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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
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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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.
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.
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