1. The Galápagos penguin is the only penguin species found north of the equator, and it is considered endangered. The population of the Galápagos penguin changes on a yearly basis as a discrete dynamic system. Suppose that initially there are 40 chicks and 40 breeding adults, that is xo = Suppose also that the yearly transition matrix is A = [0.125 0.75] (a) Which entry in the transition matrix gives the annual birthrate of chicks per adult? Which entry tells us the proportion of chicks that will survive to become adults? 40 . (b) Calculate the state vector x₁. (c) Find the eigenvalues and corresponding eigenvectors for matrix A. (d) Express xo as a linear combination of eigenvectors. (e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults. marks] (f) From your result to part (e), is the Galápagos penguin likely to survive?
1. The Galápagos penguin is the only penguin species found north of the equator, and it is considered endangered. The population of the Galápagos penguin changes on a yearly basis as a discrete dynamic system. Suppose that initially there are 40 chicks and 40 breeding adults, that is xo = Suppose also that the yearly transition matrix is A = [0.125 0.75] (a) Which entry in the transition matrix gives the annual birthrate of chicks per adult? Which entry tells us the proportion of chicks that will survive to become adults? 40 . (b) Calculate the state vector x₁. (c) Find the eigenvalues and corresponding eigenvectors for matrix A. (d) Express xo as a linear combination of eigenvectors. (e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults. marks] (f) From your result to part (e), is the Galápagos penguin likely to survive?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. The Galápagos penguin is the only penguin species found north of the equator, and it is considered
endangered. The population of the Galápagos penguin changes on a yearly basis as a discrete dynamic
[40]
system. Suppose that initially there are 40 chicks and 40 breeding adults, that is xo =
Suppose
40
also that the yearly transition matrix is A =
=
0
[o.
0.125
2
0.75
(a) Which entry in the transition matrix gives the annual birthrate of chicks per adult?
Which entry tells us the proportion of chicks that will survive to become adults?
(b) Calculate the state vector x1.
(c) Find the eigenvalues and corresponding eigenvectors for matrix A.
(d) Express xo as a linear combination of eigenvectors.
(e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults.
marks]
(f) From your result to part (e), is the Galápagos penguin likely to survive?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd8b0420-8290-4549-9f9a-309353e8d3f5%2Fe78652de-3b78-4be6-92d3-765811c58c75%2Fwgjn8cm_processed.png&w=3840&q=75)
Transcribed Image Text:1. The Galápagos penguin is the only penguin species found north of the equator, and it is considered
endangered. The population of the Galápagos penguin changes on a yearly basis as a discrete dynamic
[40]
system. Suppose that initially there are 40 chicks and 40 breeding adults, that is xo =
Suppose
40
also that the yearly transition matrix is A =
=
0
[o.
0.125
2
0.75
(a) Which entry in the transition matrix gives the annual birthrate of chicks per adult?
Which entry tells us the proportion of chicks that will survive to become adults?
(b) Calculate the state vector x1.
(c) Find the eigenvalues and corresponding eigenvectors for matrix A.
(d) Express xo as a linear combination of eigenvectors.
(e) Find the long-term distribution, after n years. State the long term ratio of chicks to adults.
marks]
(f) From your result to part (e), is the Galápagos penguin likely to survive?
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VIEWStep 2: (a) Annual birthrate of chicks per adult and proportion of chicks that will survive to become adults
VIEWStep 3: (b) State vector
VIEWStep 4: (c) To find the eigenvalues and eigenvectors
VIEWStep 5: (d) Express the initial state vector as a linear combination of the eigenvectors
VIEWStep 6: (e) Find the long-term distribution after n years
VIEWStep 7: (f) Is Galapagos penguin likely to survive?
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