Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Assume that any initial vector $\mathbf{x}_{0}$ has an eigen- vector decomposition such that the coefficient $c_{1}$ in equation $(1)$ of this section is positive. $^{3}$Determine the evolution of the dynamical system in Example 1 when the predation parameter $p$ is .2 in equation $(3)$ . (Give a formula for $\mathbf{x}_{k}$ .) Does the owl population grow or decline? What about the wood rat population?

Both decrease.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 6

Discrete Dynamical Systems

Vectors

Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

02:56

In mathematics, a vector (…

06:36

01:47

Let $w$ be the number of w…

02:20

A food web Lynx eat snowsh…

02:30

Lynx eat snowshoe hares an…

01:20

The logistic differential …

01:14

05:44

Population Growth Suppose …

01:17

Suppose that the rabbit po…

07:53

The autonomous differentia…

03:35

Immigration and Emigration…

Hi there. This question is based on the book and it lets us to, uh, eyes the evolution of the system off house and rats. And here we have basic equipment off ax zero, because one we want us, uh, all the way to see and we okay. And we also have Axe K. He goes to see one lambda walk one to the power of K Tom's. We want all the way out up to say, in Lambda and to the power of K times via Okay, on. We also have the information off the evolution off house and rests on the followed the following quitters. Old K plus one you cost too. 05 Okay. Plus you put four r k. And we also have our keep us one yuko stewed Maga to the airport to okay plus one Put one R p well, intuitively we can see that for example, if RK ecosoc Romanians at a time k, there are no rats. Then at time keep us one the number of those Well, only the half off the number of house A k which is uh which is actually cracked because if there are no rats there are no food for out, so their survival chance will decrease. And also for rest. We know that the parameter here is negative, meaning that if we have more house number rats at next time interval bill, decrease Meaning that more rascal ve eaten. So it is actually quite intriguing. And no, let's, uh, let's denote a matrix as the parameters here, which is thereupon filed their four negative 0.2 on 1.1 on the next thing we're gonna do is to compute the agent battles that had about thirst off the Matrix. A and thankfully, it is two times two matrix, so hopefully won't be too difficult. Let's first use this equation that a minus Lambda Palms liability matrix equals zero. We're gonna use this one to compute lambda, which is the age of battles off. And let's see how to do that, will it, actually, because to determinant, we have here, right? And for the content. So we have a minus lambda, so we can't just right zero point file minus Lunda right on. For this part, it would be one minus 1.1 minus loved for these two entries. Here they remain the same as a. So it is more. And now they want to. So this one that cost you there are five mile Islam that Tom's 1.1 mile Islam, then minus four times negative. What, you you cost zero. So for this equation, it is easy to solve and we can get the value. The value's for London, which is a lambda one unloved that you or that being Just write it this way. I'm the one Yukos upon seven that you because you're a benign. So these are the two agent values off the matrix eight. And now let's compute the agile electors with respect disease. So here we're gonna use a camps be you goes Lambda Tom's be to compute we which is the our agile workers. And let's just set we to be x one x two Okay, you hear from from this equation One, we have 015 Well, let's first just plugging Lunda one. You know where you quitted here. So we have thereupon file x one plus zero point for axe to write 800 v on for the next. And free should be, uh, negative. They're 12 x one, then plus 1.1 x two. Okay, so for this one issue ICO too long down times a week. So it equals two. There are about seven X one as their 10.7 x two. Okay, so here, according to disease way. See that, for example? Here, uh, negative 0.2 x one plus 0.4 x two Because yet we have another constraint. Also, you need to use it. But here, just for simplicity that that's use it as a convention. It's that the square of at swim Square of two. Because toe me in the land off the general letters, one eyes. What? So okay, according to this, we know that, um, we we want you goes to x one x two You caused to this quoted here it is ah, chu or square root off while and one over this year Routes five ok. And similarly with compute which we two forward to. We just need to plug in love that you Yukos open night into this equation here and then compute me. Okay, so similarly, we have this quantity goes to one or spirit of Jew on another one over square root off to Okay, so here we have. We won on way to now we can write our SK you goes to see one times their bond. Seven tow bar. Okay, then Tom's way. Want I just write it again? This is we want Brent. And here a plus C two times thereupon night to the ROK one oris gold up to one or square up to which is waiting. Okay, so here we have our expression No axe, Kate, which is asked in this problem. And meanwhile, we know that when K approaches to the positive infinity comparatively, we know that their opponents seven so far okay, will approach their own more quickly than therefore not of our case. So that's just assumed disc wanted to be zero because when K is very large, a 0.72 parquet will be approximation of zero were asked their opponents. A parochial will be much better, So it will be smaller than one as well, so we can just write it. It is similar to see to Tom's Airport night of ROK Tom's one. Oh, were this girl to another one of Europe's. Okay, so it is our approximation off. Ask a and it also, uh, we counsel writes XK class one right at ST Plus one. So we have another Kaposvar here, so it equals to the airport, not calms comes everything else. Except we have, say, two times there but nine power Kate here, Tom's. Wait, you That's just right. We two for simplicity. It is exactly They're born nine times axe k. So here we see that, uh, one K is very large starting from K two k plus one at the increasing rate is only there when I meaning that the number off house rats were both decreased at a rate off 0.9 or they were increased at a rich off their benign meaning that about decrease eso That's our analysis for this problem. Thank you.

View More Answers From This Book

Find Another Textbook

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a vector (from the Latin "mover") is a geometric o…

Let $w$ be the number of worms (in millions) and $r$ the number of robins (i…

A food web Lynx eat snowshoe hares, and snowshoe hares eat woody plants, suc…

Lynx eat snowshoe hares and snowshoe hares eat woody plants like willows. Su…

The logistic differential equation Suppose that the per capita growth rate o…

Population Growth Suppose that the rabbit population on Mr. Jenkins' fa…

Suppose that the rabbit population on Mr. Jenkins' farm follows the for…

The autonomous differential equations represent models for population growth…

Immigration and Emigration If population is changed either by immigration or…

03:22

Show that the signals in Exercises $3-6$ form a basis for the solution set o…

00:43

Show that if $A$ has $n$ linearly independent eigenvectors, then so does $A^…

05:21

[M] Let $A$ be the matrix in Exercise $35 .$ Construct a matrix $C$ whose co…

Let $A$ be as in Exercise 9 . Use the inverse power method with $\mathbf{x}_…

00:37

Exercises $9-14$ require techniques from Section $3.1 .$ Find the characteri…

02:50

Let each matrix in Exercises $1-6$ act on $\mathbb{C}^{2} .$ Find the eigenv…

Find the equation $y=\beta_{0}+\beta_{1} x$ of the least-squares line that b…

04:13

Refer to Exercise $4 .$ In the long run, how likely is it for the weather in…

In Exercises 17 and $18, A$ is an $m \times n$ matrix and $b$ is in $\mathbb…

In Exercises $7-12$ , use Example 6 to list the eigenvalues of $A$ . In each…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.