Almost every field has a trove of problems whose basic structure is the exploration of how a quantity (or quantities) changes over time. Some of the underlying processes are continuous (your kids growing taller) and some are arguably discrete (demographic change due to births, deaths and migration).
It turns out that in many cases we can treat the process as proceeding discretely, clock tick by clock tick, with finite changes happening each time period and accumulating over time.
INDICATIONS: we can use difference equations to model dynamic processes that either are, or can be treated as if they are, discrete (that is, unfold period by period rather than continuously)
General idea of difference equations: IF we can write an equation that tells us how the system, in general, looks one time based on how it looked the time before, AND we know what it looks like on some particular day, then we can make predictions about what it will look like over the long term.
We will sometimes speak of "where a system is" at a given point in time — what we mean is what is the value of the variable of interest describing the state of the system. For example:
- How high is the water in the reservoir?
- How much money is our rainy-day fund?
- How many people can we expect at this week's protest?
- How long is the line at the clinic?
To begin, we want to get comfortable with the mathematical notation and understand the range of phenomena to which the technique can be applied. Then we'll look at the long term behavior of systems and of more complex constellations of stocks and flows.
But first, a review of our motivation here with a few examples.
Let's draw a picture of the underlying phenomena.
NOTE: Some of these would qualify for "common sense" label — we can figure out how to do them. What DE offers is a systematic technique that can be applied across a wide range of systems. This is one of the senses in which it is a model.
Other examples. Suppose older cars pollute more and are inefficient. At a given point in time I have some fraction of older cars on the road. Every year some are replaced by new cars. And every year some formerly new cars become old cars. How big a "clunker" program should I implement if I want to get the long term proportion of old cars on the road down to a particular level?
Mills College recruits about 150 new freshwomen each year. Between first and second year some fraction of them transfer to other schools. The next year a new class is recruited, perhaps slightly larger. But then some of the sophomores transfer out before junior year. And some students are recruited as transfers. How does the school's population vary over time?