 It's often convenient to break down a process into snapshots. For example, people waiting in line for coffee, while at any given time there are always people in line, we can imagine recording the number of people in line at any given minute. For example, if I get data like this, at t equals one minute we take our picture, and we see there's eight people in line. We take another picture at t equals two, and now we see there's five people in line, and we can continue to take pictures of the people in line. So at t equals three we might see seven, at t equals four we might see seven again, and at t equals five we might see three. And while there is some number of people in line at t equals one point five, we don't know how many, and we don't actually care. A discrete time model determines a quantity at time t equals k based on the amount at time t equals k minus one. We often use discrete time models for biological populations with distinct life stages. In general, a mature or adult form of an animal can reproduce, an immature form cannot. There may be additional life stages, for example, egg becomes duckling becomes duck. In a duck population, eggs only come from adult ducks, so the number of eggs depends on the number of ducks. Meanwhile, ducklings only come from eggs, so the number of ducklings depends on the number of eggs. And finally, adult ducks come from ducklings that mature, as well as adult ducks that survive. For example, suppose that in a typical year, each adult duck lays five eggs, 90% of eggs hatch into ducklings, 80% of ducklings grow into adults, 70% of adult ducks survive into the next year. And if you begin with 100 duck eggs, let's find how many eggs, ducks, and ducklings you'll have after, well, about six years. So a large part of mathematics and life is bookkeeping, and so we'll want a way of recording the information we've found. So we'll let ET, LT, and DT be the number of eggs, ducklings, and ducks in year T, where whatever year we're in will subscript that after the letter. We'll let our start be T equals zero, and since we start with 100 eggs, we have E0 equal to 100. We have no ducklings, so the number of ducklings in year zero is going to be zero, and we have no adult ducks, so D0 is also equal to zero. Now we'll record the number of eggs, ducklings, and ducks as an ordered triple, eggs, ducklings, ducks. And so we have E0, L0, D0, that's 100, 0, 0. So let's move forward in time, and an important idea to remember here is that it's easier to know where you've come from than where you're going. So remember only adult ducks produce eggs. Since at T equals zero there are no adult ducks, then at T equals one there will be no eggs, so E1 is equal to zero. And we'll record that as the first component of our ordered triple, E1, L1, D1. Only eggs turn into ducklings. Now we're assuming 90% of the eggs hatch into ducklings. Since there are 100 eggs at T equals zero, then the number of ducklings is going to be 90% of 100, that's 90, and so the number of ducklings at time one is going to be 90. And we'll record that as the second component of our ordered triple. Now only ducklings turn into ducks, and since we've just determined there are D1 equals 90 ducklings at T equals one, this will tell us something about the number of ducks later, but not at T equals one. And here's an important thing to keep in mind when working with any discrete time model. The present is completely determined by the past. We want to know what happens at T equals one. If we don't care what's going on at T equals one, what we care about is what happened at T equals zero. So to find the number of ducks at T equals one, we need to know the number of ducklings at T equals zero. Since there aren't any ducklings at T equals zero, there won't be any ducks at T equals one, and so D1 is equal to zero. The next here, no adult ducks at T equals one mean no eggs at T equals two. No eggs at T equals one mean no ducklings at T equals two. 80% of the ducklings at T equals one become ducks at T equals two. So there are 80% of 90, 72 ducks. Now we know that 70% of the ducks at T equals one will survive until T equals two, but since there are no ducks at T equals one, these won't add any to our total. At T equals two, we have 72 adult ducks, and each produces five eggs. Yes, we'll ignore the finer points of biology. So at T equals three, we'll have 72 times five, 360 eggs. At T equals two, we have no eggs, and so we have no ducklings at T equals three. The ducks at T equals three are going to come from two sources. First, 80% of the ducklings at T equals two become ducks at T equals three. But we have no ducklings at T equals two, so these don't contribute anything. Meanwhile, 70% of the ducks at T equals two survive, and that means they become no, no, I don't think so. All right, it's just a duck. So we have 70% of 72 or about 50 ducks at T equals three. At T equals four, the 50 ducks we had earlier will produce 50 times five, 250 eggs, and the 360 eggs we had before will hatch into 324 ducklings at T equals four. And again, the ducks at T equals four come from two sources. 70% of the 50 ducks at T equals three will still be ducks at T equals four, so that's 35 ducks. And any ducklings we had at T equals three will become ducks, but again, there aren't any ducklings at this time. So we'll just have 35 ducks. At T equals five, the eggs are produced by the 35 ducks at T equals four, that's 175. The 250 eggs we have at T equals four will become 225 ducklings. Of the 35 ducks at T equals four, 70% or 25 will still be around. Additionally, 80% of the ducklings at T equals four will become ducks at T equals five. And so this adds another 259 more ducks. And so altogether we'll have 284 ducks at T equals five. And finally at T equals six, the 284 ducks will lay 1,420 eggs. The 175 eggs will become 158 ducklings. The 225 ducklings we have will become 180 ducks. And 70% of our existing ducks will still be around for a total of 379 ducks.