Logistic Growth Calculator
Calculate logistic population growth with carrying capacity K. Find population size at any time, time to reach a target population, or maximum sustainable yield.
🌳 Ecology📐 N(t) = K/(1 + ((K-N0)/N0)e^(-rt))🧬 Biology
Initial population N0
Carrying capacity K
Growth rate r (per time)
Time t
Please enter valid values.
Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Logistic Growth Calculator | — | N(t) = K/(1 + ((K-N0)/N0)e^(-rt)) | individuals |
Step-by-Step Examples
Example 1
S-curve
N0=50, K=1000, r=0.5, t=15.
- N(15) = 1000/(1+(1000-50)/50 × e^(-7.5))
- = 1000/(1+19×0.000553) = 1000/1.0105 = 990
- Near K: population essentially stabilized
✓ N(15) = 990 (99% of K)
Example 2
Early growth phase
Same population at t=3.
- N(3) = 1000/(1+19×e^-1.5) = 1000/(1+19×0.223) = 1000/5.24 = 191
✓ N(3) = 191 (19% of K, still rapid growth
Example 3
Maximum sustainable yield
K=500 population fishery.
- MSY at N = K/2 = 250
- Harvest rate = r×N/2 = r×250 per time unit
- Overfishing (N < K/2): population growth slows; collapse risk
✓ MSY = K/2 = 250 individuals at maximum productivity
Real-World Applications
Fisheries Management
Logistic model basis of Maximum Sustainable Yield (MSY) in fisheries policy.
Microbiology
Bacterial batch culture: logistic growth from inoculation to stationary phase.
Conservation
Recovery targets for endangered species often set at K/2 for fastest growth.
Invasive Species
Logistic model predicts invasion trajectory: slow start, rapid spread, then plateau at new K.
Common Mistakes to Avoid
⚠️
Population overshoots K with time lag
Real populations often overshoot K then oscillate. The simple logistic assumes immediate response to density.
⚠️
Maximum growth at N = K/2
Inflection point of logistic curve is at N = K/2. Growth rate dN/dt is maximum here.
⚠️
r must match time units
If r is per year and t is in months: convert r or t to same units.
Frequently Asked Questions
What drives logistic growth to slow? ▾
Density-dependent factors: food competition, disease transmission, predation rates, territorial behavior. As N increases, per-capita birth rate decreases and/or death rate increases.
What is the logistic equation? ▾
dN/dt = rN(1-N/K). When N << K: grows nearly exponentially. When N = K: dN/dt = 0 (equilibrium). When N > K: negative growth (overshoot correction).
Why is MSY at K/2? ▾
At N=K/2: growth rate rN(1-N/K) is maximized. Harvest at this rate is sustainable because population grows back at same rate it is harvested.
What is overshoot and collapse? ▾
When population exceeds K, food runs out and population crashes below K. Common in introduced species, lemmings, and yeast cultures.
What is a stochastic logistic model? ▾
Adds random variation to r and K. Population fluctuates randomly around K. Small populations can go extinct even when mean dynamics predict persistence.