AP Environmental Science

## Learning Outcomes

- Understand why populations cannot grow indefinitely in natural systems
- Distinguish between exponential growth and logistic growth
- Explain how **carrying capacity (K)** is set by resource availability
- Identify the difference between density-dependent and density-independent limiting factors
- Interpret population growth graphs, including overshoot and dieback
- Apply the logistic growth model qualitatively to exam questions
- Use examples to explain how environmental conditions change birth rates, death rates, immigration, and emigration

## AP Environmental Science Syllabus

For the AP Environmental Science, you are expected to understand environmental limits on growth, with a focus on the following syllabus points:

- Know that **carrying capacity (K)** is the maximum population size an environment can sustainably support over time
- Identify how **food, water, space, shelter, light, nutrients, and oxygen** can act as limiting resources
- Explain the difference between **exponential growth** and **logistic growth**
- Understand that **density-dependent** limiting factors become stronger as population density rises
- Understand that **density-independent** limiting factors affect populations regardless of density
- Explain how populations can **overshoot** carrying capacity and then experience **dieback**
- Interpret the logistic equation qualitatively, especially how growth slows as population size approaches **K**
- Recognise that carrying capacity is **dynamic**, not a fixed number forever

## Test Your Knowledge

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, look more closely at that area during your revision._

1. What is carrying capacity, and why is it not always a fixed value?
2. How does logistic growth differ from exponential growth?
3. Name two density-dependent limiting factors and two density-independent limiting factors.
4. What happens when a population overshoots its carrying capacity?
5. Why might a real population fluctuate around **K** instead of staying exactly at it?

## Introduction

Populations increase when births and immigration add individuals faster than deaths and emigration remove them. But in nature, that increase does not continue without a limit. Every habitat has finite supplies of energy and matter. Once organisms begin to compete strongly for those resources, population growth slows, levels off, or even crashes.

This subtopic matters because AP Environmental Science often tests not just vocabulary, but cause-and-effect reasoning. You need to explain how limited food, water, space, oxygen, or nutrients can change population size, and you must be able to connect those limits to graph shapes such as the **J-curve** and **S-curve**.

> **Key Term: carrying capacity (K)**
> The maximum population size an environment can sustainably support over time, given available resources and environmental conditions.

### Why environmental limits exist

A population's theoretical maximum reproductive ability is its **biotic potential**. If resources were unlimited, populations could keep growing very rapidly. In the real world, though, no environment provides endless food, water, space, light, shelter, or nutrients. This combined resistance to unlimited growth is often called **environmental resistance**.

> **Key Term: biotic potential**
> The maximum reproductive capacity of a population under ideal conditions with unlimited resources.
>
> **Key Term: environmental resistance**
> The combined effect of limiting factors that prevent a population from reaching its biotic potential.

The key idea is simple: resources are finite, but organisms reproduce. That mismatch creates limits.

Think through a deer population in a forest. At first, if the population is small, each deer may find enough food, water, and shelter. Birth rates stay high and death rates stay relatively low. As the population grows, more deer browse the same plants. Winter food becomes scarcer. More individuals are weakened by poor nutrition, making them more likely to die of starvation or disease. Birth rates may also fall because undernourished adults produce fewer healthy offspring. The same forest now supports population growth less effectively than before.

This is why carrying capacity is based on sustainability, not on the biggest population ever observed. A brief spike above the limit can happen, but it usually cannot last.

Another key point is that the limiting resource is not always the total amount of all resources combined. Under **Liebig's Law of the Minimum**, growth is controlled by the scarcest essential resource. A pond could have enough sunlight and nutrients, but if dissolved oxygen becomes too low, the aquatic population is still limited.

> **Key Term: limiting factor**
> A biotic or abiotic condition that restricts population growth, size, or distribution.

Examples of limiting resources and conditions include:

- food supply
- fresh water
- living space
- shelter
- sunlight for producers
- soil nutrients
- dissolved oxygen in water
- suitable temperature range

A common exam mistake is to say "the population got too large" as if size itself is the cause. Population size matters because it changes access to resources. The mechanism is competition, disease spread, waste buildup, or some other ecological process.

> **Test Tip:** In FRQs, do not stop at "resources became limited." Name the resource and explain the consequence: for example, "lower food availability reduced reproduction and increased mortality."

### Exponential growth, logistic growth, and what graphs show

When resources are abundant, population growth can be **exponential**. This produces a **J-shaped curve**. The bigger the population gets, the more individuals are reproducing, so growth accelerates.

> **Key Term: exponential growth**
> Population growth under abundant or effectively unlimited resources, producing a J-shaped curve.

A common model for exponential growth is:

$dN/dt = rN$

You do not usually need advanced math for APES here. Focus on what the symbols mean:

- **N** = population size
- **r** = intrinsic growth rate
- **dN/dt** = change in population over time

The model says that growth depends on how many individuals there already are. If **r** stays positive and resources remain abundant, the population keeps increasing faster and faster.

This is a useful short-term model, especially for:

- bacteria in fresh culture media
- invasive species entering a favorable habitat
- algal blooms at the beginning of nutrient enrichment
- populations recovering after a crash when resources are temporarily abundant

However, exponential growth is usually temporary in nature. Eventually, resources become limiting.

That is where **logistic growth** comes in. Logistic growth begins rapidly, but then slows as the population approaches carrying capacity. The graph becomes **S-shaped**, also called sigmoid.

> **Key Term: logistic growth**
> Population growth that starts rapidly but slows and levels off as environmental limits are reached, producing an S-shaped curve.

The logistic model is:

$dN/dt = rN((K-N)/K)$

Again, APES expects qualitative understanding more than algebraic manipulation. The important part is the factor:

$(K-N)/K$

This represents the fraction of carrying capacity still available.

- If **N** is much smaller than **K**, then much capacity remains, so growth is close to exponential.
- If **N** gets close to **K**, the factor shrinks toward zero, so growth slows.
- If **N = K**, net growth is about zero.
- If **N > K**, growth becomes negative in the model.

That last point matters. If a population rises above carrying capacity, the model predicts decline because the environment cannot support that number sustainably.

A useful way to read the logistic curve is in stages:

1. **Lag phase**: population is small, so total increase is still modest
2. **Rapid growth phase**: resources are plentiful, births exceed deaths strongly
3. **Slowing phase**: competition intensifies and growth rate falls
4. **Plateau around K**: births and deaths are roughly balanced

Students often confuse the plateau with "nothing is happening." In fact, many births and deaths may still occur. The population size is stable only because gains and losses are balancing overall.

> **Test Tip:** If a graph shows a smooth J-curve, think unlimited resources. If it shows a leveling S-curve, think carrying capacity and density-dependent limitation.

### Density-dependent and density-independent limits

Not all limiting factors work in the same way. APES expects you to separate them into two groups.

**Density-dependent limiting factors** become stronger as population density increases. The more crowded the population, the greater their effect.

> **Key Term: density-dependent limiting factor**
> A limiting factor whose impact increases as population density rises.

Common density-dependent examples include:

- competition for food
- competition for space or territory
- disease transmission
- parasitism
- predation
- waste buildup
- overcrowding stress

These all make ecological sense. Disease spreads more easily in crowded populations because individuals contact one another more often. Predators may gather where prey are dense. Territorial conflict increases when too many individuals occupy a small area.

For example, a crowded fish population in a pond may experience lower dissolved oxygen and higher waste concentration. Those conditions weaken individuals, making disease outbreaks more severe. The resulting increase in mortality pushes the population back down.

By contrast, **density-independent limiting factors** affect populations regardless of how crowded they are.

> **Key Term: density-independent limiting factor**
> A limiting factor that affects population size regardless of population density.

Common density-independent examples include:

- drought
- flood
- wildfire
- hurricane
- extreme temperatures
- volcanic eruption
- pollution event
- habitat destruction

A wildfire can reduce a population whether it was dense or sparse. A severe freeze may kill exposed organisms across an area regardless of local crowding.

This does not mean the impact is identical for every species. It means the main cause is not density. A drought may hurt one species more than another, but it is still considered density-independent because it is not driven by crowding.

A classic APES contrast is:

- influenza spreading rapidly in a crowded hospital = density-dependent
- drought reducing rainforest area = density-independent

Sometimes human activity can shift how a factor operates. Forest destruction caused by natural drought is density-independent. Forest destruction caused by rising human population pressure for ranching or logging links more strongly to density-dependent human factors.

### Overshoot, dieback, and fluctuating populations

Real populations often do not rise neatly to carrying capacity and stop. Instead, they may **overshoot** the limit.

> **Key Term: overshoot**
> A situation in which a population temporarily exceeds carrying capacity.

Overshoot happens because population responses often lag behind environmental change. Organisms do not stop reproducing instantly when resources begin to run short. Stored resources, delayed mortality, and existing young individuals can all keep numbers rising briefly.

If the overshoot depletes resources badly enough, the population may then experience **dieback**, a rapid decline.

> **Key Term: dieback**
> A sharp population decline after overshoot, usually caused by resource depletion and rising mortality.

A deer population can illustrate this well:

1. predators are removed or reduced
2. deer survive and reproduce successfully
3. deer numbers rise rapidly
4. winter browse becomes depleted
5. malnutrition spreads
6. disease and starvation increase
7. the population crashes

If the vegetation recovers, the deer population may later recover too. If the habitat is badly damaged, the new carrying capacity may be lower than before.

That is why some overshoot graphs show different outcomes:

- mild overshoot followed by recovery and oscillation around **K**
- severe overshoot followed by stabilization around a lower **K**
- catastrophic overshoot followed by local extinction

Bacteria in a closed flask provide another good example. They may grow almost exponentially at first because nutrients are abundant. Later, nutrients are depleted and wastes accumulate. The population then levels off or crashes.

Natural populations also often **oscillate around K** instead of sitting exactly at it. Reasons include:

- seasonal variation in resources
- predator-prey interactions
- weather changes
- disease cycles
- time lags in reproduction and mortality
- disturbance events

So logistic growth is best treated as a model, not a perfect picture. The S-curve helps explain the pattern, but actual natural systemss are messier.

### Applying limits to APES questions and FRQs

On the exam, you may be asked to classify factors, explain graph shapes, or connect resource availability to population outcomes.

A strong response usually follows this structure:

1. identify the limiting factor
2. state whether it is density-dependent or density-independent
3. explain how it changes birth rate, death rate, immigration, or emigration
4. connect it to carrying capacity or graph behavior

For example, if a question asks why a paramecium population levels off in culture, a strong answer would be:

- food or dissolved oxygen became limiting
- as the population increased, per capita resource availability decreased
- growth slowed because birth rate fell and/or death rate rose
- the population approached a new carrying capacity

For a wildfire question:

- wildfire is density-independent
- it reduces population regardless of prior density
- habitat loss increases mortality and may reduce shelter or food
- carrying capacity after the fire may be lower until recovery occurs

If asked why logistic growth is more realistic than exponential growth, explain that exponential growth assumes unlimited resources, while logistic growth includes environmental limits and carrying capacity.

If asked why carrying capacity changes, mention shifting conditions such as drought, habitat degradation, nutrient loss, pollution, or changing oxygen levels.

A frequent calculation-linked concept is the **Rule of 70**, used to estimate doubling time:

doubling time ≈ 70 / growth rate (%)

A population growing at 5% per year doubles in about 14 years. This is useful because it shows how quickly exponential growth can become unsustainable if resources do not expand too.

## Key Point Checklist

_This article has covered the following key knowledge points:_

- Carrying capacity **(K)** is the maximum sustainable population size an environment can support
- Carrying capacity depends on resource availability and environmental conditions
- Carrying capacity is dynamic and can rise or fall over time
- Exponential growth occurs under abundant resources and produces a **J-shaped** curve
- Logistic growth includes environmental limits and produces an **S-shaped** curve
- In the logistic model, growth slows as population size **N** approaches **K**
- Density-dependent factors include competition, disease, predation, parasitism, and waste buildup
- Density-independent factors include drought, floods, fires, storms, pollution, and habitat destruction
- Overshoot occurs when a population temporarily exceeds carrying capacity
- Dieback is the rapid decline that can follow overshoot
- Real populations often fluctuate around **K** rather than staying exactly at it
- Strong APES answers explain how limiting factors affect birth rate, death rate, immigration, or emigration

## Key Terms and Concepts

- carrying capacity (K)
- biotic potential
- environmental resistance
- limiting factor
- exponential growth
- logistic growth
- density-dependent limiting factor
- density-independent limiting factor
- overshoot
- dieback


Population Growth and Resource Availability - Environmental limits on growth