 So let's take another look at a Leslie model and see what information we can glean from it. So I suppose I have an animal population with three stages. We'll call them egg, nymph, and adult. And our vector gives us the number of individuals in each of the three stages at some time. And suppose that our population has a certain transition matrix. Let's do two things. First of all, let's describe the transition for each of the three stages and then find m squared. So first, let's deconstruct our transition matrix. Remember the entries of the transition matrix give us the coefficients of the linear formulas that describe each of the vector components. So our vector components give the number of eggs, nymphs, and adults at any time. So the number of eggs at time n plus 1 is going to be 0 times the number of eggs at time n plus 0 times the number of nymphs at time n plus 10 times the number of adults at time n. And one way we can interpret this equation is that every adult produces 10 eggs. Our second line tells us the number of nymphs at time n is 1 times the number of eggs at time n plus 0 times the number of nymphs at time n plus 0 times the number of adults at time n. And we can read that as every egg turns into 1 nymph. And our last line tells us that the number of adults at time n plus 1 is 0 times the number of eggs plus 1 times the number of nymphs plus 1 times the number of adults. adults, and what that says is that every nymph turns into an adult, moreover, every existing adult remains an adult. And so our Leslie Matrix, our transition matrix, describes an animal whose life cycle can be described as every adult produces 10 eggs, every egg turns into a nymph, every nymph turns into an adult, and every existing adult remains an adult. Biologically speaking, this isn't realistic. We can expect some attrition among the eggs, not every egg is going to survive, not every nymph is going to survive, and adults die or get eaten. But it's a starting point. So how about M squared? So remember, M is the transition matrix that describes how we go from the number of eggs, nymphs, and adults at one point in time to the number of eggs, nymphs, and adults in the next point in time. If I apply M to the output, I get the number of eggs, nymphs, and adults at the point in time after that. And so that means that M squared, which corresponds to applying the transition M twice, is going to tell me the number of eggs, nymphs, and adults two time intervals after my starting point. And what this means is that M squared is going to give me a way to determine these values from my starting values. So let's try and find these numbers. The number of eggs at time n plus 2 is 10 times the number of adults at the previous point in time. So that's 10 times an plus 1. What I know that an plus 1 is 0en plus 1yn plus 1an. So I'll drop that into my equation, and I have the number of eggs at time n plus 2 in terms of the number of eggs, nymphs, and adults at time n. And this is what I want for my first row of M squared. What about yn plus 2? Well, I know the number of nymphs is equal to the number of eggs in the previous time interval, so I know that yn plus 2 is equal to 1en plus 1. And I know what en plus 1 looks like. It's 10 times the number of adults in the stage before that. And so now I have yn plus 2 in terms of the number of eggs, nymphs, and adults at time n. And that's going to give me the second row of the transition matrix. And finally, the number of adults at time n plus 2 is going to be equal to the number of nymphs and adults at time n plus 1. And we know the formulas for each of these. So after all the dots settles, we'll have our formula for the number of adults at time n plus 2 in terms of the number of eggs, nymphs, and adults at time n.