=-bd(x-x_{\ast }{)}^2$: its predator-prey cross terms cancel. Its sublevel sets are compact in the positive quadrant. The only invariant subset of $\{\dot{V}=0\}$ is the coexistence equilibrium point, because staying on $x=x_{\ast }$ requires $y=y_{\ast }$. The LaSalle invariance principle therefore proves convergence to coexistence for every positive initial state.

Assume well-mixed populations, exponential prey growth without density regulation, predator mortality without food, and bilinear encounters. The Lotka-Volterra predator-prey model isIts coexistence equilibrium is $X_{\ast }=d/c$, $Y_{\ast }=a/b$. The Jacobian matrix there has eigenvalues $\pm i\sqrt{ad}$, so it is a center equilibrium rather than an attracting equilibrium. More globally,is a first integral: substituting the two equations into $\dot{H}$ gives zero. It has a strict minimum at coexistence, diverges at the boundary and at infinity, and its other positive-quadrant level sets are closed periodic orbits. Small oscillations have frequency $\omega =\sqrt{ad}$. In scaled variables $U=\sqrt{d}(X-X_{\ast })/X_{\ast }$ and $V=\sqrt{a}(Y-Y_{\ast })/Y_{\ast }$, the linearization is $\dot{U}=-\omega V$, $\dot{V}=\omega U$.

Model an annual fixed quota as a pulse removing $h>0$ prey, $X^{+}=X^{-}-h$, $Y^{+}=Y^{-}$, with $h<X^{-}$. Between pulses the original equations apply. In logarithmic coordinates the flow has zero divergence, but the pulse has determinant $X^{-}/(X^{-}-h)>1$. Therefore a periodic annually harvested trajectory has return map determinant greater than one, and cannot have both multipliers inside the unit circle. It is unstable, even when a leading rotation-plus-translation approximation looks neutral. Near a nonresonant small-quota forced orbit this expansion destabilizes the original center equilibrium; resonant pulse timing can add secular effects. Quotas larger than the remaining prey are not biologically realizable and require stopping the model. This is fixed-quota harvesting of Lotka-Volterra populations.

The common continuous approximation to an annual quota gives $\dot{X}=X(a-bY)-h$. Its positive equilibrium is $X_{\ast }=d/c$, $Y_h=(a-h/X_{\ast })/b$, requiring $h<a{X}_{\ast }$. Its Jacobian matrix has trace $h/X_{\ast }>0$ and determinant $bc{X}_{\ast }{Y}_h>0$, so this approximation also destabilizes coexistence. At small $h$ it is an unstable focus. A quota is a larger per-capita mortality when prey are scarce, which is the adverse feedback responsible for the result.

For abundance-triggered pulses, retain the same fixed prey removal and specify the crossing direction. The exact event-triggered harvesting by conserved energy calculation isIf prey abundance triggers the event at $X=X_c$, this increment is the same at every event. A sufficiently high prey threshold with a modest quota gives $\Delta H<0$: the orbit's oscillation energy decreases until it no longer crosses the threshold, after which harvesting stops. If $X_c\le X_{\ast }$, then $\Delta H>0$ and harvesting increases oscillations. At the balanced threshold satisfying $ch=d\log [X_c/(X_c-h)]$, energy is unchanged and the ideal orbit is neutral. Thus the useful rule is harvesting at high prey abundance, not simply using any fixed threshold; the equality case supplies no asymptotic restoration.

A predator threshold $Y=Y_c$ does not hold prey abundance fixed. On a rising-predator crossing, $X>X_{\ast }$, so a sufficiently small harvest initially lowers $H$; on a falling-predator crossing $X<X_{\ast }$, it raises $H$. Nevertheless the rising-predator rule is not a robust stabilizer in this conservative model. Its balanced return orbit has preharvest prey abundanceobtained by equating $cX-d\log X$ before and after the pulse. On the return section the derivative isIts magnitude exceeds one because $e^z-e^{-z}>2z$ for $z>0$. The balanced predator-triggered cycle is consequently unstable. At leading small-oscillation order its multiplier is $-1$, explaining a marginal alternation rather than stable damping. If the trigger is crossed in both directions, or a variable rather than fixed quota is used, a different return rule results and must be analyzed separately.

For conservation, the ideal model shows why fixed removals can be dangerous and why prey-based high-abundance feedback can be safer than a calendar quota or an unspecified predator threshold. These conclusions depend on the chosen pulse rule and on the absence of density dependence, refuges and stochasticity. They do not justify a universal numerical safe quota. A harvest policy needs a stopping threshold and the correct phase-dependent feedback, not merely a small nominal annual catch.

Along a solution, the chain rule and antisymmetry of $K$ givebecause $v^{T}Kv=0$ for every real vector $v$. ThereforeThe closed level curves around the strict minimum at the coexistence equilibrium are the periodic population cycles of the Lotka-Volterra predator-prey model.

At a nonzero fixed point, both bracketed per-capita growth rates vanish:Hence the coexistence equilibrium of this Lotka-Volterra predator-prey model is