Lotka-Volterra predator-prey model

The Lotka-Volterra predator-prey model couples exponential prey growth and predator mortality through a bilinear encounter term. Its positive coexistence equilibrium is surrounded by closed conserved-energy curves, so the populations oscillate periodically in the idealized model.

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 $0<s<1$.

The annual pulse-breeding predator-prey model is an area-preserving map in logarithmic population coordinates: $u^{\prime }=u+\log r-\kappa e^v$, then $v^{\prime }=v+\log s+\log (1+b{e}^{u^{\prime }})$. Each step is a shear of determinant one. At a positive fixed point, the ordinary-population Jacobian matrix has determinant one and trace $2-(1-s)\log r$. Eigenvalues lie on the unit circle for $0<(1-s)\log r<4$, but this linear neutrality does not prove nonlinear stability at every resonance. Area preservation excludes an isolated attracting positive periodic cycle.

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.

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.

This map equals an explicit unit-step update of a logistic predator-prey differential equation. Its coexistence fixed point is $(b,(1-a-b)/a)$, requiring $a+b<1$. Its trace and determinant are $T=2-b/a$ and $D=(1-2b)/a$. The Schur stability criterion gives stability precisely when $b>(1-a)/2$ and $b<a+1/3$. The coexistence flip at $b=a+1/3$ is subcritical on $1/9<a<1/3$, with effective cubic coefficient $-9/[(3a+1{)}^2(9a-1)]$ when the $-1$ eigenvector has first component one. The unit-circle crossing $a+2b=1$ is a supercritical Neimark–Sacker bifurcation away from the strong resonances of a planar map $a=1/7,1/5$; in the eigenvector normalization $q_1=1$ its cubic radial coefficient is $-1/[a^2(1-a)]$. The whole nonnegative quadrant is not forward invariant, because $x^{\prime }$ can become negative. Local fixed-point stability is therefore not a global population-dynamics classification.

// Destination: dynamical-systems.bigb