How to use the simulator
- City A pop P1: 50k to 2,000k in steps of 50k (starts at 800k). Populations are in thousands.
- City B pop P2: the same range (starts at 400k).
- Distance d: 20 to 500 km in steps of 10 km (starts at 150 km).
Four outputs update:
- Predicted interaction I. The headline number, with the substitution written out underneath, for example "= (800 × 400) / 150²".
- Verdict line. Below the sliders, a sentence labels the result as a strong pull (I above 100), a moderate interaction (above 10) or a weak interaction (10 or less).
- Map panel. Cities A and B are drawn as circles whose area tracks population, joined by a flow line that thickens as interaction grows and labelled with the distance.
- Distance-decay graph. Interaction plotted against distance from 20 to 500 km, with a dot at your current distance.
The vertical axis of the graph rescales whenever you change a population, so the curve keeps the same shape. Population moves the whole curve up or down, but only distance moves you along it. The defaults give I = 14.22, a moderate interaction.
Because the populations are in thousands, I is a relative index rather than a count of trips or migrants. Use it to compare pairs of places, not as an absolute figure.
The key ideas
The gravity model borrows its form from Newton's law of gravitation. The interaction between two places is proportional to the product of their populations and inversely proportional to the distance between them:
- Double one population and I doubles. Double both and I is four times as large.
- Double the distance and I falls to one quarter, because . Triple it and I falls to one ninth.
- Some textbooks divide by rather than . Either version gives the same pattern: interaction rises with population size and falls with distance.
The model is a numerical version of distance decay: the farther apart two places are, the less contact they have. It is used to estimate migration, trade, commuting and the pull of shopping centres. Related ideas:
- Friction of distance. Distance costs time and money, so it discourages interaction.
- Time-space compression. Faster transport and communication weaken the effect of distance, so real-world interaction falls off more slowly than the model predicts.
- Ravenstein's laws of migration. Most migrants travel short distances, and large cities attract migrants, which matches the model's two inputs.
The model leaves out a lot. It ignores political borders, shared language or colonial ties, and intervening opportunities: a closer place offering the same jobs can capture migrants before they reach a larger city.
Worked example
Problem: City A has 1,200,000 people. City B has 300,000 people and is 200 km away. City C has 600,000 people and is 100 km from A. Which city should A interact with more, and by how much?
Step 1: A and B. Work in thousands, as the simulator does. .
Step 2: A and C. .
Step 3: Compare. , so A should interact with C about eight times as much as with B. C's larger population accounts for a factor of 2, and being half as far away accounts for a factor of .
Check it in the simulator: set P1 to 1,200k, P2 to 300k and d to 200 km. The readout shows I = 9 and calls it a weak interaction. Now set P2 to 600k and d to 100 km. The readout jumps to 72, the verdict changes to moderate, and the flow line on the map gets much thicker.
Common mistakes on the AP exam
- Treating distance as linear. In the squared version, doubling distance cuts interaction to a quarter, not a half.
- Adding populations instead of multiplying. The model uses . With the product, one very small place keeps the interaction small, however big the other place is.
- Reading I as a real count. The output is a comparative index, useful for ranking pairs of places, not a prediction of exactly how many people will move.
- Ignoring what the model leaves out. If a question asks why real flows differ from the prediction, name a specific factor: an international border, a shared language, intervening opportunities or an improved transport link.
- Mixing up distance decay and time-space compression. Distance decay is the drop in interaction with distance. Time-space compression is the shrinking of that effect as technology improves.
When the AP exam uses this
Distance decay and time-space compression are spatial concepts from Unit 1 (topic 1.4), and the gravity model comes back in Unit 2 (topic 2.10, causes of migration) to explain migration patterns alongside Ravenstein's laws. It can also come up in Unit 6 when questions ask about the market areas that cities and shopping centres draw on. Questions usually ask which pair of places will interact most, or how a change in transport technology would alter the predicted flow.