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Carrying
Capacity

Explore how ecosystems set population limits, the math behind logistic growth, and why understanding \( K \) is critical for conservation and sustainability.

AP Environmental Science

What Is Carrying Capacity?

Carrying capacity, symbolized as \( K \), is the maximum population size of a species that an environment can sustain indefinitely, given the available food, water, habitat, and other resources.

Limiting Factors That Determine \( K \)

Density-Dependent
Competition, predation, disease, parasitism: effects intensify as population density rises.
Density-Independent
Natural disasters, climate events, human destruction: affect populations regardless of density.

Exponential vs Logistic Growth

Time
Population
\( K \)
Exponential (J-curve)
Logistic (S-curve)

The Logistic Growth Equation

$$ \frac{dN}{dt} = r_{\max} \cdot N \cdot \left(\frac{K - N}{K}\right) $$

\( N \)
Current population size
\( r_{\max} \)
Maximum per capita growth rate
\( K \)
Carrying capacity of the environment
\( \frac{K - N}{K} \)
Braking fraction: approaches 0 as \( N \to K \)

Understanding the Braking Term

Population (\( N \))
Growth Rate (\( \frac{dN}{dt} \))
\( K/2 \)
\( K \)
Max growth

r-Selected vs K-Selected Species

r-Selected K-Selected
Offspring count Many Few
Parental care Little to none Extensive
Body size Small Large
Lifespan Short Long
Growth pattern Boom-and-bust Near \( K \), stable
Examples Insects, bacteria Elephants, whales

What Happens When Populations Overshoot \( K \)?

Time
Population
\( K \)
Overshoot
Die-off

When a population temporarily exceeds \( K \), resources are depleted faster than they regenerate. This triggers a die-off (population crash), after which the population may oscillate around \( K \) or stabilize, depending on how severely resources were degraded.

How \( K \) Can Shift Over Time

Increase \( K \)
New food sources, habitat restoration, fewer predators, favorable climate shifts
Decrease \( K \)
Habitat destruction, pollution, drought, invasive species, over-harvesting

Explain why K changes with available resources.

Use this illustration to explore the concept described in its caption.

The same meadow is shown in green summer and snowy winter. Dashed conditional resource ceilings show 100 and 60 hypothetical deer-equivalent resource units rather than counts of the illustrated deer.
Seasonal changes in accessible forage can change the resource ceiling for the same deer habitat. The illustrative carrying-capacity bands are conditional rather than fixed species limits.

How to Solve Logistic Growth Problems

1
Identify the variables. Find \( N \), \( r_{\max} \), and \( K \) from the problem statement.
2
Compute the braking fraction. Calculate \( \frac{K - N}{K} \) to see how close the population is to capacity.
3
Plug into the equation. \( \frac{dN}{dt} = r_{\max} \cdot N \cdot \frac{K - N}{K} \).
4
Interpret the result. A positive value means growth; near zero means the population is close to \( K \).

Worked Example: Deer Population

A forest can support \( K = 500 \) deer. The current population is \( N = 100 \) and the maximum per capita growth rate is \( r_{\max} = 0.5 \) per year. What is the population growth rate?

Worked Example: Near Capacity

Now suppose the same forest has \( N = 450 \) deer, with \( K = 500 \) and \( r_{\max} = 0.5 \). How does the growth rate compare?

Worked Example: Maximum Growth at \( \frac{K}{2} \)

With the same parameters (\( K = 500 \), \( r_{\max} = 0.5 \)), what is \( \frac{dN}{dt} \) when \( N = 250 \)?

Real-World Applications

Wildlife Management
Harvest quotas are set to keep populations near \( \frac{K}{2} \), where regrowth is fastest.
Fisheries Science
Maximum Sustainable Yield (MSY) equals \( \frac{dN}{dt} \) at \( N = \frac{K}{2} \).
Conservation
Habitat restoration directly increases \( K \), allowing endangered species to recover.

Common Mistakes to Avoid

✕ Treating \( K \) as Fixed
\( K \) changes with environmental conditions. A drought can lower \( K \); habitat restoration can raise it.
✕ Confusing Growth Rate with Population
\( \frac{dN}{dt} \) is the rate of change, not the population size. A large \( N \) near \( K \) can have a small \( \frac{dN}{dt} \).
✕ Forgetting Maximum Growth at \( \frac{K}{2} \)
The fastest growth occurs at half the carrying capacity, not when the population is smallest.

Question 1 of 5

A population of rabbits has \( K = 1{,}000 \), \( N = 500 \), and \( r_{\max} = 0.3 \). What is \( \frac{dN}{dt} \)?

Question 2 of 5

At what population size does the logistic growth rate reach its maximum?

Question 3 of 5

A severe drought reduces available water in a grassland. What is the most likely effect on the resident bison population?

Question 4 of 5

Which of the following is a density-dependent limiting factor?

Question 5 of 5

A fish population has \( K = 10{,}000 \), \( r_{\max} = 0.4 \), and \( N = 9{,}500 \). The population growth rate \( \frac{dN}{dt} \) is: