The science
Each case-study file integrates dN/dt = rN(1 − N/K). This lab keeps that growth term and adds what a manager actually has to decide: how hard to harvest, and what happens if the population is small enough to suffer an Allee effect.
Equations as delivered in the case-study files
dN/dt = rN(1 − N/K) [individuals/s]; N(t) = K/(1 + (K/N₀ − 1)e^(−rt)).
dN/dt = rN(1 − N/K) [cells/s]; N(t)=K/[1+(K/N₀−1)exp(−rt)] [cells].
dN/dt = (r/60)N(1 − N/K) [cells/s]; N(t) = K/[1 + ((K/N₀) − 1)e^(−rt/60)] [cells].
dN/dt = r N (1 − N/K) [individuals/s] ; N(t) = K / (1 + (K/N₀ − 1) exp(−rt)) [individuals] ; r [1/s]; K [individuals]
dN/dt = rN(1−N/K) [animals/s] ; N(t)=K/(1+(K/N₀−1)e^(−rt)) [animals] ; r is converted day⁻¹ to s⁻¹; 1 playback second = 1 simulated day
What this lab adds
dN/dt = r N (1 − N/K) · α(N) − e r N − c r K/4, α = (N − A)/(K − A) for A > 0 else 1
Proportional harvest alone: logistic with r′ = r(1 − e), K′ = K(1 − e); yield at equilibrium = e r K(1 − e)
Maximum sustainable yield rK/4 occurs at e = 1/2 and N = K/2
Assumptions in the delivered files
- One homogeneous population with fixed carrying capacity. Rates and short timescales are illustrative, not field-calibrated.
- Deterministic, continuous, single-species logistic growth with constant carrying capacity; illustrative fast-growing culture, no discrete cell events, metabolic lag, contamination or spatial variation.
- Constant parameters between decisions and timeline events. The numerical core uses physical seconds and the displayed physical units (including metres and kilograms where relevant). Time acceleration is explicitly noted when simulation time is faster than playback time. Idealised equations are for e
Honest simplifications of this lab
- Continuous, deterministic and unstructured: no age classes, no randomness, no space.
- Time units follow each file (seconds, minutes or days); rates in the files are illustrative, not field-calibrated.
- The Allee factor (N − A)/(K − A) is one simple strong-Allee form; other forms give different tipping points.
Independent check The page compares the integrator with the logistic closed form (also with harvest via r(1−e), K(1−e)), the MSY identity and collapse beyond rK/4; the test file repeats them with an independent Python RK4.
Educational model. Case-study presets reuse the parameter sets, titles and captions of ChatGPT-generated simulations delivered for MySimulator; the physics core here is a richer shared model.