01 / 21

Population Growth
& Resource Availability

Explore how populations grow, what limits them, and the mathematical models that predict change over time.

AP Environmental Science

What is a Population?

A population is a group of organisms of the same species living in a defined area at a given time. Understanding population dynamics is central to environmental science: it shapes how we manage ecosystems, predict food supply, and plan for sustainability.

Population Size
Total number of individuals (\( N \)) at a given time
Growth Rate
Change in \( N \) per unit time: \( \frac{\Delta N}{\Delta t} \)
Density
Individuals per unit area: \( D = \frac{N}{A} \)

Factors Affecting Population Change

Four key processes drive population change. The net growth rate can be expressed as:

Inputs
Births and immigration add individuals to the population
Outputs
Deaths and emigration remove individuals from the population

Why Resource Availability Matters

Every population depends on finite resources: food, water, space, and shelter. When resources are abundant, populations grow rapidly. When resources become scarce, growth slows or reverses. This dynamic is the foundation of ecological sustainability.

Time Quantity Resources declining Population growing Crossover point

Exponential Growth Model

When resources are unlimited, populations grow exponentially. Each generation multiplies by a constant factor, producing the classic J-shaped curve.

Time
Population (\(N\))
J-curve Slow start Rapid growth

Logistic Growth Model

In reality, resources are limited. The logistic model introduces carrying capacity (\( K \)): the maximum population an environment can sustain indefinitely.

Time
Population (\(N\))
\(K\)
S-curve Lag phase Exponential phase Stabilization

Understanding Carrying Capacity (\( K \))

Carrying capacity is not fixed: it can shift due to changes in resource availability, technology, disease, or environmental conditions.

1
When \( N \ll K \): Resources are abundant: the term \( \frac{K-N}{K} \approx 1 \), so growth is nearly exponential.
2
When \( N = \frac{K}{2} \): Growth rate is at its maximum. This is the inflection point of the S-curve.
3
When \( N \to K \): Resources are depleted: \( \frac{K-N}{K} \to 0 \), and growth effectively stops.
4
When \( N > K \): Population overshoots carrying capacity: \( \frac{dN}{dt} \) becomes negative (population declines).

Exponential vs Logistic

Time
\(N\)
\(K\)
Exponential (J) Logistic (S)
Exponential Logistic
Equation \( \frac{dN}{dt} = r_{\max} N \) \( \frac{dN}{dt} = r_{\max} N \left(\frac{K-N}{K}\right) \)
Curve shape J-shaped S-shaped (sigmoid)
Resources Unlimited Limited
Long-term Unbounded growth Stabilizes at \( K \)

Density-Dependent vs Density-Independent

Limiting factors constrain population growth, but they differ in how they operate. Understanding this distinction is critical for the AP exam.

Density-Dependent
Effect intensifies as population density increases: competition, predation, disease, parasitism
Density-Independent
Effect is constant regardless of density: natural disasters, climate events, human destruction

r-Selected vs K-Selected Species

Species adopt different reproductive strategies depending on their environment and how they relate to carrying capacity.

r-Selected K-Selected
Offspring Many, small Few, large
Parental care Little or none Extensive
Maturation Fast Slow
Population size Fluctuates widely Stable, near \( K \)
Examples Bacteria, insects, rodents Elephants, whales, humans

How to Apply the Logistic Model

1
Identify the variables: Find \( N \) (current population), \( K \) (carrying capacity), and \( r_{\max} \) (growth rate) from the problem.
2
Calculate the brake term: Compute \( \frac{K - N}{K} \). This tells you what fraction of growth capacity remains.
3
Multiply all terms: \( \frac{dN}{dt} = r_{\max} \times N \times \frac{K - N}{K} \). This is the population change per unit time.
4
Interpret the result: Positive = population growing. Zero = at carrying capacity. Negative = population overshooting and declining.

Example 1: Basic Logistic Growth

A deer population has \( N = 200 \), \( K = 1000 \), and \( r_{\max} = 0.5 \) per year. What is the growth rate?

1
Brake term: \( \frac{K - N}{K} = \frac{1000 - 200}{1000} = \frac{800}{1000} = 0.8 \)
2
Plug in: \( \frac{dN}{dt} = 0.5 \times 200 \times 0.8 = 80 \) deer per year
3
Interpret: The population is still well below \( K \), so growth is strong: 80 more deer per year.

Example 2: Near Carrying Capacity

Now the deer population has grown to \( N = 900 \). Same \( K = 1000 \) and \( r_{\max} = 0.5 \).

1
Brake term: \( \frac{1000 - 900}{1000} = \frac{100}{1000} = 0.1 \)
2
Plug in: \( \frac{dN}{dt} = 0.5 \times 900 \times 0.1 = 45 \) deer per year
3
Interpret: Despite having more individuals (\( N = 900 \) vs 200), growth is slower (45 vs 80) because resources are nearly exhausted.

Example 3: Overshoot Scenario

A rabbit population temporarily reaches \( N = 1200 \) due to a mild winter, while \( K = 1000 \) and \( r_{\max} = 0.3 \).

1
Brake term: \( \frac{1000 - 1200}{1000} = \frac{-200}{1000} = -0.2 \)
2
Plug in: \( \frac{dN}{dt} = 0.3 \times 1200 \times (-0.2) = -72 \) rabbits per year
3
Interpret: The negative value means the population is declining: 72 fewer rabbits per year until \( N \) drops back toward \( K \).

Common Mistakes

✕ Confusing r and K
\( r_{\max} \) is the per capita growth rate; \( K \) is the maximum population size. They are independent variables.
✕ Ignoring the Brake
Students often compute \( r_{\max} \times N \) and forget the \( \frac{K-N}{K} \) factor, yielding exponential results instead of logistic.
✕ Max Growth at K
Growth rate is zero at \( N = K \). The maximum growth rate actually occurs at \( N = \frac{K}{2} \), the inflection point.

Quick Reference Summary

Concept Key Formula / Idea When to Use
Exponential growth \( \frac{dN}{dt} = r_{\max} N \) Unlimited resources
Logistic growth \( \frac{dN}{dt} = r_{\max} N \frac{K-N}{K} \) Limited resources
Carrying capacity \( K \) = max sustainable \( N \) Predicting equilibrium
Max growth point \( N = \frac{K}{2} \) Inflection of S-curve
Overshoot \( N > K \implies \frac{dN}{dt} < 0 \) Population decline

Question 1 of 5

A population of fish has \( N = 500 \), \( K = 2000 \), and \( r_{\max} = 0.4 \). What is \( \frac{dN}{dt} \)?

Question 2 of 5

In the logistic model, at what population size is the growth rate maximized?

Question 3 of 5

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

Question 4 of 5

A population has \( N = 1500 \) and \( K = 1000 \). What does the logistic model predict?

Question 5 of 5

Which reproductive strategy describes a species with few offspring, slow maturation, and extensive parental care?