Logistic predator-prey model ($\dot{x}=x(a-bx-cy),\dot{y}=y(dx-e)$)

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