AP Environmental Science: Population Ecology (Exponential vs Logistic Growth, Carrying Capacity)

AP Environmental Science – Population Ecology (Exponential vs Logistic Growth, Carrying Capacity)

AP Environmental Science: Population Ecology Study Guide

Topic: Exponential vs. Logistic Growth, Carrying Capacity

What This Is

Population ecology explains how populations (groups of the same species) grow, shrink, or stabilize over time. On the AP exam, you’ll analyze graphs, calculate growth rates, and predict how limiting factors (like food, space, or disease) affect populations. Real-world example: The reintroduction of wolves to Yellowstone National Park in 1995 caused elk populations to shift from exponential growth (unlimited resources) to logistic growth (hitting carrying capacity) due to predation and habitat limits. This restored balance to the ecosystem.

Key Terms & Concepts

• Population: A group of individuals of the same species living in the same area at the same time.
• Exponential growth (J-curve): Population growth with no limits, where the growth rate accelerates over time. Formula: dN/dt = rN
• dN/dt = change in population size over time
• r = intrinsic growth rate (birth rate – death rate)
• N = current population size

• Example: Bacteria doubling every 20 minutes in a petri dish.

• Logistic growth (S-curve): Population growth that slows as it approaches carrying capacity due to limiting factors. Formula: dN/dt = rN (1 – N/K)

• K = carrying capacity (max population the environment can support)
• (1 – N/K) = fraction of carrying capacity still available

• Example: Deer populations in a forest leveling off due to food scarcity.

• Carrying capacity (K): The maximum population size an environment can sustain indefinitely without degrading the habitat.

• Example: A pond can support 500 fish; adding more leads to starvation and disease.

• Limiting factors: Resources or conditions that restrict population growth. Two types:

• Density-dependent: Effects worsen as population density increases (e.g., disease, competition, predation).

• Density-independent: Effects occur regardless of population size (e.g., natural disasters, climate change).

• Overshoot: When a population temporarily exceeds carrying capacity, leading to a crash (dieback).

• Example: Reindeer on St. Matthew Island (1944) grew exponentially, then starved when they overgrazed lichen.

• K-selected species: Species with low growth rates, few offspring, and high parental care (e.g., elephants, humans).

• r-selected species: Species with high growth rates, many offspring, and little parental care (e.g., insects, weeds).

• Age structure diagrams: Graphs showing the distribution of ages in a population (pyramid = growing, rectangle = stable, inverted pyramid = declining).

• Survivorship curves: Graphs showing the % of a population surviving to different ages.
• Type I: High survival until old age (humans, K-selected).
• Type II: Constant mortality rate (birds).
• Type III: High early mortality (oysters, r-selected).

Step-by-Step: Analyzing Population Growth Graphs

How to tackle an AP FRQ or MCQ about population graphs:

1. Identify the curve type:
2. J-curve = exponential growth (unlimited resources).

3. S-curve = logistic growth (hits carrying capacity).

4. Locate key points:

5. Exponential phase: Steep upward slope (high r).
6. Inflection point: Where growth starts slowing (N = K/2).

7. Carrying capacity (K): Where the curve levels off.

8. Calculate growth rate (if given data):

9. Use dN/dt = rN for exponential growth.

10. Use dN/dt = rN (1 – N/K) for logistic growth.

11. Predict future trends:

12. If N < K: Population will grow.
13. If N = K: Population is stable.

14. If N > K: Overshoot-dieback likely.

15. Explain limiting factors:

16. Density-dependent (e.g., food, disease) or density-independent (e.g., drought, fire)?

Example Problem: A population of 1,000 rabbits has an intrinsic growth rate (r) of 0.2 per year. The carrying capacity (K) is 5,000 rabbits. What is the growth rate when N = 2,000? Solution: dN/dt = rN (1 – N/K) = 0.2 × 2,000 × (1 – 2,000/5,000) = 400 × 0.6 = 240 rabbits/year.

Common Mistakes

• Mistake: Confusing exponential and logistic growth. Correction: Exponential = unlimited resources (J-curve); logistic = limited resources (S-curve). Why? Exponential growth can’t continue forever—real populations hit limits.

• Mistake: Assuming carrying capacity is fixed. Correction: K can change due to environmental shifts (e.g., drought lowers K for deer). Why? Carrying capacity depends on resource availability.

• Mistake: Ignoring units in growth rate calculations. Correction: Always check if r is per year, month, etc. Why? A growth rate of 0.1/month-0.1/year!

• Mistake: Forgetting that r can be negative (population decline). Correction: If death rate > birth rate, r is negative. Why? Populations can shrink (e.g., endangered species).

• Mistake: Misidentifying density-dependent vs. independent factors. Correction: Density-dependent = worsens with crowding (e.g., disease); density-independent = affects all equally (e.g., hurricane). Why? The AP exam loves testing this distinction.

AP Exam Insights

1. Graph interpretation is key: You’ll see S-curves and J-curves on MCQs and FRQs. Know how to:
2. Label K, inflection point, and exponential phase.

3. Explain why a population overshoots K (e.g., delayed response to limiting factors).

4. FRQ traps:

5. "Explain the difference between exponential and logistic growth"-Must mention limiting factors and carrying capacity.

6. "Predict what happens if K decreases"-Population will crash (dieback) until it stabilizes at the new K.

7. MCQ tricks:

8. Questions may show a population graph with a sudden drop and ask if it’s due to a density-dependent or independent factor.

9. Watch for units in growth rate calculations (e.g., r = 0.05/year vs. 0.05/month).

10. Real-world applications:

11. Human population growth: Currently exponential but slowing (logistic phase).
12. Invasive species: Often grow exponentially at first (e.g., zebra mussels in the Great Lakes).

Quick Check Questions

1. MCQ: A population of 500 deer has a carrying capacity of 1,000. If the intrinsic growth rate (r) is 0.1/year, what is the growth rate when N = 600? A) 24 deer/year B) 30 deer/year C) 60 deer/year D) 100 deer/year Answer: A) 24 deer/year. Explanation: dN/dt = 0.1 × 600 × (1 – 600/1,000) = 60 × 0.4 = 24.

2. FRQ (short): Explain why a population might temporarily exceed its carrying capacity. What is this phenomenon called? Answer: A population may overshoot K if there’s a delay between resource depletion and population response (e.g., animals reproduce before food runs out). This leads to a dieback as resources become scarce.

3. MCQ: Which of the following is a density-independent limiting factor? A) Competition for food B) Predation C) Hurricane D) Disease Answer: C) Hurricane. Explanation: Hurricanes affect populations regardless of size; the others worsen with crowding.

Last-Minute Cram Sheet

1. Exponential growth: J-curve, dN/dt = rN, unlimited resources.
2. Logistic growth: S-curve, dN/dt = rN (1 – N/K), hits carrying capacity (K).
3. Carrying capacity (K): Max population an environment can support long-term.
4. Overshoot-dieback: Population exceeds K, then crashes.
5. Density-dependent factors: Disease, competition, predation (worse with crowding).
6. Density-independent factors: Natural disasters, climate change (affect all equally).
7. K-selected species: Few offspring, high parental care (e.g., elephants).
8. r-selected species: Many offspring, low parental care (e.g., insects).
9. Inflection point: N = K/2 (growth rate starts slowing).
10. Human population: Currently exponential but slowing (logistic phase).