Logistic Population with Seasonal Harvesting dy/dt = y(1 − y/K) − H(t)

Reveals seasonal limit-cycle oscillation above critical biomass vs. sudden population collapse below the saddle-node separatrix.

Normalized Quiver Slope Field (Click Anywhere on Plane to Drop New RK4 Trajectory)

Initial-Value Trajectory Summary (RK4 vs. Euler)

dy/dxy*(1 - y/3) - 0.55*(1 + 0.6*sin(2*x)) , y(0) = 2.6

RK4 y(8)2.1312 (Euler: 2.11452)

All trajectories exponentially approach horizontal equilibrium asymptote y = 1 regardless of initial temperature y(0).

Autonomous phase portrait contrasting unstable extinction equilibrium y = 0 against stable carrying capacity y = 3.

Initial states y(0) > 0 snap to upper attractor y = +2 while y(0) < 0 snap to y = −2, modeling Schmitt triggers and genetic switches.

Shows transient initial offset decaying away until all RK4 trajectories lock onto the same phase-lagged sinusoidal steady state.

Compares Euler’s 1st-order drift against 4th-order Runge–Kutta (RK4) across symmetric Gaussian peak trajectories.

Non-separable quadratic ODE appearing in optimal LQR control theory and WKB quantum tunneling approximations.