For this problem. We have been given a matrix that shows how the population of northern spotted owls is changing year by year. And we're watching three different subgroups of female owls juveniles, sub adults, which are 1 to 2 years old and then full adults two years and older. And we're looking at this. The Matrix A that I have here shows the percentages, um, that we need to use in order to find our population. So if I have Matrix B, that shows how many female owls in each of those three subgroups 900 juveniles, 500 sub adults, 2600 adults if I multiply a Times B, that shows me the number off owls in each of those subsections the following year. So after one year I go from my population has given to this new population, and I want to take a look at for each of the next five years. So let's go five years out. This is your one now for year two. I'll take this new matrix that we have and multiply it by a so I'll have a times what we just found a times B. So this is after year one after year, too. What about your three? Well, I'm going to take my Matrix A and then multiplied by this result that I just got so for years, three. I will get these numbers. Your one year, two year three. How about your four A. B and year five. Great five. Okay, now, let's just take a look at what we have here now. We started with 900 juveniles. Now we had a little bit of a dip, and then it increased. So this is not perfectly steady moving in one direction. But as we go on, those numbers do seem to be declining. What about our sub adults? What looks like we must have had a very large population of sub adults because it went down rapidly that first year and then kind of stabilized a little, but definitely stabilized lower. And how about our adults and went up a little bit again. We had a large group of sub adults, but now it is going back down again, and we are. So we went up really high, and it's starting to decline. In fact, we've been told that our, um the overall um Matrix so we can use to find long term growth of this population. If I want to find the end plus one year of juveniles, sub adults and adults, I'm going to take the current numbers that we have. And I'm gonna multiply that by 0.9835 n. Well, that number is less than one. So overall, every year that I check these populations I'm expecting from one year to the next to go down, not by a lot. You know, one would make a stable population the same number from one year to the next. It was bigger than one. Our population would be growing. But this means that my spotted our population is in decline. This is less than one. Well, what if we take a look at our original graph? I'm gonna come back up here and look at a so that's 0.18 number there in that second row first column that says that Onley 18% of juveniles become sub adults. The rest of them don't make it past that first year. Yeah, What could cause that number? Well, maybe it's the habitat. There isn't. Ah, good habitat for the babies. to grow in. Maybe the population of predator versus prey is a little bit out of whack. Maybe there are too many animals that eat small birds that that's causing a problem here. Maybe dogs or cats that are on the loose or foxes or something that might damage these babies. Maybe there's simply not enough food for them to, uh, go from egg to mature adults. Maybe there's something in the environment that's making them sick. We don't know, but there's a lot of things that could cause that problem. What would happen, though, if we targeted that number that we put the management habitat management in place may be reduced pollution. Um, took a look at the environment and try to make some changes, and we could increase that from 18% of 40%. What would that change? Will now take a look What happens to my numbers? Well, I start with 900 juveniles, and again, it's a little bit of a dip, But look at those numbers. 9 23 9 52 till your five. I'm over 1000 juveniles sub adults again, there's a small gap. I think there were more sub adults than usual in this first year. But now, once it kind of stabilizes, those numbers are increasing. And what about my adults? 27 99 28 87? I'm over 3100 by the end of this five year period. In fact, if I add up all of the birds at the end of my five year period, I have 1004 juveniles, 390 sub adults and 3130 adults. So I have gone from a population of 4000 at the beginning of my study, up to 4524. And that's with the 40% if I had added up those same numbers back when it was only 18%. The some of those numbers after five years was 3639 a decrease rather than increase. So that number is very important to the population of this spotted owl. Getting that number increased from 18 to 40 will make a huge impact on the overall health and well being of the spotted owl population