Population Ecology- Practice Exercises and Solutions

What This Article Covers

Population ecology is the study of how populations of organisms change over time. If you're taking ecology, environmental science, or AP Biology, you'll need to solve problems involving population growth, density, and models. This guide gives you actual practice exercises with worked solutions — no filler.

Core Concepts You Need to Know

Before jumping into problems, make sure these terms are solid:

• Population: A group of individuals of the same species living in the same area
• Population density: Number of individuals per unit area or volume
• Population growth rate: Change in population size over time (births + immigration minus deaths - emigration)
• Carrying capacity (K): Maximum population size an environment can sustain indefinitely
• Exponential growth: Growth without limits — population multiplies by a constant factor
• Logistic growth: Growth that slows as population approaches carrying capacity

Population Growth Models: Exercise and Solutions

Most exams test your ability to identify and calculate exponential vs. logistic growth. Here's how it works.

Exercise 1: Identifying Growth Models

A bacteria population starts with 100 individuals. Under ideal conditions, it doubles every 20 minutes.

Question: Is this exponential or logistic growth?

Solution: This is exponential growth. The population grows without any limiting factors. There's no carrying capacity mentioned, and the growth rate stays constant regardless of population size.

Exercise 2: Calculating Exponential Growth

A population of 500 rabbits has a per capita growth rate (r) of 0.3 per year. What will the population be after 3 years?

Solution:

Use the formula: Nt = N0 × e^(rt)

• N0 = 500 (starting population)
• r = 0.3 (growth rate)
• t = 3 (years)
• e = 2.718 (constant)

Calculation:

Nt = 500 × 2.718^(0.3 × 3)
Nt = 500 × 2.718^0.9
Nt = 500 × 2.46
Nt ≈ 1,230 rabbits

The population nearly triples in 3 years. That's the nature of exponential growth — small rates create massive increases over time.

Exercise 3: Logistic Growth Calculation

A lake has a carrying capacity of 2,000 fish. Starting with 200 fish, the population grows at r = 0.5 per year. What is the population growth rate when the population is at 500?

Solution:

Use the logistic growth formula: dN/dt = rN × (1 - N/K)

• r = 0.5
• N = 500
• K = 2,000

Calculation:

dN/dt = 0.5 × 500 × (1 - 500/2000)
dN/dt = 250 × (1 - 0.25)
dN/dt = 250 × 0.75
dN/dt = 187.5 fish per year

Notice the growth rate is lower than if the population kept growing exponentially (which would be 250/year at this point). The "1 - N/K" term represents the environmental resistance slowing growth as you approach carrying capacity.

Population Density Calculations

Density problems are straightforward — they're just ratios. Students lose points here from unit conversion errors.

Exercise 4: Basic Density Calculation

A forest area of 50 km² contains 1,250 oak trees. What is the population density?

Solution:

Density = Population / Area
Density = 1,250 / 50
Density = 25 oak trees per km²

Exercise 5: Density with Unit Conversion

A field mouse population occupies 15 hectares. Scientists trap and count 180 mice. What is the density in individuals per square meter?

Solution:

First, convert hectares to square meters:
1 hectare = 10,000 m²
15 hectares = 150,000 m²

Then calculate density:
Density = 180 / 150,000
Density = 0.0012 mice per m²

Or expressed as: 1 mouse per 833 m²

This is a common exam trap. Always check what units the question asks for before calculating.

Mark-Recapture Problems

This is a standard method for estimating animal populations. The formula is:

N = (M × C) / R

• N = estimated population size
• M = number of individuals initially captured and marked
• C = total number of individuals captured in second sample
• R = number of marked individuals recaptured

Exercise 6: Mark-Recapture Estimation

Wildlife biologists capture 80 deer, tag them, and release them. One week later, they capture 100 deer, of which 25 are already tagged. Estimate the deer population.

Solution:

N = (M × C) / R
N = (80 × 100) / 25
N = 8,000 / 25
N = 320 deer

This method assumes the marked animals mix evenly with the population, no births/deaths occur during the study period, and animals have equal capture probability.

Population Growth Rate Calculations

Exercise 7: Intrinsic Growth Rate

A population of 1,000 birds has 150 births and 50 deaths in one year. Immigration is 20, emigration is 10. What is the growth rate?

Solution:

Growth rate = (Births + Immigration) - (Deaths + Emigration)
Growth rate = (150 + 20) - (50 + 10)
Growth rate = 170 - 60
Growth rate = +110 individuals per year

Per capita growth rate (r) = 110 / 1,000 = 0.11 per year

Age Structure Diagram Interpretation

Age structure diagrams show population distribution across age groups. You'll need to read them to predict future growth trends.

Exercise 8: Reading Age Structure Diagrams

Examine an age structure diagram with these characteristics:

• Wide base (large proportion of young)
• Narrowing rapidly toward older ages
• Approximately triangular shape

Question: What does this diagram indicate about the population?

Solution:

This is a rapidly growing population. The wide base indicates high birth rates, and the narrow top indicates high mortality or short life expectancy. This pattern is typical of developing countries.

Compare this to a diagram that's more rectangular — that indicates a stable population with low growth. A diagram narrowing at the base indicates a declining population.

Quick Reference: Population Ecology Formulas

Common Mistakes on Exams

• Confusing exponential and logistic growth — Exponential has no ceiling; logistic hits a carrying capacity
• Forgetting unit conversions — km² to m², hectares to km²
• Using the wrong formula — Check what you're solving for before picking an equation
• Mark-recapture assumptions — Know the three assumptions or you'll miss conceptual questions
• Misreading carrying capacity — K is not a limit the population cannot exceed; it's where growth rate becomes zero

Practice Problems: Test Yourself

Try these before checking the answers below.

Problem 1: A bacterial colony grows from 100 to 400 cells in 2 hours under ideal conditions. What is the per capita growth rate (r)?

Problem 2: A pond has a carrying capacity of 5,000 frogs. If the population is currently 2,000 with r = 0.4, what is the current growth rate?

Problem 3: Biologists tag 50 fish in a lake. A second catch of 80 fish contains 10 tagged individuals. Estimate the fish population.

Answers

Problem 1:
400 = 100 × e^(r × 2)
4 = e^(2r)
ln(4) = 2r
r = 1.386 / 2 = 0.693 per hour

Problem 2:
dN/dt = 0.4 × 2,000 × (1 - 2000/5000)
dN/dt = 800 × (0.6)
dN/dt = 480 frogs per year

Problem 3:
N = (50 × 80) / 10
N = 400 fish

Getting Started with Population Ecology Problems

Here's your step-by-step approach:

1. Identify what the question is asking — Population size? Growth rate? Density? This determines your formula
2. List known variables — Write down N₀, r, K, t, or whatever values are given
3. Convert units if needed — Do this before calculating, not after
4. Apply the correct formula — Check the table above if you're unsure
5. Check your answer — Does the number make biological sense? Exponential growth should increase faster over time. Logistic growth should approach but not exceed K dramatically

Work through 10-15 practice problems and you'll recognize the patterns. Population ecology problems follow predictable structures — once you see the setup, the solution is usually straightforward arithmetic.