Biology · Introductory biology · Worked example

Find the producer energy a top predator needs

The tertiary consumers in a forest build $6 kJ / m^{2}$ of new tissue a year. How much must the producers make, at $10 %$ efficiency per step? How does the answer change at the measured average of $5.92 %$ ?

Count the steps

Tertiary consumers are level 4, three steps above the producers. Write $E$ for the producers’ production in $kJ / m^{2}$ a year. Each step keeps $0.1$ of the level below, so three steps keep ${0.1}^{3}$ of $E$ .

E {(0.1)}^{3} = 6

Evaluate the power

${0.1}^{3} = 0.001$ , so the tertiary level keeps one thousandth of the producers’ production.

0.001 E = 6

Divide

Divide both sides by $0.001$ to leave $E$ alone.

E = \frac{6}{0.001}

Evaluate

The producers must make $6, 000 kJ / m^{2}$ a year.

E = 6000

Use the measured average

At $5.92 %$ per step the producers would need $6 / {0.0592}^{3}$ , nearly $29, 000 kJ / m^{2}$ a year. A drop in efficiency from $10 %$ to about $6 %$ means almost five times as much plant production for the same predators.

\frac{6}{{0.0592}^{3}} \approx 28919

Check by going up

Going up from $6, 000$ at $10 %$ a step gives $600$ , $60$ and $6$ , as required. The trophic pyramid in Biology shows those four levels and a $10 %$ transfer at each step.

Result

$6, 000 kJ / m^{2}$ a year at $10 %$ per step. At the measured $5.92 %$ the producers would need nearly $29, 000 kJ / m^{2}$ a year, almost five times as much.

Your turn

1. One like the example

Secondary consumers in a grassland build $50 kJ / m^{2}$ a year. At $10 %$ per step, how much must the producers make?

Show the answer and explanation

$5, 000 kJ / m^{2}$ a year.

Two steps, so $E {(0.1)}^{2} = 50$ , $0.01 E = 50$ and $E = 50 / 0.01 = 5, 000$ .

\begin{array}{l} E {(0.1)}^{2} = 50 \\ 0.01 E = 50 \\ E = \frac{50}{0.01} \\ E = 5000 \end{array}

2. A variation

A coastal sea makes $120 kJ / m^{2}$ of phytoplankton a year, and each step passes on $8 %$ . How much reaches the fourth trophic level?

Show the answer and explanation

About $0.0614 kJ / m^{2}$ a year.

Three steps up, so $120 \times {0.08}^{3}$ $= 120 \times 0.000512$ $\approx 0.0614$ . Multiplying by $0.08 \times 3$ instead of ${0.08}^{3}$ is the slip to avoid.

120 {(0.08)}^{3} \approx 0.0614

3. Somewhere new

People can eat grain directly or eat cattle fed on grain. At $10 %$ per step, how much more grain does the second diet need for the same energy, and how much more would a diet two steps up need?

Show the answer and explanation

Ten times as much one step up, and a hundred times two steps up.

Eating one step higher divides the energy by the efficiency, $1 / 0.1 = 10$ , and two steps divide it by ${0.1}^{2}$ , giving $100$ . The $10 %$ is the rule of thumb, so the true factor depends on the animal. Mammals such as cattle keep a warm body and spend much of what they absorb on body heat.

\begin{array}{l} \frac{1}{0.1} = 10 \\ \frac{1}{{0.1}^{2}} = 100 \end{array}

Keep exploring

The trophic pyramid tool opens with producers at $6, 000 kJ / m^{2}$ a year and $600$ , $60$ and $6$ above them, each transfer $10 %$ .

Return to the concept →

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After reading

Study it, then try it

Practice the key terms as flashcards, then open the example in its tool, change a value and keep a useful result on your board.

Study the 8 key terms Open Food web & energy flow Solve for E in Math Open worked example on a board Ecology reference

Your existing work stays on this device. Examples open as editable copies.

Count the steps

Evaluate the power

Divide

Evaluate

Use the measured average

Check by going up

Result

Your turn

1. One like the example

5,000 kJ⁢/m25,000 kJ/m2 a year.

2. A variation

About 0.0614 kJ⁢/m20.0614 kJ/m2 a year.

3. Somewhere new

Ten times as much one step up, and a hundred times two steps up.

Keep exploring

Study it, then try it

$5, 000 kJ / m^{2}$ a year.

About $0.0614 kJ / m^{2}$ a year.