Logistic Growth Equation
The logistic growth equation dN/dt = rN(1-N/K) models population growth that slows as it approaches the environment carrying capacity.
The Formula
The logistic growth equation models population growth that is limited by resources. Unlike exponential growth (which assumes unlimited resources), logistic growth slows down as the population approaches the carrying capacity.
The term (1 − N/K) acts as a "braking factor." When N is small compared to K, growth is nearly exponential. As N approaches K, growth rate drops toward zero.
Variables
| Symbol | Meaning |
|---|---|
| dN/dt | Rate of population change over time |
| N | Current population size |
| r | Intrinsic rate of natural increase (per capita growth rate) |
| K | Carrying capacity (maximum sustainable population) |
The Integrated Form
This gives the population size at any time t, where N₀ is the initial population. The result is the classic S-shaped (sigmoid) curve.
Example 1
A deer population has r = 0.5 per year, current population N = 200, and carrying capacity K = 1000. What is the growth rate?
dN/dt = rN(1 − N/K)
dN/dt = 0.5 × 200 × (1 − 200/1000)
dN/dt = 100 × (1 − 0.2)
dN/dt = 100 × 0.8
dN/dt = 80 deer per year
Example 2
A bacterial culture starts with N₀ = 100, has r = 1.0 per hour, and K = 10,000. What is the population after 5 hours?
Use the integrated form: N(t) = K / (1 + ((K − N₀)/N₀) × e⁻ʳᵗ)
Calculate the constant: (K − N₀)/N₀ = (10000 − 100)/100 = 99
N(5) = 10000 / (1 + 99 × e⁻⁵)
e⁻⁵ ≈ 0.00674
N(5) = 10000 / (1 + 99 × 0.00674) = 10000 / (1 + 0.667)
N(5) ≈ 5,997 bacteria (already past the halfway point of the S-curve)
When to Use It
The logistic growth model applies to any population with limited resources.
- Ecology: modeling wildlife populations in a bounded habitat
- Microbiology: bacterial and yeast culture growth
- Epidemiology: spread of disease in a finite population
- Business: market adoption of new products (S-curve)
- Fisheries management: setting sustainable harvest levels
Maximum growth rate occurs at N = K/2 (the inflection point of the S-curve). This is why wildlife managers often aim to keep populations near half of carrying capacity for maximum sustainable yield.