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In an isolated region of the Kruger National Park, a population of black wolves, x(t), and a population of silver foxes, y(t), compete for survival. For each population, one unit represents 100 individuals. The two species have a common, limited food supply, which consists mainly of mice. The interaction of the two species can be modelled by the following system of differential equations: dx = 2x(1) -xy, dt dy dt = = 3y(1-3) — 2xy, where the proportionality constants were obtained from observation. (i) Find the nullclines of the system for x ≥ 0 and y ≥ 0.

In an isolated region of the Kruger National Park, a population of black wolves, x(t), and a population of silver foxes, y(t), compete for survival. For each population, one unit represents 100 individuals. The two species have a common, limited food supply, which consists mainly of mice. The interaction of the two species can be modelled by the following system of differential equations: dx = 2x(1) -xy, dt dy dt = = 3y(1-3) — 2xy, where the proportionality constants were obtained from observation. (i) Find the nullclines of the system for x ≥ 0 and y ≥ 0.

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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hey can u help me with question 4

(iii) State the Ro value for this model and explain what it represents. (iv) Find the disease free equilibrium point. (v) Find the Jacobian of the system. (vi) Show that the disease free equilibrium point is stable if Ro < 1 comment on the infections and unstable if Ro> 1 comment on the infections Question 4: In an isolated region of the Kruger National Park, a population of black wolves, x(t), and a population of silver foxes, y(t), compete for survival. For each population, one unit represents 100 individuals. The two species have a common, limited food supply, which consists mainly of mice. The interaction of the two species can be modelled by the following system of differential equations: dx | | dt dy dt 2x(1-1/2) - ) - xy, = 3y(1)-2xy, where the proportionality constants were obtained from observation. (i) Find the nullclines of the system for x ≥ 0 and y ≥ 0. 2 (ii) Find all of the equilibrium solutions. (iii) Using linearization, determine the nature of the equilibrium solution that lies strictly in the first quadrant. That is, determine the stability of the equilibrium solution for x > 0 and y > 0. What is the long-term situation for the foxes and wolves? Can the two species survive together? (iv) Sketch the nullclines of the system in the xy-phase plane. Use arrows to indicate the direction of the vector field in the xy-phase plane. Be sure to indicate the direction of the vector field on the nullclines. Sketch the trajectory in the xy-phase plane that begins at (1,2).
Transcribed Image Text:(iii) State the Ro value for this model and explain what it represents. (iv) Find the disease free equilibrium point. (v) Find the Jacobian of the system. (vi) Show that the disease free equilibrium point is stable if Ro < 1 comment on the infections and unstable if Ro> 1 comment on the infections Question 4: In an isolated region of the Kruger National Park, a population of black wolves, x(t), and a population of silver foxes, y(t), compete for survival. For each population, one unit represents 100 individuals. The two species have a common, limited food supply, which consists mainly of mice. The interaction of the two species can be modelled by the following system of differential equations: dx | | dt dy dt 2x(1-1/2) - ) - xy, = 3y(1)-2xy, where the proportionality constants were obtained from observation. (i) Find the nullclines of the system for x ≥ 0 and y ≥ 0. 2 (ii) Find all of the equilibrium solutions. (iii) Using linearization, determine the nature of the equilibrium solution that lies strictly in the first quadrant. That is, determine the stability of the equilibrium solution for x > 0 and y > 0. What is the long-term situation for the foxes and wolves? Can the two species survive together? (iv) Sketch the nullclines of the system in the xy-phase plane. Use arrows to indicate the direction of the vector field in the xy-phase plane. Be sure to indicate the direction of the vector field on the nullclines. Sketch the trajectory in the xy-phase plane that begins at (1,2).
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