Population Size

Lessons

Senior 3 · Biology

Population Size

5 lessons

Lesson 4 of 5

The Population Growth Curve and Carrying Capacity

25 min

Yeast in a bottle

When banana juice is left to ferment to make beer, a few yeast cells in the juice start to divide. For a while nothing seems to happen. Then the juice froths as the yeast multiplies rapidly. After a few days the bubbling slows and stops. The yeast population has passed through the same stages that every population passes through when it enters a new habitat: slow start, rapid growth, levelling off.

Definition

Population growth curve: a graph of population size against time.

Lag phase: the first phase, when the population grows slowly because there are few individuals to reproduce and they are adjusting to the new conditions.

Log (exponential) phase: the phase of rapid growth, when resources are plentiful and the population doubles in equal intervals of time.

Carrying capacity ( $K$ ): the largest population that an environment can support for a long time with the food, water, space and other resources available.

Stationary phase: the phase when the population stays roughly constant at the carrying capacity, because births (plus immigration) equal deaths (plus emigration).

The S-shaped (sigmoid) curve

Figure 1: The phases of population growth. *The death phase happens only in a closed container, such as a flask, where food runs out and wastes build up.

Lag phase. Few individuals; they are settling in, and in microorganisms making the enzymes they need for the new food. Growth is slow.
Log (exponential) phase. Plenty of food and space, few limiting factors: births far exceed deaths, and the more individuals there are, the faster the population grows.
Slowing phase. Limiting factors begin to act (Lesson 3): food runs short, wastes build up, predators and disease increase. The death rate rises and the birth rate falls.
Stationary phase. Births equal deaths; the population levels off at the carrying capacity, rising and falling a little around it.
Death (decline) phase. In a closed culture, food runs out and toxic wastes, such as the alcohol made by yeast, accumulate: deaths exceed births and the population falls.

Investigation: the growth of a yeast population

A little yeast was added to a flask of sugar solution at 25 °C. Every two hours a drop was taken, stained with methylene blue, and the yeast cells in a fixed small volume were counted under a microscope. (Illustrative results.)

Time / h	0	2	4	6	8	10	12	14	16	18	20
Number of cells	10	27	68	158	309	478	598	659	684	694	698

Figure 2: The yeast data plotted: an S-shaped curve that levels off at about 700 cells.

Worked example 1: reading the yeast growth curve

Use the table and Figure 2. (a) Between which times is the log phase? (b) Calculate the mean rate of increase between 6 h and 10 h. (c) Compare it with the rate between 16 h and 20 h. (d) Estimate the carrying capacity.

(a) The curve is steepest from about 4 h to 10 h: that is the log phase.
(b) $\text{rate} = \frac{\text{change in number}}{\text{change in time}} = \frac{478 - 158}{10 - 6} = \frac{320}{4} = 80 \text{ cells per hour}$
(c) $\frac{698 - 684}{20 - 16} = \frac{14}{4} = 3.5$ cells per hour: almost no growth, because the population is in the stationary phase.
(d) The curve levels off at about 700 cells: this is the carrying capacity of the flask.

Worked example 2: how fast is exponential growth?

In ideal conditions a bacterium divides into two every 20 minutes. Starting with one bacterium, how many will there be after 3 hours if nothing limits the growth?

Number of divisions: $n = \frac{3 \times 60}{20} = 9$ .
Each division doubles the number: $N = N_0 \times 2^{n} = 1 \times 2^{9} = 512$

After 3 hours: 512 bacteria. After 24 hours (72 divisions) the number would be $2^{72}$ , about $4.7 \times 10^{21}$ bacteria, which would weigh thousands of tonnes: all from one cell in one day. Real populations never grow like this for long: limiting factors bend the curve into an S.

J-shaped and S-shaped growth

Figure 3: Exponential (J-shaped) growth compared with real (S-shaped) growth. Environmental resistance is the difference between what the population could reach and what it does reach.

Some populations do grow in a J-shape for a short time, then crash: algae in a lake that suddenly receives fertiliser, or locusts after heavy rains. The crash comes when the food is used up or conditions change. This "boom and bust" pattern shows what happens when growth overshoots the carrying capacity.

Warning

Common mistakes. (1) Saying the population "stops reproducing" in the stationary phase: individuals still breed, but births are balanced by deaths. (2) Drawing the stationary phase as a death phase: in a natural habitat the population levels off; it falls only in a closed container. (3) Reading the carrying capacity from the steepest part of the curve instead of the level part.

Population Growth Models: Exponential, Logistic... Explained! (Brain Boost). Watch how the J-shaped exponential model and the S-shaped logistic model differ, and where the carrying capacity appears.

Tip

Memory hook: "Lazy, Leaping, Levelling". Lag (slow start), Log (leaping up), Levelling off at K. In an exam, name the phase and give its cause: few individuals; plentiful resources; limiting factors balance births and deaths.

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Practice: growth curves

Name the phases of the S-shaped growth curve in order.
Define carrying capacity.
Explain why growth is slow in the lag phase.
Use the yeast table to calculate the mean rate of increase between 2 h and 6 h.
A bacterium divides every 30 minutes. How many bacteria come from one bacterium after 4 hours, with no limiting factors?
Explain why the yeast population in a closed flask eventually falls.
Sketch and label a J-shaped and an S-shaped curve on the same axes, and mark the environmental resistance.
A farmer adds 20 goats to an island with no goats. Predict and explain the shape of the goat population curve over 20 years.

Show answersHide answers

Lag phase, log (exponential) phase, slowing phase, stationary phase (and, in a closed container, the death phase).
The largest population an environment can support for a long time with the resources available.
There are few individuals to reproduce, and they are adjusting to the new conditions (in microorganisms, making the enzymes needed for the new food).
$\frac{158 - 27}{6 - 2} = \frac{131}{4} = 32.75$ cells per hour.
$n = \frac{4 \times 60}{30} = 8$ divisions; $N = 2^{8} = 256$ .
The sugar is used up and toxic alcohol builds up, so deaths exceed births.
As in Figure 3: the J-curve rises ever more steeply; the S-curve levels off at the carrying capacity; the gap between them is the environmental resistance.
S-shaped: slow growth at first (lag); rapid growth while grass is plentiful (log); then growth slows as food runs short, and the population levels off at the carrying capacity of the island. If the goats overgraze, the carrying capacity itself may fall and the population may crash.

Key points

Phases: lag → log (exponential) → slowing → stationary at the carrying capacity (→ death, in a closed container).
Exponential growth: $N = N_0 \times 2^{n}$ for $n$ doublings; it never lasts.
Environmental resistance bends the J-curve into an S-curve.

Sources and credits

19.2: Population Growth and Regulation, from Concepts of Biology by Samantha Fowler, Rebecca Roush and James Wise (OpenStax), via LibreTexts Biology, licensed under CC BY 4.0. Adapted by Ganzaa. (sections "Exponential Growth", "Logistic Growth" and "Carrying Capacity")
Photo: Saccharomyces cerevisiae Loeffler's Methylene Blue Staining, Cboerner, CC0, via Wikimedia Commons.
Video: Population Growth Models- Exponential, Logistic... Explained!, Brain Boost, embedded from YouTube.

Written by Ganzaa: Figures 1 to 3, the yeast investigation and its illustrative results, the worked examples and the practice questions.

1Yeast in a bottle

2The S-shaped (sigmoid) curve

3Investigation: the growth of a yeast population

4J-shaped and S-shaped growth

Yeast in a bottle

The S-shaped (sigmoid) curve

Investigation: the growth of a yeast population

J-shaped and S-shaped growth