AP-ENVSCI-3.4-3.5

U3.2 Carrying Capacity and Population Growth

Master AP Environmental Science carrying capacity (K), exponential vs logistic growth, density-dependent and independent limiting factors, and growth-rate math.

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What this lesson covers

Every population — deer, algae, or humans — pushes against the limits of its environment. In this lesson you'll learn how ecologists describe those limits with carrying capacity (KK) and two contrasting growth models. You'll also learn to tell apart the factors that slow a population as it crowds versus factors that strike regardless of density.

These ideas show up constantly on the AP exam, both in multiple-choice questions about graph shapes and in FRQs that ask you to calculate a growth rate or explain why a population crashed. By the end, you'll be able to define KK, read J-curves and S-curves, sort limiting factors, and plug numbers into the population growth-rate formula with confidence.

Carrying Capacity (K)

Carrying capacity (KK) is the maximum population size that a particular environment can support indefinitely, given the available resources such as food, water, shelter, and space. It is not a fixed universal number — it depends on the species and the specific habitat, and it can change if resources change. A drought, a wildfire, or the introduction of a competitor can lower KK; a wetter year or a new food source can raise it.

A key exam idea is that populations do not simply stop growing exactly at KK. Instead they tend to oscillate around it. When a population temporarily exceeds KK, resources become scarce, deaths outpace births, and the population declines back toward KK — sometimes sharply. This is called overshoot followed by a dieback (or crash).

Students often confuse KK with the highest number a population ever reaches. On a logistic growth curve, KK is the value the curve levels off at (the horizontal asymptote), not a momentary peak. Another common misconception: KK is permanent. Remember that environmental degradation — soil erosion, pollution, resource depletion — can permanently reduce the carrying capacity of an ecosystem, meaning it supports fewer organisms afterward.

Exponential vs Logistic Growth

Two models describe how populations grow. Exponential growth occurs when resources are unlimited: the population grows by a constant percentage each time interval, producing a J-shaped curve that gets steeper and steeper. This happens with r-selected species colonizing a new area, bacteria in fresh medium, or an invasive species with no predators. The formula is dNdt=rN\frac{dN}{dt} = rN, where rr is the per-capita growth rate and NN is population size.

Logistic growth occurs when resources are limited. Growth starts fast but slows as the population approaches KK, producing an S-shaped (sigmoidal) curve. The formula is dNdt=rN(K−N)K\frac{dN}{dt} = rN\frac{(K-N)}{K}. The term (K−N)K\frac{(K-N)}{K} shrinks toward zero as NN nears KK, so growth stalls.
FeatureExponentialLogistic
Curve shapeJ-shapedS-shaped
ResourcesUnlimitedLimited
Effect of KNoneGrowth slows near K
Typical speciesr-selectedK-selected
A subtle point: in logistic growth the population grows fastest at the inflection point, roughly N=K/2N = K/2, where there are enough individuals reproducing but resources are not yet scarce. Many exam questions ask you to identify this point of maximum growth rate.

Density-Dependent vs Density-Independent Factors

Limiting factors are conditions that restrict population growth. The AP exam wants you to sort them into two categories.

Density-dependent factors intensify as population density increases. When individuals are crowded, competition for food and space rises, predators concentrate where prey is dense, and disease and parasites spread more easily because hosts contact each other more often. Territorial behavior and buildup of waste also fit here. These factors are the mechanism behind logistic growth — they push a population back toward KK.

Density-independent factors affect a population regardless of its density. Classic examples are natural disasters and weather events: floods, fires, droughts, extreme cold, hurricanes, and volcanic eruptions. A flood drowns the same fraction of a sparse population as a dense one; the crowding does not matter.
TypeDepends on crowding?Examples
Density-dependentYesCompetition, predation, disease, waste
Density-independentNoFloods, fire, drought, cold snaps
A common trap: pollution and habitat destruction from human activity are often density-independent in the way the exam presents them, while resource competition is always density-dependent. When answering, ask yourself: would this factor hit harder if there were more individuals packed together? If yes, it is density-dependent.

Computing Population Growth Rate

The AP exam frequently asks for a population growth rate, and you should know two calculations.

The simplest uses births, deaths, immigration, and emigration. The growth rate as a change in number is (births+immigration)−(deaths+emigration)(\text{births} + \text{immigration}) - (\text{deaths} + \text{emigration}). To express it as a percentage per year, user=(B−D)N×100r = \frac{(B - D)}{N} \times 100where BB is births, DD is deaths, and NN is the starting population (immigration and emigration can be added when given).

A handy shortcut is the rule of 70: the doubling time in years for a population growing at a steady percentage rate is approximately 70r\frac{70}{r}, where rr is the annual percent growth rate. A population growing at 2% per year doubles in about 702=35\frac{70}{2} = 35 years.

Common mistakes on the exam: forgetting to divide by the original population when a percentage is requested, mixing up per-capita rate rr with total change dNdt\frac{dN}{dt}, and using the wrong starting NN. Always write your formula, plug in units, and check whether the question asks for a raw number of individuals or a percentage. Show your work fully on FRQs — the AP rubric awards points for correct setup even if arithmetic slips.

Key terms

Carrying capacity (K).
The maximum population size an environment can sustain indefinitely given its available resources.
Exponential growth.
Growth at a constant per-capita rate under unlimited resources, producing a J-shaped curve; modeled by dNdt=rN\frac{dN}{dt}=rN.
Logistic growth.
Growth that slows as population nears carrying capacity, producing an S-shaped curve; modeled by dNdt=rN(K−N)K\frac{dN}{dt}=rN\frac{(K-N)}{K}.
Overshoot and dieback.
When a population temporarily exceeds K, then crashes as resources run short and deaths exceed births.
Density-dependent factor.
A limiting factor whose effect strengthens as population density rises, such as competition, predation, or disease.
Density-independent factor.
A limiting factor that affects a population regardless of its density, such as floods, fire, or drought.
Per-capita growth rate (r).
The rate of population change per individual, calculated from birth and death rates.
Rule of 70.
An estimate of doubling time in years equal to 70 divided by the percent annual growth rate.

Worked example

A population of 5,000 rabbits experiences 900 births and 400 deaths in one year. There is no immigration or emigration. Calculate the annual percent growth rate, and estimate the doubling time.
Start with the growth-rate formula: r=(B−D)N×100r = \frac{(B - D)}{N} \times 100.

Identify the values: births B=900B = 900, deaths D=400D = 400, and starting population N=5000N = 5000.

Subtract deaths from births: 900−400=500900 - 400 = 500 net new rabbits.

Divide by the starting population and multiply by 100: r=5005000×100=10%r = \frac{500}{5000} \times 100 = 10\% per year.

Now estimate doubling time with the rule of 70: doubling time =70r=7010=7= \frac{70}{r} = \frac{70}{10} = 7 years.

So the rabbit population grows at 10% per year and would double in about 7 years if the rate stayed constant. Note that this exponential estimate holds only while resources are plentiful — as the population approaches carrying capacity, density-dependent factors would slow this growth.

Practice questions

Which of the following best describes a density-dependent limiting factor?
  1. A hurricane that floods a coastal marsh
  2. An unusually cold winter that freezes a pond
  3. The spread of a contagious disease through a crowded herd
  4. A volcanic eruption that buries a forest

Answer: The spread of a contagious disease through a crowded herd

Disease spreads more easily when individuals are packed closely, so its impact grows with population density — the definition of a density-dependent factor. Hurricanes, cold snaps, and volcanic eruptions strike regardless of how crowded the population is, making them density-independent.
On a logistic growth curve, at approximately what population size is the growth rate (number of individuals added per unit time) the greatest?
  1. At N near 0
  2. At N = K/2
  3. At N = K
  4. At N greater than K

Answer: At N = K/2

In the logistic model, growth is fastest at the inflection point around half the carrying capacity. There are enough reproducing individuals to add many offspring, but resources are not yet limiting. Near N=0N = 0 there are too few reproducers, and near N=KN = K the (K−N)K\frac{(K-N)}{K} term drives growth toward zero.
A population of 20,000 deer has a birth rate of 1,200 and a death rate of 800 in one year, with 100 individuals immigrating and 300 emigrating. Calculate the population growth rate as a percentage and explain what the result means.

Answer: The growth rate is 1% per year.

Net change equals (B+immigration)−(D+emigration)=(1200+100)−(800+300)=1300−1100=200(B + \text{immigration}) - (D + \text{emigration}) = (1200 + 100) - (800 + 300) = 1300 - 1100 = 200. Divide by the starting population and multiply by 100: 20020000×100=1%\frac{200}{20000} \times 100 = 1\%. A 1% growth rate means the deer population is increasing slowly; using the rule of 70, it would take about 70 years to double if this rate held steady.

FAQ

What is the difference between exponential and logistic growth?
Exponential growth assumes unlimited resources and produces a J-shaped curve that keeps getting steeper. Logistic growth accounts for limited resources, so growth slows as the population nears carrying capacity, producing an S-shaped curve that levels off at KK.
How do I know if a limiting factor is density-dependent or density-independent?
Ask whether the factor hits harder when individuals are crowded. Competition, predation, disease, and waste buildup worsen with crowding, so they are density-dependent. Weather and natural disasters like floods, fires, and droughts affect a population regardless of its density, so they are density-independent.
Does a population stop growing exactly at carrying capacity?
No. Populations tend to oscillate around KK. They can overshoot it, then crash (dieback) when resources run short, then recover. Carrying capacity is the level a population fluctuates around, not a hard ceiling it hits and freezes at.
How do I calculate a population growth rate on the AP exam?
Use r=(B−D)N×100r = \frac{(B - D)}{N} \times 100 for a percentage, adding immigration to births and emigration to deaths when given. Always divide by the starting population NN when a percentage is requested, and use the rule of 70 (70r\frac{70}{r}) to estimate doubling time.

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