Lesson 3 of 8

Predator and prey.

About 30 minutes · 6 practice questions

Read the full article: Ecosystems and population dynamics

Foxes and rabbits on one meadow rise and fall in a cycle: four quarters of a loop round a balance point, the foxes lagging behind the rabbits, and the surprise in the averages: good years for rabbits raise foxes, and better hunters stay few.

Step 1 of 6 · Four quarters

Step 1 of 6

Four quarters

A population is all the individuals of one species in one place. The Scene boils an ecosystem down to two populations: wild rabbits, which eat grass that never runs out, and foxes, which eat rabbits. The fox is the predator and the rabbit the prey. In year $0$ ⁠ the meadow has $40$ ⁠ rabbits and $6$ ⁠ foxes, and each figure is one animal.

How each population grows depends on the size of the other. The rabbits multiply while there are fewer than $15$ ⁠ foxes, and the foxes multiply while there are more than $120$ ⁠ rabbits. Those two lines split the loop into four quarters. Years is locked at year $2.8$ ⁠ for now; read the note in the Scene and predict the next quarter.

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Step 2 of 6

The foxes lag

Drag Years slowly back and forth round years $3$ ⁠ to $5$ ⁠, and watch the first chart: two waves, the foxes’ a little behind the rabbits’. The second chart shows the same run another way, rabbits across and foxes up, and the point goes round it anticlockwise.

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Step 3 of 6

The balance point

The cross inside the loop is the balance point: if the meadow ever had exactly $120$ ⁠ rabbits and $15$ ⁠ foxes, births and deaths would balance for both and nothing would change. It is also each population’s average over a whole turn. The model is the Lotka–Volterra equations,

$\frac{dR}{dt} = aR - bRF, \qquad \frac{dF}{dt} = k\,bRF - mF$ ⁠

and when both rates are zero, $R^* = \dfrac{m}{kb}$ ⁠ and $F^* = \dfrac{a}{b}$ ⁠. Here $a$ ⁠ is Rabbit birth rate, $m = 0.8$ ⁠ is the foxes’ death rate, $k = 0.1$ ⁠ is the share of a cub each rabbit caught raises, and $b$ ⁠ is set by Hunting success.

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Step 4 of 6

Breaking even

The foxes stop growing exactly when the cubs born replace the foxes that die. You can ask this of a single fox: how many rabbits must it catch?

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Step 5 of 6

Bumper years

The two balance-point formulas say something unexpected: the rabbits’ birth rate $a$ ⁠ appears only in the foxes’ average, and the foxes’ death rate only in the rabbits’. Before changing anything, predict what happens to the averages.

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Step 6 of 6

Better hunters

Rabbit birth rate is fixed, and Hunting success is free. What happens to the foxes when they learn to hunt better? Use the formula for $F^*$ ⁠ to know which way to go.

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Still of the Ecosystems and population dynamics 3D scene

Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Predict, then look

Year $2.8$ ⁠: the meadow has nearly $300$ ⁠ rabbits, more than the balance point of $120$ ⁠, and only $7$ ⁠ foxes, fewer than $15$ ⁠. The note in the Scene says the foxes are raising more cubs. What happens next?

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Practice · Choose

The chart shows the foxes peaking about a year after the rabbits: the rabbits are already falling, and the foxes are still rising. Why?

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Practice · Predict, then look

The rabbits’ average over a whole turn is the balance point, $R^* = \dfrac{m}{kb}$ ⁠, with $m = 0.8$ ⁠ and $k = 0.1$ ⁠, and $b$ ⁠ is Hunting success as a fraction, times $\tfrac13$ ⁠. Now $b = 0.2 \times \tfrac13$ ⁠, and $R^* = 120$ ⁠. Predict: how many rabbits will there be on average when Hunting success rises to 30%?

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Practice · Calculate

In the model each fox dies at a rate of $0.8$ ⁠ a year, and each rabbit it catches raises $0.1$ ⁠ of a cub. How many rabbits must a fox catch a year for the foxes to raise exactly enough cubs to replace those that die?

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Practice · Predict, then look

Under Rabbit birth rate, bumper years: the rate rises from $1$ ⁠ to $1.5$ ⁠ a year. What goes up on average over a whole turn?

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Practice · Reach the target

Rabbit birth rate is fixed at $1$ ⁠, and on average the meadow has $15$ ⁠ foxes. Change Hunting success until the foxes’ average falls to $7.5$ ⁠, half, then press Check.