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Population
Ecology

Explore how populations grow, interact, and are regulated by environmental factors: from exponential surges to logistic limits.

AP Biology

What Is a Population?

A population is a group of individuals of the same species living in the same area at the same time. Population ecology studies how population size, density, and structure change over time and what factors drive those changes.

Population Size
Total number of individuals (\( N \)) in a defined area
Population Density
Number of individuals per unit area or volume
Dispersion
Spatial pattern: clumped, uniform, or random

Key Population Variables

Four processes determine how a population's size changes over time. The change in population size (\( \Delta N \)) follows:

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

Dispersion Patterns

Clumped Uniform Random

Exponential Growth Model

When resources are unlimited, populations grow at their maximum rate. The per capita rate of increase (\( r \)) stays constant, producing a characteristic J-shaped curve.

\( r_{\max} \)
Intrinsic rate of natural increase: maximum per capita growth rate
\( N \)
Current population size at time \( t \)

Logistic Growth Model

In reality, environments have limited resources. As populations approach the carrying capacity (\( K \)), growth slows, producing an S-shaped curve.

J-Curve vs S-Curve

Time Population Size (N) K Exponential Logistic

Carrying Capacity: K

Carrying capacity (\( K \)) is the maximum population size an environment can sustain indefinitely given the available resources. \( K \) is not fixed: it can change with seasons, habitat alteration, or human activity.

1
When \( N \ll K \): growth is nearly exponential: the braking factor \( \approx 1 \)
2
When \( N = K/2 \): growth rate is at its maximum (\( \Delta N / \Delta t \) peaks)
3
When \( N = K \): growth rate = 0, population is at equilibrium
4
When \( N > K \): the braking factor becomes negative: population declines

Density-Dependent vs Density-Independent

Density-Dependent Density-Independent
Definition Effect intensifies as population density increases Effect is the same regardless of density
Examples Competition, predation, disease, parasitism Floods, fires, drought, volcanic eruptions
Pattern Keeps population near K Can cause sudden crashes at any population size
Type Usually biotic Usually abiotic

r-Selected vs K-Selected Species

r-Selected K-Selected
Offspring Many, small Few, large
Parental Care Little or none Extensive
Maturation Rapid Slow
Lifespan Short Long
Growth Curve J-shaped (boom-bust) S-shaped (near K)
Examples Bacteria, insects, mice Elephants, whales, humans

Survivorship Curves

Relative Age Survivors (log) Type I Type II Type III
Type I
Low early mortality, high late mortality (humans, elephants)
Type II
Constant mortality rate throughout life (songbirds, rodents)
Type III
Very high early mortality, few survivors (oysters, fish)

Applying the Logistic Model

1
Identify the variables: Determine \( N \) (current size), \( K \) (carrying capacity), and \( r_{\max} \) (intrinsic growth rate)
2
Calculate the braking factor: Compute \( \dfrac{K - N}{K} \). If \( N = 200 \) and \( K = 1000 \), then braking factor = \( \dfrac{800}{1000} = 0.8 \)
3
Multiply: \( \dfrac{dN}{dt} = r_{\max} \times N \times \text{braking factor} \)
4
Interpret: Positive result = population growing. Zero = at equilibrium. Negative = population declining.

Example 1: Exponential Growth

A bacterial colony starts with \( N = 500 \) cells. The intrinsic growth rate is \( r_{\max} = 0.4 \) per hour. Assuming unlimited resources, what is the growth rate?

Example 2: Logistic Growth

A deer population has \( N = 400 \), carrying capacity \( K = 1000 \), and \( r_{\max} = 0.5 \) per year. Find the growth rate.

1
Braking factor: \( \dfrac{K - N}{K} = \dfrac{1000 - 400}{1000} = 0.6 \)
2
Plug into formula: \( \dfrac{dN}{dt} = 0.5 \times 400 \times 0.6 = \mathbf{120} \) deer/year

Example 3: Maximum Growth at \( N = K/2 \)

Same deer population (\( r_{\max} = 0.5 \), \( K = 1000 \)). Now \( N = 500 \) (exactly half of \( K \)).

Compare: at \( N = 400 \), growth was 120. At \( N = 500 \), growth is 125. At \( N = 900 \), growth would be only \( 0.5 \times 900 \times 0.1 = \) 45.

Common Mistakes

✕ Confusing r and K
\( r_{\max} \) is the growth rate. \( K \) is the carrying capacity. They are independent parameters: a high \( r \) does not mean high \( K \).
✕ Max Growth at K
Maximum growth rate occurs at \( N = K/2 \), not at \( K \). At \( N = K \), growth rate is zero.
✕ K Is Fixed Forever
Carrying capacity changes with environmental conditions. Habitat loss, climate change, or new resources all shift \( K \).

Question 1 of 5

In the logistic growth equation \( \dfrac{dN}{dt} = r_{\max} N \left(\dfrac{K-N}{K}\right) \), what does the term \( \left(\dfrac{K-N}{K}\right) \) represent?

Question 2 of 5

A population has \( r_{\max} = 0.3 \), \( K = 2000 \), and \( N = 1000 \). What is \( \dfrac{dN}{dt} \)?

Question 3 of 5

Which of the following is a density-dependent factor that regulates population size?

Question 4 of 5

An organism that produces thousands of offspring with minimal parental care, matures rapidly, and has a short lifespan is best described as:

Question 5 of 5

At what population size does the logistic growth model predict the maximum rate of population increase?