Populations — AP Environmental Science APES Study Guide

For: AP Environmental Science candidates sitting AP Environmental Science.

Covers: Exponential vs logistic population growth, demographic transition, carrying capacity and resource limits, and human population dynamics per the AP Environmental Science Course and Exam Description.

You should already know: Algebra 1, basic biology and chemistry.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Environmental Science style for educational use. They are not reproductions of past College Board papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official College Board mark schemes for grading conventions.

1. What Are Populations?

A population is a group of individuals of the same species living in the same defined geographic area at the same time, capable of interbreeding and producing fertile offspring. This is the core unit of study for ecological dynamics, and makes up 10-15% of your AP Environmental Science exam score as outlined in Unit 2 of the official CED. Common synonyms used in exam questions include local species cohort, breeding group, and conspecific community. All calculations in this unit use standard algebraic notation you are already familiar with from your prerequisites.

2. Population growth — exponential vs logistic

All population growth is measured using the per capita growth rate $r$, the average number of offspring produced per individual in the population minus average deaths per individual, per unit time (usually per year for most species tested on the exam). Two core growth models appear on every AP ES exam:

Exponential Growth

Exponential growth occurs when resources are completely unlimited, with no predation, disease, or competition to restrict reproduction. Under these ideal conditions, the population grows at its maximum possible intrinsic growth rate $r_{max}$. The formula for exponential population size at time $t$ is: $$N_t=N_0{e}^{r_{max}t}$$ Where $N_0$ = initial population size, $t$ = time elapsed, $e$ = Euler's constant (~2.718). This growth produces a characteristic J-shaped curve when plotted over time, and is typical of r-selected species (mosquitoes, weeds, bacteria) in the early colonization phase of a new habitat.

Worked Example: A colony of invasive zebra mussels colonizes a new lake with no natural predators. The initial population is 50 mussels, and $r_{max}=0.3$ per month. What is the population size after 6 months, rounded to the nearest whole individual? $$N_6=50\ast e^{(0.3\ast 6)}=50\ast e^{1.8}\approx 50\ast 6.05=302\text{ mussels}$$

Logistic Growth

In natural ecosystems, resources are always limited, so exponential growth cannot continue indefinitely. Logistic growth models incorporate the carrying capacity $K$ (maximum population the habitat can support) to show slowing growth as the population approaches resource limits. The formula for logistic population growth rate is: $$\frac{dN}{dt}=r_{max}N\left(1-\frac{N}{K}\right)$$ When $N$ is very small relative to $K$, $\frac{N}{K}\approx 0$, so growth is nearly exponential. When $N=K$, $\left(1-\frac{N}{K}\right)=0$, so population growth stops entirely. This produces a characteristic S-shaped (sigmoid) growth curve.

Worked Example: A forest has a carrying capacity of 800 red squirrels, current population is 200, and $r_{max}=0.2$ per year. How many squirrels will be added to the population this year? $$\frac{dN}{dt}=0.2\ast 200\ast \left(1-\frac{200}{800}\right)=0.2\ast 200\ast 0.75=30\text{ new squirrels}$$

3. Demographic transition

The Demographic Transition Model (DTM) is a framework that tracks changes in crude birth rate (CBR, births per 1000 people per year) and crude death rate (CDR, deaths per 1000 people per year) as a country develops economically. It is tested frequently on both multiple choice and FRQ sections of the exam. The 4 core DTM stages are:

1. Stage 1 (Pre-industrial): High CBR (no access to contraception, children used as family labor) and high CDR (poor healthcare, limited food access, high infant mortality). Population growth is near 0%, total population is low. No countries remain in Stage 1 today, only small isolated indigenous communities.
2. Stage 2 (Developing/Industrializing): CDR drops sharply due to widespread access to clean water, vaccines, and basic healthcare, while CBR remains high. Population growth is very high (1.5-3% per year). Most low-income countries (Nigeria, Afghanistan) are in Stage 2.
3. Stage 3 (Industrialized): CBR drops sharply due to access to contraception, increased women's education, urbanization, and reduced need for child labor. CDR remains low, so population growth slows but remains positive. Upper-middle income countries (Brazil, India) are in Stage 3.
4. Stage 4 (Post-industrial): CBR falls to or below replacement level (2.1 children per woman), so CBR equals or is lower than CDR. Population growth is 0% or negative, with an aging population structure. Most high-income countries (Japan, Germany, Sweden) are in Stage 4.

Worked Example: Country Y has a CBR of 38 per 1000 and CDR of 13 per 1000. What DTM stage is it in, and what is its annual population growth rate? Growth rate = $\frac{38-13}{10}=2.5\%$ per year. This high positive growth, paired with high CBR and low CDR, confirms Stage 2.

4. Carrying capacity and resource limits

Carrying capacity $K$ is defined as the maximum number of individuals of a species that an ecosystem can support indefinitely, given available food, water, shelter, and habitat. Examiners frequently test the following key rules about $K$:

1. $K$ is not fixed: It increases or decreases if resource availability changes. For example, a drought that kills 50% of grass in a grassland will reduce $K$ for grazing zebras by ~50%, while reforestation of abandoned farmland will increase $K$ for native songbirds.
2. Overshoot and dieback: If a population temporarily exceeds $K$ (overshoot), resources will be depleted faster than they can regenerate, leading to a sharp population decline (dieback) until the population falls well below the new, lower $K$.
3. Limiting factors: The factors that determine $K$ are split into two categories, a common exam question topic:

• Density-dependent: Effects increase as population size increases (food scarcity, disease transmission, predation, intraspecific competition)
• Density-independent: Effects are identical regardless of population size (wildfires, hurricanes, volcanic eruptions, toxic pollution)

Worked Example: A small wetland has a carrying capacity of 2000 frogs. A flood washes 2600 frogs into the wetland, leading to a 40% dieback from food scarcity. What is the new frog population after dieback? New population = $2600\ast (1-0.4)=1560$ frogs, which is below the original $K$, so slow growth will resume in the next breeding season.

5. Human population dynamics

As of 2026, the global human population is ~8.1 billion, making human population dynamics a core applied topic for AP Environmental Science. Key metrics and rules for this section include:

1. Historical growth trend: Pre-1800, global population remained under 1 billion, with near-zero growth matching Stage 1 DTM. The Industrial Revolution (1800-1950) shifted most countries to Stage 2, pushing population to 2.5 billion by 1950. The Green Revolution and widespread global access to antibiotics and vaccines after 1950 led to exponential growth, with the population reaching 8 billion by 2022. Global growth rate is now slowing to ~1% per year as more countries shift to Stage 3 and 4.
2. Rule of 70: A simple formula to estimate doubling time (years required for a population to double at constant growth rate): $$\text{Doubling Time}=\frac{70}{\text{Annual Growth Rate (\%)}}$$
3. Age structure diagrams: These plot population share by age group, and are used to predict future growth:

• Pyramid shape (wide base, large share of population <15 years old): High future growth, Stage 2 country
• Column shape (even age distribution): Stable slow growth, Stage 3 country
• Inverted pyramid (narrow base, large share of population >65 years old): Negative future growth, Stage 4 country

Worked Example: The global human population growth rate in 2026 is 0.9%. If this rate remains constant, when will the global population reach 16.2 billion? Doubling time = $70/0.9\approx 78$ years, so the population will double to ~16.2 billion by 2026 + 78 = 2104.

6. Common Pitfalls (and how to avoid them)

• Wrong move: Using linear growth instead of exponential/logistic formulas for population calculations. Why? Students confuse fixed absolute growth with per capita growth rates. Correct move: Always confirm if the question specifies unlimited resources (exponential) or limited resources (logistic), and use the correct per capita growth formula.
• Wrong move: Assuming carrying capacity $K$ is a fixed permanent value. Why? Textbook examples use fixed $K$ for basic calculations, but real-world $K$ changes with resource availability. Correct move: If a question mentions a change to the ecosystem (drought, deforestation, pollution), adjust $K$ first before calculating growth rates.
• Wrong move: Forgetting to divide CBR/CDR by 10 when calculating percentage growth rate. Why? CBR/CDR are reported per 1000 people, not per 100. Correct move: Growth rate % = $(CBR-CDR)/10$, double check that a CBR of 20 per 1000 equals 2% annual growth to avoid unit errors.
• Wrong move: Stating that all countries must progress through DTM stages in order at the same speed. Why? The DTM is a model, not a universal rule. Correct move: Note that external factors (war, climate disasters, government family planning policy) can shift countries between stages faster or skip stages entirely.
• Wrong move: Classifying drought as a density-dependent limiting factor. Why? Students assume all resource-related factors are density-dependent. Correct move: Ask if the factor's effect increases with population size. Drought reduces water availability for all individuals regardless of how many are present, so it is density-independent.

7. Practice Questions (AP Environmental Science Style)

Question 1

A population of r-selected ragweed colonizes an abandoned construction site with unlimited sunlight, water, and soil nutrients. The initial population is 30 ragweed plants, and the intrinsic growth rate $r_{max}$ is 0.5 per month. a) What type of growth will the ragweed exhibit in the first 4 months? Justify your answer. b) Calculate the population size after 4 months, rounded to the nearest whole plant. c) After 6 months, the construction site is sprayed with herbicide that kills 80% of the ragweed population regardless of its size. Is this a density-dependent or density-independent limiting factor? Justify your answer.

Solution

a) Exponential growth. The population has unlimited resources and no limiting factors during early colonization, so it will grow at its maximum intrinsic rate. b) Use the exponential growth formula: $N_t=N_0{e}^{rt}$ $N_4=30\ast e^{(0.5\ast 4)}=30\ast e^2\approx 30\ast 7.389=222$ plants c) Density-independent. The herbicide kills a fixed percentage of the population regardless of how large the population is, so its effect does not increase with higher population density.

Question 2

Country M has a crude birth rate of 12 per 1000 and crude death rate of 11 per 1000. Country N has a crude birth rate of 42 per 1000 and crude death rate of 17 per 1000. a) Calculate the annual population growth rate for both countries, as a percentage. b) Identify the DTM stage for each country. c) Calculate the doubling time for Country N, if its growth rate remains constant.

Solution

a) Country M growth rate = $(12-11)/10=0.1\%$ per year. Country N growth rate = $(42-17)/10=2.5\%$ per year. b) Country M is Stage 4 (low birth rate, low death rate, near-zero growth). Country N is Stage 2 (high birth rate, low death rate, high positive growth). c) Doubling time for Country N = $70/2.5=28$ years.

Question 3

A mountain ecosystem has a carrying capacity of 900 bighorn sheep. The current sheep population is 300, and the maximum per capita growth rate $r_{max}$ is 0.2 per year. a) Calculate the annual population growth rate (number of sheep added) for the current year using the logistic growth formula. b) A wildfire destroys 30% of the sheep's grazing habitat. What is the new carrying capacity for bighorn sheep? c) Assuming the sheep population remains at 300 after the fire, calculate the new annual growth rate.

Solution

a) Logistic growth formula: $\frac{dN}{dt}=r_{max}N\left(1-\frac{N}{K}\right)$ $\frac{dN}{dt}=0.2\ast 300\ast (1-300/900)=0.2\ast 300\ast 0.667\approx 40$ sheep added. b) New K = $900\ast (1-0.3)=630$ sheep. c) New growth rate = $0.2\ast 300\ast (1-300/630)=0.2\ast 300\ast 0.524\approx 31$ sheep added.

8. Quick Reference Cheatsheet

9. What's Next

The population concepts you mastered in this guide are foundational for the rest of the AP Environmental Science syllabus. They connect directly to Unit 3 (Ecosystems) where you will analyze how population dynamics shape food webs and trophic cascades, Unit 5 (Land and Water Use) where you will evaluate how human population growth drives deforestation, agricultural expansion, and freshwater scarcity, and Unit 9 (Global Change) where you will calculate the impact of population growth on greenhouse gas emissions and climate change. Carrying capacity and demographic transition are also two of the most frequently tested topics on FRQs, so practicing these calculations will directly boost your exam score.

If you have questions about any of the formulas, DTM stages, or practice questions in this guide, you can ask Ollie, our AI tutor, for personalized explanations, additional practice problems, or step-by-step walkthroughs of tricky exam questions. You can also find more study guides for other AP Environmental Science topics on the homepage to build your full exam prep plan before test day.