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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
hey can u help me with question 4
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).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
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,
