Carrying Capacity
Carrying capacity (K) is the maximum population size of a species that a given environment can sustain indefinitely, given the available food, water, habitat, and other essential resources. When a population reaches K, growth rate slows to zero because limiting factors such as resource competition, predation, and disease increase death rates and decrease birth rates. The concept is central to the logistic growth model, which describes how populations grow rapidly when small and stabilize as they approach K.
Key Formula
dN/dt = rN × (K - N) / K
LaTeX: \frac{dN}{dt} = rN\left(\frac{K - N}{K}\right)
| Symbol | Meaning | Unit |
|---|---|---|
| dN/dt | Rate of change of population size with time | individuals/time |
| r | Intrinsic (maximum) rate of natural increase | per unit time |
| N | Current population size | individuals |
| K | Carrying capacity of the environment | individuals |
Worked Example
Problem
A deer population in a forest has an intrinsic growth rate r = 0.4 per year and the forest's carrying capacity K = 500 deer. Currently there are N = 200 deer. Calculate the instantaneous rate of population growth (dN/dt).
Solution
Step 1: Write the logistic growth equation: dN/dt = rN × (K − N)/K. Step 2: Substitute values: dN/dt = 0.4 × 200 × (500 − 200) / 500. Step 3: Calculate the numerator bracket: (500 − 200) = 300. Step 4: dN/dt = 0.4 × 200 × 300 / 500. Step 5: dN/dt = 0.4 × 200 × 0.6 = 0.4 × 120 = 48.
Answer
dN/dt = 48 deer per year (population is growing toward carrying capacity)
Comparison of Exponential vs. Logistic Population Growth
| Feature | Exponential Growth | Logistic Growth |
|---|---|---|
| Growth rate | Constant (always r) | Decreases as N → K |
| Limiting factors | None assumed | Included (food, space, disease) |
| Long-term trajectory | Unlimited increase (J-curve) | Stabilises at K (S-curve) |
| Equation | dN/dt = rN | dN/dt = rN(K−N)/K |
| Real-world fit | Bacteria in ideal lab conditions | Most wildlife populations |
Interactive Tools
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Related Terms
Ecological Succession
Ecological succession is the process of change in the species composition of an ecological community over time, following a disturbance or the formation of a new habitat. Primary succession occurs on bare, previously uncolonised substrate (e.g., newly formed volcanic rock), beginning with pioneer species such as lichens and mosses that gradually modify the environment for subsequent species. Secondary succession occurs in areas where a community has been disturbed but soil remains (e.g., after a forest fire), proceeding more rapidly to a stable climax community because soil and seed banks persist.
Biodiversity
Biodiversity refers to the variety of life on Earth at all its levels — from genes and species to ecosystems — and the ecological processes that support this variety. It is typically measured at three levels: genetic diversity (variation within species), species diversity (number and abundance of species in an area), and ecosystem diversity (variety of habitats, communities, and ecological processes). High biodiversity generally confers ecosystem stability, resilience to disturbance, and a wider range of ecosystem services such as food, medicine, and clean water for human societies.
Consumer (Ecology)
A consumer is any organism in an ecosystem that obtains energy by feeding on other organisms rather than producing its own food through photosynthesis or chemosynthesis. Consumers are classified into primary consumers (herbivores that eat producers), secondary consumers (carnivores that eat herbivores), and tertiary consumers (carnivores that eat other carnivores). They play a critical role in energy transfer through food chains and food webs, regulating population sizes of prey species.
From the nautical term "carrying capacity" used to describe how much cargo a ship could hold, metaphorically applied to ecology. The concept was formalised by Pierre-François Verhulst in 1838 when he published the logistic equation.