Science
Strong Allee Effect Population Growth & Critical Extinction Threshold
Calculate cubic strong Allee population growth rate dN/dt = r * N * (1 - N / K) * (N / A - 1), per-capita growth rate, and extinction-vortex vs carrying-capacity trajectory.
Model minimum viable populations, mate-finding failure, and cooperative defense!
Standard Verhulst logistic growth dN/dt = r * N * (1 - N / K) assumes per-capita growth peaks as density approaches zero—yet real endangered species that rely on Broadcast spawning, pack hunting, pollinator attraction, or predator dilution suffer positive density dependence at low numbers (Warder Clyde Allee's Effect)!
Introducing a Critical Allee Threshold A (0 < A < K) yields the cubic Strong Allee Equation: dN/dt = r * N * (1 - N / K) * (N / A - 1)! Above N > A (and below K), growth is positive and converges stably to carrying capacity K; if poaching or habitat loss pushes N below A, the factor (N / A - 1) turns negative and drives deterministic collapse toward extinction (N -> 0)!
| Population Size N (r = 0.40/yr, K = 1,000, Allee A = 150) | Logistic Factor (1 - N/K) | Allee Factor (N/A - 1) | Growth dN/dt & Fate |
|---|---|---|---|
| Sub-Threshold Herd (N = 100 < A) | 0.90 | -0.3333 (Negative!) | -12.00 / yr -> Extinction Vortex! |
| Recovering Population (N = 300 > A) | 0.70 | +1.0000 | +84.00 / yr -> Grows Toward K |
| Carrying Capacity (N = 1,000 = K) | 0.00 | +5.6667 | 0.00 / yr -> Stable Equilibrium |
Cubic Strong Allee Growth Rate dN/dt Formula
Evaluates dN/dt = r * N * (1 - N / K) * (N / A - 1) after validating 0 < A < K.
=IF(OR(B3 <= 0, B4 <= 0, B4 >= B3), "Invalid Parameters (Require 0 < A < K)", ROUND(B2 * B5 * (1 - B5 / B3) * (B5 / B4 - 1), 2))Demographic Trajectory Regime (Extinction Vortex vs Viable Growth)
Classifies current abundance N (B5) against critical threshold A (B4) and carrying capacity K (B3).
=IFS(B5 <= 0, "Extinct (N = 0)", B5 < B4, "Extinction Vortex (N < Allee Threshold)", B5 = B4, "Unstable Allee Tipping Point", B5 < B3, "Viable Recovery Toward K", TRUE, "At or Above Carrying Capacity K")Try the example
Use the original values or change them to test your understanding.