homework 7 Solutions University of California, Los Angeles CHICANO 30B
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Course
CHICANO 30B
Institution
University Of California, Los Angeles
Homework 7 Solutions March 2, 2018
Chapter 6.5.
Exercise 6.5.1.
If there are two noninteracting populations, one of which is growing at 20% a year and the other at
25% a year, derive the matrix that describes the dynamics of the system and simulate a trajectory
of this system.
We know that fo...
2018 chapter 65 exercise 651 if there are two noninteracting populations
one of which is growing at 20 a year and the other at 25 a year
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University Of California, Los Angeles
CHICANO 30B
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Homework 7 Solutions
March 2, 2018
Rubric
2 problems were graded according to the following scale for a total of 10 points.
These two problems were selected for the following reason:
• Exercise 6.5 FE 1: The problem covers comprehensively major topics of the course, including
analyzing the significance of eigenvalues and determining long term behavior
• Additional exercise: it was specifically stated that you would be expected to do a similar
problem like this on the final exam
Exercise 6.5. FE 1: (6 points)
Part a: (2 points)
• 2 points: Able to state a reasonable explanation on the significance of most (if not all) of the
values
• 1 point: State some explanations that carry little-to-no biological significance
• 0 point: Blank answer
Part b: (2 points)
• 2 points: Able to identify the long-term behavior and state a reasonable explanation that
involves the discussion of the dominant eigenvalue
• 1 point: An answer with not much explanation OR an explanation that does NOT involve
the discussion of the dominant eigenvalue
• 0 point: Blank answer
1
,LS 30B Homework 7 Solutions March 2, 2018
Part c: (2 points)
• 2 points: Able to recognize the significance of the eigenvector associated with the dominant
eigenvalue and compute the number of young swans accordingly
• 1 point: An answer that does not involve the use of the eigenvector associated with the
dominant eigenvalue
• 0 point: Blank answer
Additional Exercise: (4 points)
Computing the fu,v matrix: (2 points)
• 2 points: Able to compute the eigenvalues and put them in a diagonal form
• 1 point: Unable to compute any eigenvalues
• 0 point: Blank answer
Computing the T matrix: (2 points)
• 2 points: Able to compute the eigenvectors and put them in the appropriate order, according
to the student’s fu,v matrix
• 1 point: Unable to compute any eigenvectors OR the T matrix does NOT match with the
student’s fu,v matrix (they should lose points for the latter mistake because it is better to
learn that now than make that same mistake on the final exam)
• 0 point: Blank answer
2
, LS 30B Homework 7 Solutions March 2, 2018
Chapter 6.5.
Exercise 6.5.1.
If there are two noninteracting populations, one of which is growing at 20% a year and the other at
25% a year, derive the matrix that describes the dynamics of the system and simulate a trajectory
of this system.
We know that for these two non-interacting populations, A and B, we have:
An+1 = An + 20% · An = 1.2 · An
Bn+1 = Bn + 25% · Bn = 1.25 · Bn
Thus, we have the following equations:
An+1 = 1.2 · An + 0 · Bn
Bn+1 = 0 · An + 1.25 · Bn
Therefore,
An+1 1.2 · An + 0 · Bn 1.2 0 An
= = ·
Bn+1 0 · An + 1.25 · Bn 0 1.25 Bn
Thus, the matrix that describes the dynamics of the system is:
1.2 0
0 1.25
Trajectories for two noninteracting populations
with A growing at 20% a year and B growing at 25% a year
3
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