The unit on People in Motion looks at mathematical models that describe human interactions across space – or even across space and time. When the movement involves humans, we tend to call it “migration” but the mathematical model that describe movement of people in space can also work well when used to describe the flow of goods between regions (trade) or even the diffusion of ideas across space.

Human geographers use the term “spatial interaction” to describe flows of people, flows of trade or flows of information and ideas across space. Physicists use mathematics (fluid dynamics) to describe flows- many of the same mathematical models will work equally well when describing spatial interactions.

Ullman originally developed his three-factor model to describe trade but it works equally well in describing migration or flows of information.

The inverse square law is a fundamental principle of physics and is used to describe how the intensity of light or heat falls off with the square of distance. The exact same relationship, known in human geography as “distance decay” also follows the inverse square law. If we double the distance between two regions, their interactions should decrease by a factor of four. Increase the distance by a factor of four and interactions should fall to one-sixteenth of their former level.

The basic equation for gravity (seen here) can also be modified to model situations in human geography. In our example, we have GDP in the numerator to indicate the likelihood of trade. We could also use population in the numerator to describe the amount of migration. Notice the inverse relationship between distance (always in the denominator) with the final amount of interaction. In my NAFTA example, Mexico and Canada are both large, populous nations with big economies. The numerator will be large and the denominator small, meaning the amount of trade or migration should be high.

We can apply our mathematical models to real-world situations. We often notice competing pharmacies on opposite street corners. Why do competitors spend millions of dollars to put their locations so near to one another? An economist named Hotelling has the answer.

In the simplified Hotelling model, the is one linear street and convenience is the only reason for patronizing one shop over another. We very quickly can intuit that the most logical placement for our red and blue competitors is to locate next door to each other in the middle of the street. The same logic applies to the pharmacies. They are often located across the street from each other in areas where the street bisects a populous neighborhood. Their industry is such that one pharmacy is similar to any other and thus, convenience is the main draw for customers. Locating across the street from each other maximizes their access to the side of the street they are on.

Human movement is biased in three ways – we prefer short journeys over long ones and we follow the smoothest path. We also prefer to stay “in network” – that is, we are biased in favor of movement within areas with which we have some affiliation or connection.