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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 – 1.00r – 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.

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 – 1.00r – 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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 2EQ: 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed...
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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?
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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