3. A team of scientists are studying two species of deer in a park who are competing for the same grass. Let r be the population of Sika deer, and y be the population of White-tailed deer. After observing many interactions between the two species and monitoring population for a handful of cycles, the scientists form the following dynamical system to model future populations: dr r(2.20-0.22r- 2.00y) dt dy y(2.40 1.00r- 0.20g) dt (a) Find all the fixed points for this system (b) For each fixed point, find the eigenvalues and eigenvectors of the Jacobian matrix and thus characterise each fixed point. (c) Plot the phase portrait for this system (d) Assume the population of X in the absence of Y in the model above, can be modelled by the logistic model as follows: dr dt k What are the values of r and k in this model?
3. A team of scientists are studying two species of deer in a park who are competing for the same grass. Let r be the population of Sika deer, and y be the population of White-tailed deer. After observing many interactions between the two species and monitoring population for a handful of cycles, the scientists form the following dynamical system to model future populations: dr r(2.20-0.22r- 2.00y) dt dy y(2.40 1.00r- 0.20g) dt (a) Find all the fixed points for this system (b) For each fixed point, find the eigenvalues and eigenvectors of the Jacobian matrix and thus characterise each fixed point. (c) Plot the phase portrait for this system (d) Assume the population of X in the absence of Y in the model above, can be modelled by the logistic model as follows: dr dt k What are the values of r and k in this model?
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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Transcribed Image Text:3. A team of scientists are studying two species of deer in a park who are competing
for the same grass. Let z be the population of Sika deer, and y be the population
of White-tailed deer. After observing many interactions between the two species
and monitoring population for a handful of cycles, the scientists form the following
dynamical system to model future populations:
dr
= r(2.20 – 0.22r - 2.00y)
dt
dy
y(2.40 – 1.00z – 0.20y)
dt
(a) Find all the fixed points for this system
(b) For each fixed point, find the eigenvalues and eigenvectors of the Jacobian
matrix and thus characterise each fixed point.
(c) Plot the phase portrait for this system
(d) Assume the population of X in the absence of Y in the model above, can be
modelled by the logistic model as follows:
dr
= rx - -r
dt
What are the values of r and k in this model?
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