Carrying Capacity and Logistic Growth
Reference for logistic growth dN/dt = rN(1-N/K) and carrying capacity K.
Includes worked examples for wildlife populations, bacteria, and ecosystem limits.
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
Key Notes
- Maximum growth rate occurs at N = K/2, not at N = 0 — at half the carrying capacity, the (1 − N/K) term equals 0.5 while N is already substantial; this is why fisheries managers aim to harvest enough to keep populations near K/2 for maximum sustainable yield
- K is not a fixed constant — carrying capacity changes with habitat loss, climate change, food availability, or human intervention; populations can temporarily overshoot K (use resources faster than they replenish) and then crash, rather than stopping smoothly at K
- The logistic model assumes a single limiting resource in a simple environment — real ecosystems have multiple interacting species; predator-prey interactions, competition, and disease are modeled by the Lotka-Volterra equations, which extend logistic growth
- Allee effect: some species require a minimum viable population size for mate finding, group defense, or cooperative foraging — below this threshold the effective r becomes negative and the population spirals to extinction even though K > 0