Carrying Capacity and Logistic Growth
Model population growth with resource limits using the logistic growth equation and carrying capacity.
The Formula
dN/dt = r × N × (1 - N/K)
The logistic growth equation models a population that grows quickly at first, then slows as it approaches the carrying capacity. Unlike exponential growth, it accounts for limited resources.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| dN/dt | Rate of population change | individuals per time |
| r | Intrinsic growth rate | per time period |
| N | Current population size | individuals |
| K | Carrying capacity (maximum sustainable population) | individuals |
| (1 - N/K) | Fraction of capacity remaining | (unitless) |
Example 1
A population of 100 deer (K = 500, r = 0.3). Find the growth rate.
dN/dt = 0.3 × 100 × (1 - 100/500)
= 30 × (1 - 0.2) = 30 × 0.8
= 24 deer per time period
Example 2
Same population when N = 450 (near carrying capacity)
dN/dt = 0.3 × 450 × (1 - 450/500)
= 135 × (1 - 0.9) = 135 × 0.1
= 13.5 deer per time period (growth has slowed dramatically)
When to Use It
Use the logistic growth equation when:
- Modeling populations with limited food, space, or other resources
- Predicting when a population will stabilize
- Studying wildlife management and conservation
- Analyzing bacterial growth in a culture with finite nutrients