AP Environmental Science

## Learning Outcomes

- Understand why populations cannot grow indefinitely in real natural environmentss
- Explain how a finite resource base leads to carrying capacity
- Distinguish between exponential growth and logistic growth
- Identify how limiting factors change birth rate, death rate, and fecundity
- Explain overshoot and dieback using ecological reasoning and examples
- Apply the logistic growth model conceptually, including the role of **r**, **N**, and **K**
- Distinguish density-dependent from density-independent limiting factors

## AP Environmental Science Syllabus

For the AP Environmental Science, you are expected to understand how a finite resource base limits population growth, with a focus on the following syllabus points:

- Know that resources such as food, water, space, nutrients, and waste-absorption capacity are limited
- Define **carrying capacity (K)** as the maximum population size an environment can support sustainably
- Explain why exponential growth is temporary and logistic growth is more realistic in natural systems
- Identify J-shaped and S-shaped population growth curves and interpret what they show
- Understand how resource limitation changes mortality, fecundity, and population growth rate
- Explain overshoot and dieback as outcomes when populations exceed carrying capacity
- Distinguish between density-dependent and density-independent limiting factors
- Recognise that carrying capacity can change when environmental conditions or resource supply change

## 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. Why can a population not keep growing exponentially forever in nature?
2. What is the difference between a J-shaped growth curve and an S-shaped growth curve?
3. How does a finite resource base lead to carrying capacity?
4. What is meant by overshoot, and why can it lead to dieback?
5. Name two density-dependent limiting factors and two density-independent limiting factors.

## Introduction

A major idea in population ecology is that every population depends on a limited supply of resources. Even when a species reproduces quickly, it still needs food, water, habitat space, nutrients, and suitable environmental conditions. Because those resources are finite, population growth must eventually slow.

For AP Environmental Science, this topic is tested through graph interpretation, cause-and-effect reasoning, and application of terms such as **carrying capacity**, **logistic growth**, and **limiting factor**. You need to explain not just what these terms mean, but how they connect in a system.

> **Key Term: finite resource base**
> The idea that the total resource supply available to a population is limited over time, so populations cannot grow indefinitely.

## Introduction to finite limits and carrying capacity

### Why a finite resource base matters

A population can only survive if enough resources are available for its members. At low population size, resources may seem abundant. There may be plenty of food per individual, low competition for nesting sites, and little buildup of waste. Under those conditions, births can exceed deaths by a wide margin.

The key problem is that population growth changes the resource balance. If the number of organisms doubles, the amount of food, water, and space available to each individual usually falls unless resource supply also increases. In real environmentss, resource supply may fluctuate a little, but it does not rise without limit.

This is why ecologists do not treat population growth as a purely biological issue. Reproduction matters, but environmental support matters just as much. A species may have high **biotic potential**, but that potential is checked by environmental resistance.

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

Carrying capacity is often misunderstood. It is not the same as the current population size. If a forest can support 500 deer but currently has 320, then 320 is the present population and 500 is the carrying capacity. The population can still grow, but not forever.

Carrying capacity also is not fixed forever. If drought reduces plant growth, then food supply falls and carrying capacity may decrease. If habitat restoration improves water availability and plant productivity, carrying capacity may rise.

A useful chain to remember is:

- finite resource base
- limiting factors
- carrying capacity
- logistic growth

Each part causes the next. The environment has limited resources. Those limits act as limiting factors. Limiting factors set carrying capacity. Carrying capacity shapes the way population growth changes over time.

> **Key Term: limiting factor**
> Any environmental resource or condition that restricts the growth, abundance, or distribution of a population.

Liebig's Law of the Minimum helps explain why this matters. Population growth is controlled not by the total amount of all resources combined, but by whichever necessary resource is in shortest supply. A pond may have enough sunlight and nutrients, but if dissolved oxygen becomes too low, that factor controls survival.

Examples of limiting factors include:

- food supply
- water availability
- shelter
- habitat space
- soil nutrients
- dissolved oxygen in aquatic systems
- waste accumulation
  > **Test Tip:** If a multiple-choice question asks what most directly limits population growth, look for the scarcest necessary resource, not the resource that seems most obvious overall.

### How resource scarcity changes birth and death rates

As resources become scarce, populations do not just "stop." Instead, demographic processes change.

First, mortality may rise. Organisms weakened by poor nutrition may die from starvation, disease, or predation. Second, fecundity may fall. Fecundity means reproductive output, such as the number of offspring produced. Organisms under stress may produce fewer offspring or reproduce less often.

This distinction matters. A slowing population can result from:

- increased death rate
- decreased birth rate or fecundity
- both at the same time

That is why population growth curves are really summaries of changing survival and reproduction. A flatter curve is not just a shape on a graph. It shows that environmental limits are changing the balance between births and deaths.

## Growth models: from exponential to logistic

### Exponential growth and why it is temporary

When resources are effectively unlimited, populations can grow exponentially. Exponential growth means the population increases at a constant per-capita rate. The more individuals there are, the more births occur, so the curve gets steeper over time.

This produces a J-shaped graph.

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

Classic APES examples include bacteria in a petri dish with abundant food or a newly introduced species in an environment with little competition or predation. At first, such populations can grow very fast.

Conceptually, exponential growth can be modeled as:

$$
\frac{dN}{dt} = rN
$$

Where:

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

The key idea is that growth depends on the number of organisms already present. If **r** stays constant and nothing limits resources, the population grows faster and faster.

But this model is unrealistic for long periods in nature. Food runs out. Waste accumulates. Space becomes crowded. Disease spreads more easily. Predators may concentrate where prey are abundant. Exponential growth is therefore usually short-lived in real natural systemss.

A common AP mistake is assuming exponential growth is the normal long-term pattern. It is not. It is the early phase when a population is still well below environmental limits.

### Logistic growth and the effect of carrying capacity

Logistic growth is the more realistic pattern for most natural populations. It begins with rapid growth when the population is small, but growth slows as the population approaches carrying capacity.

This produces an S-shaped curve.

> **Key Term: logistic growth**
> Population growth that is rapid at low population size but slows as the population approaches carrying capacity, producing an S-shaped curve.

The conceptual logistic model is:

$$
\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)
$$

You are not usually asked in APES to derive this mathematically, but you should understand what it means.

Break the equation into parts:

- **rN** is the exponential part
- $\left(1 - \frac{N}{K}\right)$ is the limiting part

If **N** is much smaller than **K**, then $\frac{N}{K}$ is close to 0, so the bracketed term is close to 1. Growth is then close to exponential.

If **N** gets close to **K**, then $\frac{N}{K}$ gets close to 1, so the bracketed term gets close to 0. Growth slows toward zero.

If **N = K**, then:

$$
1 - \frac{N}{K} = 1 - 1 = 0
$$

So net population growth is about zero.

This does not mean no births occur. It means births and deaths are roughly balanced.

Worked example:

Suppose:

- $r = 0.4$
- $N = 100$
- $K = 500$

Step 1: Calculate $\frac{N}{K}$

$$
\frac{100}{500} = 0.2
$$

Step 2: Calculate $1 - \frac{N}{K}$

$$
1 - 0.2 = 0.8
$$

Step 3: Calculate $rN$

$$
0.4 \times 100 = 40
$$

Step 4: Multiply both parts

$$
\frac{dN}{dt} = 40 \times 0.8 = 32
$$

So the population is growing by 32 individuals per unit time in this simplified model.

Now compare with $N = 450$, keeping the same $r$ and $K$:

Step 1:

$$
\frac{450}{500} = 0.9
$$

Step 2:

$$
1 - 0.9 = 0.1
$$

Step 3:

$$
0.4 \times 450 = 180
$$

Step 4:

$$
180 \times 0.1 = 18
$$

Even though the population is larger, the growth rate has slowed because it is much closer to carrying capacity.

> **Test Tip:** In FRQs, explain logistic growth as a process: abundant resources at low population size allow rapid growth, then competition and other limiting factors intensify, causing growth to slow near **K**.

Common student errors:

- confusing **K** with current population size
- assuming growth stops suddenly at carrying capacity
- forgetting that logistic growth can still be rapid early on

## Overshoot, dieback, and changing carrying capacity

### Why populations can exceed carrying capacity

Real populations often do not level off neatly at carrying capacity. There can be a time lag between resource depletion and the population's response. Organisms continue reproducing for a while even though the environment is already under strain.

This leads to overshoot.

> **Key Term: overshoot**
> A condition in which a population temporarily exceeds the carrying capacity of its environment.

Once overshoot happens, resources may be consumed faster than they can regenerate. Food shortages intensify, disease spreads, conflict may increase, and habitat quality may decline. When population size then falls sharply, the decline is called dieback.

> **Key Term: dieback**
> A rapid population decline that follows overshoot, usually because resources have been depleted or habitat quality has been reduced.

A helpful real-world case is the reindeer on St. Matthew Island, Alaska. Reindeer were introduced there in **1944**. With abundant lichen and no major predators, the population rose from about **29** to around **6,000** by **1963**. But lichen grows very slowly. The herd overshot the island's resource base. After resource depletion and a severe winter, the population crashed to about **42** by **1966** and then disappeared.

This example matters because it shows that overshoot can look like success at first. A rapidly growing population may seem healthy, but if it is consuming resources faster than those resources recover, the growth is unstable.

Three broad outcomes are possible after overshoot:

- mild depletion followed by recovery and oscillation around **K**
- severe depletion with a permanently reduced carrying capacity
- catastrophic collapse and possible extinction

This is why carrying capacity is dynamic. It can shrink if habitat is damaged. Overgrazing, drought, nutrient depletion, or waste buildup can all reduce future support for the population.

### Density-dependent and density-independent controls

Some limiting factors become stronger as population density rises. These are density-dependent.

> **Key Term: density-dependent factors**
> Limiting factors whose effects become stronger as population density increases.

Examples:

- competition for food
- disease transmission
- parasitism
- predation
- crowding stress
- territorial conflict

These factors help explain why logistic growth slows as population size rises. The larger the population, the stronger these pressures become.

Other factors act regardless of how dense the population is. These are density-independent.

> **Key Term: density-independent factors**
> Limiting factors that affect populations regardless of their density.

Examples:

- drought
- floods
- wildfire
- hurricanes
- extreme temperature
- pollution events

These do not depend on crowding. A severe wildfire can kill organisms whether the population was dense or sparse.

This distinction is often tested. Predation and disease are generally density-dependent. Drought and storms are density-independent.

A subtle but common point: density-independent factors can still lower carrying capacity. A drought can reduce plant growth and water supply, which means the environment can support fewer organisms afterward.

## Key Point Checklist

_This article has covered the following key knowledge points:_

- A finite resource base means populations cannot grow forever because food, water, space, nutrients, and waste-absorption capacity are limited
- Carrying capacity (**K**) is the maximum population size an environment can support sustainably
- Carrying capacity is not the same as current population size
- Carrying capacity can rise or fall when environmental conditions change
- Exponential growth produces a J-shaped curve and occurs only when resources are effectively unlimited
- Logistic growth produces an S-shaped curve and is more realistic for natural populations
- In the logistic model, growth slows as **N** approaches **K** because the term $(1 - N/K)$ becomes smaller
- Resource scarcity can increase mortality, reduce fecundity, or do both
- Overshoot happens when a population temporarily exceeds carrying capacity
- Dieback is the rapid decline that can follow overshoot and resource depletion
- Density-dependent factors include competition, disease, and predation
- Density-independent factors include drought, flood, wildfire, and extreme weather

## Key Terms and Concepts

- finite resource base
- carrying capacity (K)
- limiting factor
- exponential growth
- logistic growth
- overshoot
- dieback
- density-dependent factors
- density-independent factors


Population Growth and Resource Availability - Finite resource base