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3. Consider a predator-prey model, z (r-ray), r r, K, a>0, (1) y(-b+cz), b, c> 0. (2) The variable z represents the density of the prey species and the variable y the predator. (a) Find all equilibrium points of this system. (b) In the absence of the predator, solve the differential equation (1) with initial condition z(0) = Io- Then, calculate lim z(t). (c) In equations (1) and (2), let r=1, a = 1, b=0.5, c = 1 and K = 4. i. Plot the nullclines of the system on the same set of axes. ii. Sketch the direction field of the system. iii. Use the direction field to determine the nature and stability of the equilibrium points. iv. Use the Jacobian matrix method to confirm the results obtained above regarding the nature and stability of equilibrium points. 004-7

3. Consider a predator-prey model, z (r-ray), r r, K, a>0, (1) y(-b+cz), b, c> 0. (2) The variable z represents the density of the prey species and the variable y the predator. (a) Find all equilibrium points of this system. (b) In the absence of the predator, solve the differential equation (1) with initial condition z(0) = Io- Then, calculate lim z(t). (c) In equations (1) and (2), let r=1, a = 1, b=0.5, c = 1 and K = 4. i. Plot the nullclines of the system on the same set of axes. ii. Sketch the direction field of the system. iii. Use the direction field to determine the nature and stability of the equilibrium points. iv. Use the Jacobian matrix method to confirm the results obtained above regarding the nature and stability of equilibrium points. 004-7

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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3. Consider a predator-prey model, dr = x (r - r ² − ay), r, K, a > 0, (1) dt dy = y(-b+cz), b, c> 0. (2) dt The variable z represents the density of the prey species and the variable y the predator. I (a) Find all equilibrium points of this system. = 10. (b) In the absence of the predator, solve the differential equation (1) with initial condition z(0) Then, calculate lim z(t). (c) In equations (1) and (2), let r = 1, a = 1, b=0.5, c = 1 and K = 4. i. Plot the nullclines of the system on the same set of axes. ii. Sketch the direction field of the system. iii. Use the direction field to determine the nature and stability of the equilibrium points. iv. Use the Jacobian matrix method to confirm the results obtained above regarding the nature and stability of equilibrium points. | |
Transcribed Image Text:3. Consider a predator-prey model, dr = x (r - r ² − ay), r, K, a > 0, (1) dt dy = y(-b+cz), b, c> 0. (2) dt The variable z represents the density of the prey species and the variable y the predator. I (a) Find all equilibrium points of this system. = 10. (b) In the absence of the predator, solve the differential equation (1) with initial condition z(0) Then, calculate lim z(t). (c) In equations (1) and (2), let r = 1, a = 1, b=0.5, c = 1 and K = 4. i. Plot the nullclines of the system on the same set of axes. ii. Sketch the direction field of the system. iii. Use the direction field to determine the nature and stability of the equilibrium points. iv. Use the Jacobian matrix method to confirm the results obtained above regarding the nature and stability of equilibrium points. | |
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