Exponential and Logistic Growth

Essential Question

Why do populations stop growing exponentially?

A few bacteria can become millions in a day, but a forest of trees does not grow without end. In this lesson, you’ll see how births, deaths, and movement change a population, and how to read the J-shaped and S-shaped growth graphs. By the end, you should be able to explain why growth slows as a population nears its carrying capacity.

How Populations Change

• Births and immigration add individuals
• Deaths and emigration remove individuals
• New size = start + births + immigration − deaths − emigration
• Growth means the population gets larger over time

A population is a group of individuals of one species living in the same area. Its size changes for four reasons. Births and immigration (individuals moving in) add to the population. Deaths and emigration (individuals moving out) take away from it. If more individuals are added than removed, the population grows. If fewer are added than removed, it shrinks. If they are equal, the size stays about the same.

Exponential Growth

• Each generation adds more individuals than the last
• Growth speeds up over time
• Graph is J-shaped
• Possible only when resources are plentiful and nothing limits growth

When every individual has plenty of food, space, and other resources, a population can grow faster and faster, because a larger population produces more offspring each generation. This is exponential growth, and it makes a J-shaped graph. A few bacteria in a fresh dish of nutrients can grow this way for a while. Exponential growth cannot continue for long in nature, because resources run out.

Logistic Growth and Carrying Capacity

• Growth slows as resources become limited
• Graph is S-shaped
• Carrying capacity (K): the largest population an environment can support
• Population levels off near K

Most populations cannot keep growing exponentially. As the population gets larger, food, space, and other resources get used up, so fewer offspring survive and growth slows. The result is logistic growth, which makes an S-shaped graph. The population grows slowly at first, grows fastest in the middle, and then levels off near the carrying capacity (K), the largest population size that the environment can support over time. On a graph like this one, the steepest part of the curve shows where the population is increasing fastest, and the flat part shows where growth has nearly stopped.

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Worked Examples

Try each problem on your own first, then open the solution to check your work.

Example 1: Change in a Population

A population of 500 mice lives in a barn. In one year, 80 mice are born, 20 mice move in, 30 mice die, and 10 mice move out. Find the population size at the end of the year, and the percent change in the population.

Show the solution

1. Add what comes in: 80 births + 20 immigrants = 100 mice
2. Add what goes out: 30 deaths + 10 emigrants = 40 mice
3. New size: 500 + 100 − 40 = 560 mice
4. Change: 560 − 500 = 60 mice
5. Percent change: 60 mice ÷ 500 mice × 100 = 12%
6. Reasoning: more mice were added than removed, so the population grew.
7. Answer: 560 mice, a 12% increase.

Example 2: Yeast in a Flask

Students grew yeast in a flask with a fixed amount of sugar and counted the cells every 2 days. The graph shows the data (sample data). Describe the shape of the graph. Estimate the carrying capacity. During which 2-day interval did the population increase the most?

Show the solution

1. Shape: the graph is S-shaped. It rises slowly, then steeply, then levels off.
2. Carrying capacity: the population levels off at about 500 thousand cells, so K is about 500 thousand cells.
3. Change in each interval: 20, 50, 100, 140, 100, 50, 20, and 5 thousand cells.
4. Fastest growth: 140 thousand cells, between day 6 and day 8.
5. Reasoning: as the yeast used up the sugar, growth slowed and the population leveled off near K.
6. Answer: S-shaped; K is about 500 thousand cells; fastest growth from day 6 to day 8 (140 thousand cells).

Example 3: Using the Logistic Model

The graph shows logistic growth of a population. At which point, A, B, or C, is the population increasing fastest? At which point is growth slowest? Explain why growth is slow at point C.

Show the solution

1. Fastest: point B, where the curve is steepest.
2. Slowest: point C, where the curve is nearly flat. (Point A is also slow, but the curve is steeper there than at C.)
3. Reasoning: at C the population is near the carrying capacity. Resources are limited, so few offspring survive and the number added is about equal to the number lost.
4. Answer: B is fastest and C is slowest. At C the population is near K, where limited resources slow growth.

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Practice

Calculate how a population changes, tell J-shaped growth from S-shaped growth, and read a growth graph to find carrying capacity. Try it on your own first, then check your work with the answer key.

• Exponential and Logistic Growth Practice Worksheet (PDF)
• Exponential and Logistic Growth Practice Answer Key (PDF)

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Lesson Slides

• 2.3 Exponential and Logistic Growth Slides (PDF)

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Big Ideas

• Population change = births + immigration − deaths − emigration
• Exponential growth is J-shaped and needs unlimited resources
• Logistic growth is S-shaped and levels off
• Carrying capacity (K) is the largest population the environment can support
• Growth is fastest at the steepest part of the S-curve

Back to the essential question: populations stop growing exponentially because resources are limited. As a population gets larger, food and space run short, fewer offspring survive, and growth slows until the population levels off near the carrying capacity. Next, you’ll look at the factors that set the carrying capacity.

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