Gompertz Asymmetric Growth ODE dy/dt = α·y·ln(K/y) — Inflection at K/e

Unlike symmetric logistic growth (inflection at K/2), Gompertz growth inflects earlier at y = K/e ≈ 0.368K.

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

Initial-Value Trajectory Summary (RK4 vs. Euler)

dy/dx0.55*y*ln(3.5 / max(0.05, abs(y))) , y(0) = 0.25

RK4 y(7)3.308862 (Euler: 3.31979)

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.