Question

Northern Spotted Owl Population In an attempt to save the endangered northern spotted owl, the U.S. Fish and Wildlife Service imposed strict guidelines for the use of 12 million acres of Pacific Northwest forest. This decision led to a national debate between the logging industry and environmentalists. Mathematical ecologists have created a mathematical model to analyze population dynamics of the northern spotted owl by dividing the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years), and adult (over 2 years old). By analyzing these three subgroups, it is possible to use the number of females in each subgroup at time n to estimate the number of females in each group at any time with the following matrix equation: $$ \left[\begin{array}{c}{j_{n+1}} \\ {s_{n+1}} \\ {a_{n+1}}\end{array}\right]=\left[\begin{array}{ccc}{0} & {0} & {0.33} \\ {0.18} & {0} & {0} \\ {0} & {0.71} & {0.94}\end{array}\right]\left[\begin{array}{c}{j_{n}} \\ {s_{n}} \\ {a_{n}}\end{array}\right] $$ where $j_{n}$ is the number of juveniles, $s_{n}$ is the number of subadults, and $a_{n}$ is the number of adults at time $n .$ Source: Conservation Biology. a. If there are currently 4000 female northern spotted owls made up of 900 juveniles, 500 subadults, and 2600 adults, use a graphing calculator or spreadsheet and matrix operations to determine the total number of female owls for each of the next 5 years. (Hint: Round each answer to the nearest whole number after each matrix multiplication.) b. With advanced techniques from linear algebra, it is possible to show that in the long run, the following holds. $$ \left[\begin{array}{l}{j_{n+1}} \\ {s_{n+1}} \\ {a_{n+1}}\end{array}\right]=0.98359\left[\begin{array}{c}{j_{n}} \\ {s_{n}} \\ {a_{n}}\end{array}\right] $$ What can we conclude about the long-term survival of the northern spotted owl? c. Notice that only 18 percent of the juveniles become subadultsAssuming that, through better habitat management, this number could be increased to 40 percent, rework part a. Discuss possible reasons why only 18 percent of the juveniles become subadults. Under the new assumption, what can you conclude about the long-term survival of the northern spotted owl?

Name: Problem 52, Chapter 2: Finite Mathematics and Calculus with Applications
Uploaded: 2020-10-02T15:06:03-08:00
Duration: 6 min 35 s
Channel: Jacquelyn Trost

   Northern Spotted Owl Population In an attempt to save the endangered northern spotted owl, the U.S. Fish and Wildlife Service imposed strict guidelines for the use of 12 million acres of Pacific Northwest forest. This decision led to a national debate between the logging industry and environmentalists. Mathematical ecologists have created a mathematical model to analyze population dynamics of the northern spotted owl by dividing the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years), and adult (over 2 years old). By analyzing these three subgroups,
it is possible to use the number of females in each subgroup at time n to estimate the number of females in each group at any time with the following matrix equation: 
$$
\left[\begin{array}{c}{j_{n+1}} \\ {s_{n+1}} \\ {a_{n+1}}\end{array}\right]=\left[\begin{array}{ccc}{0} & {0} & {0.33} \\ {0.18} & {0} & {0} \\ {0} & {0.71} & {0.94}\end{array}\right]\left[\begin{array}{c}{j_{n}} \\ {s_{n}} \\ {a_{n}}\end{array}\right]
$$
where $j_{n}$ is the number of juveniles, $s_{n}$ is the number of subadults, and $a_{n}$ is the number of adults at time $n .$ Source: Conservation Biology. 

a. If there are currently 4000 female northern spotted owls made up of 900 juveniles, 500 subadults, and 2600 adults, use a graphing calculator or spreadsheet and matrix operations to determine the total number of female owls for each of the next 5 years. (Hint: Round each answer to the nearest whole number after each matrix multiplication.) 
b. With advanced techniques from linear algebra, it is possible to show that in the long run, the following holds.
$$
\left[\begin{array}{l}{j_{n+1}} \\ {s_{n+1}} \\ {a_{n+1}}\end{array}\right]=0.98359\left[\begin{array}{c}{j_{n}} \\ {s_{n}} \\ {a_{n}}\end{array}\right]
$$ 
What can we conclude about the long-term survival of the northern spotted owl? 
c. Notice that only 18 percent of the juveniles become subadultsAssuming that, through better habitat management, this number could be increased to 40 percent, rework part a. Discuss possible reasons why only 18 percent of the juveniles become subadults. Under the new assumption, what can you conclude about the long-term survival of the northern spotted owl?

Finite Mathematics and Calculus with Applications

Margaret L. Lial,… 9th Edition

Chapter 2, Problem 52

Instant Answer

Solved by Expert Jacquelyn Trost

10/02/2020

Step 1

We can do this by multiplying the given matrix by the current population vector for each year. The given matrix is: $$ A=\left[\begin{array}{ccc}{0} & {0} & {0.33} \\ {0.18} & {0} & {0} \\ {0} & {0.71} & {0.94}\end{array}\right] $$ and the current population Show more…

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Northern Spotted Owl Population In an attempt to save the endangered northern spotted owl, the U.S. Fish and Wildlife Service imposed strict guidelines for the use of 12 million acres of Pacific Northwest forest. This decision led to a national debate between the logging industry and environmentalists. Mathematical ecologists have created a mathematical model to analyze population dynamics of the northern spotted owl by dividing the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years), and adult (over 2 years old). By analyzing these three subgroups, it is possible to use the number of females in each subgroup at time n to estimate the number of females in each group at any time with the following matrix equation: $$ \left[\begin{array}{c}{j_{n+1}} \\ {s_{n+1}} \\ {a_{n+1}}\end{array}\right]=\left[\begin{array}{ccc}{0} & {0} & {0.33} \\ {0.18} & {0} & {0} \\ {0} & {0.71} & {0.94}\end{array}\right]\left[\begin{array}{c}{j_{n}} \\ {s_{n}} \\ {a_{n}}\end{array}\right] $$ where $j_{n}$ is the number of juveniles, $s_{n}$ is the number of subadults, and $a_{n}$ is the number of adults at time $n .$ Source: Conservation Biology. a. If there are currently 4000 female northern spotted owls made up of 900 juveniles, 500 subadults, and 2600 adults, use a graphing calculator or spreadsheet and matrix operations to determine the total number of female owls for each of the next 5 years. (Hint: Round each answer to the nearest whole number after each matrix multiplication.) b. With advanced techniques from linear algebra, it is possible to show that in the long run, the following holds. $$ \left[\begin{array}{l}{j_{n+1}} \\ {s_{n+1}} \\ {a_{n+1}}\end{array}\right]=0.98359\left[\begin{array}{c}{j_{n}} \\ {s_{n}} \\ {a_{n}}\end{array}\right] $$ What can we conclude about the long-term survival of the northern spotted owl? c. Notice that only 18 percent of the juveniles become subadultsAssuming that, through better habitat management, this number could be increased to 40 percent, rework part a. Discuss possible reasons why only 18 percent of the juveniles become subadults. Under the new assumption, what can you conclude about the long-term survival of the northern spotted owl?

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Key Concepts

Matrix Population Models

Matrix population models use matrices to represent the transitions between different stages or age classes within a population over time. In these models, the population is divided into distinct groups, and the entries in the matrix describe the rates of survival, reproduction, and progression from one stage to the next. This mathematical framework enables ecologists to use matrix multiplication to project the future state of the population based on its current structure.

Eigenvalue Analysis and Long-term Dynamics

Eigenvalue analysis is a linear algebra technique used to study the long-term behavior of matrix models. The dominant eigenvalue indicates the asymptotic growth rate of the population; if it is greater than one, the population will grow, equal to one it remains stable, and if less than one, it will decline over time. This analysis helps in understanding the sustainability and persistence of a species in the long run.

Parameter Sensitivity in Population Dynamics

Parameter sensitivity examines how changes in demographic parameters, such as survival rates or reproductive success, affect the overall population growth rate. By identifying which parameters have the greatest influence on population dynamics, researchers can target management strategies more effectively. Adjusting these parameters, even slightly, can lead to significant changes in the long-term viability of a population.

Conservation and Habitat Management

Conservation strategies and habitat management play a critical role in influencing the parameters of population models. Effective management, such as improving survival rates or increasing successful transitions between life stages, can be essential for the conservation of endangered species. By linking mathematical predictions with real-world habitat interventions, conservation biologists can design strategies that enhance the long-term survival of species.

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Transcript

00:01 For this problem, we have been given a matrix that shows how the population of northern spotted owls is changing year by year.

00:09 And we're watching three different subgroups of female owls, juveniles, subadults, which are one to two years old, and then full adults, two years and older.

00:21 And we're looking at the matrix a that i have here shows the percentages that we need to use in order to find our population.

00:30 So if i have matrix b, that shows how many female owls in each of those three subgroups, 900 juveniles, 500 sub -adults, 2 ,600 adults.

00:42 If i multiply a times b, that shows me the number of owls in each of those subsections the following year.

00:52 So after one year, i go from my population as given to this new population.

00:58 And i want to take a look at for each of the next five years.

01:02 So let's go five years out.

01:03 This is year one.

01:05 Now for year two, i'll take this new matrix that we have and multiply it by a.

01:11 So i'll have a times what we just found, a times b.

01:17 So this is after year one, after year two.

01:21 What about year three? well, i'm going to take my matrix a and then multiply by this result that i just got.

01:27 So for year three, i will get these numbers.

01:33 Year one, year two, year three.

01:36 How about year four? a, a, a, a, b.

01:42 And year five.

01:48 Okay.

01:49 Now, let's just take a look at what we have here.

01:54 Now, we started with 900 juveniles.

01:57 Now, we had a little bit of a dip and then it increased.

02:00 So this is not perfectly steady moving in one direction.

02:04 But as we go, we go on, those numbers do seem to be declining.

02:10 What about our sub -adults? well, it 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.

02:24 And how about our adults? it went up a little bit, because again, we had a large group of sub -adults, but now it is going back down again.

02:33 And we are, so we went up really high, and it's starting to decline.

02:37 In fact, we've been told that our, the overall matrix that we can use to find long -term growth of this population, if i want to find the n plus one year of juveniles, sub -adults, and adults, i'm going to take the current numbers that we have, and i'm going to multiply that by 0 .935n.

03:09 Well, that number is less than one.

03:13 So overall, every year that i check these populations, i'm expecting from one year to the next to go down.

03:21 Not by a lot.

03:22 You know, one would make a stable population, the same number from one year to the next.

03:26 If it's bigger than one, our population would be growing...

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