Consider the following nonlinear predator-prey model: 6y₁ - 2y - y192, where dy₁ dt dy2 dt where y₁ (t) is the population of the prey species, and y2(t) is the population of the predator species. We can write this in vector form as y' = F(y), = y1y2 - 2y2, F(y) = - ( 1 (m. 12)) - (61 = (112). (a) Find the steady states of this system, i.e. the vectors (31,32) so that F(₁,32): (0,0). Hint: there are three steady states. (b) Calculate the total derivative DF(₁,32) = 6y₁-2y-9132 9192-2y2 (əfi əfi дуı дуг მ2 მ2 dy dy₂/ as a function of y₁ and y2. (c) For each steady state (y1, 32) from (a), find the eigenvalues of DF (y1, 92) and classify the steady state as stable or unstable.
Consider the following nonlinear predator-prey model: 6y₁ - 2y - y192, where dy₁ dt dy2 dt where y₁ (t) is the population of the prey species, and y2(t) is the population of the predator species. We can write this in vector form as y' = F(y), = y1y2 - 2y2, F(y) = - ( 1 (m. 12)) - (61 = (112). (a) Find the steady states of this system, i.e. the vectors (31,32) so that F(₁,32): (0,0). Hint: there are three steady states. (b) Calculate the total derivative DF(₁,32) = 6y₁-2y-9132 9192-2y2 (əfi əfi дуı дуг მ2 მ2 dy dy₂/ as a function of y₁ and y2. (c) For each steady state (y1, 32) from (a), find the eigenvalues of DF (y1, 92) and classify the steady state as stable or unstable.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:5. Consider the following nonlinear predator-prey model:
= 6y₁2yY1Y2,
dy₁
dt
dy2
dt
where
= y1y2 - 2y2,
where y₁ (t) is the population of the prey species, and y2(t) is the population of the
predator species. We can write this in vector form as
y' = F(y),
6y₁-2y1-9192
F(y) = - (5 (37-302)) = (0934-201² - 31.12).
(91,
22
(a) Find the steady states of this system, i.e. the vectors (y1, 92) so that F(y₁, y2) =
(0,0). Hint: there are three steady states.
(b) Calculate the total derivative
DF (y1, y2) =
əfi Əf₁
дуг
Əy₁
af₂ Əf2
дуı
дуг
as a function of y₁ and y2.
(c) For each steady state (y1, 92) from (a), find the eigenvalues of DF(y1, 92) and classify
the steady state as stable or unstable.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
