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For u = 0 we obtain the logistic DE on the line: i= r – r. Draw ihe phase line for this system. Identify and classify the equilibria.

For u = 0 we obtain the logistic DE on the line: i= r – r. Draw ihe phase line for this system. Identify and classify the equilibria.

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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7:55 "Z O l O 17% Let's put everything together! What happens when we take a logistic model for population growth and make the carrying-capacity depend on time? (1- H: (1+ sin (t))) – x² We can also view this equation as an autonomous system on the plane: (1- H. (1+ sin ( Ô = 1 (0))) – r² Using the methods we developed in class, we can analyze the behavior one piece at a time! Note: There are two pages. For u = 0 we obtain the logistic DE on the line: i = x – x². Draw the phase line for this system. Identify and classify the equilibria. 1 2. For u = 0 the (r, 0) system becomes decoupled. Identify all periodic solutions and explain if they are stable or unstable. For 0 < µ < } and e > 0 sufficiently small, consider the closed and bounded “washer" region of the plane given by e <r< 1 and 0 <0< 2n. Show that solutions that start in this region stay in this region for all time and explain why this means that a closed orbit exists. 3. Explain why using the linearization theorem on the (r, 0) system will NOT help us determine the stability of any fixed points. 4.
Transcribed Image Text:7:55 "Z O l O 17% Let's put everything together! What happens when we take a logistic model for population growth and make the carrying-capacity depend on time? (1- H: (1+ sin (t))) – x² We can also view this equation as an autonomous system on the plane: (1- H. (1+ sin ( Ô = 1 (0))) – r² Using the methods we developed in class, we can analyze the behavior one piece at a time! Note: There are two pages. For u = 0 we obtain the logistic DE on the line: i = x – x². Draw the phase line for this system. Identify and classify the equilibria. 1 2. For u = 0 the (r, 0) system becomes decoupled. Identify all periodic solutions and explain if they are stable or unstable. For 0 < µ < } and e > 0 sufficiently small, consider the closed and bounded “washer" region of the plane given by e <r< 1 and 0 <0< 2n. Show that solutions that start in this region stay in this region for all time and explain why this means that a closed orbit exists. 3. Explain why using the linearization theorem on the (r, 0) system will NOT help us determine the stability of any fixed points. 4.
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