EE 407 Homework 2 solution

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2 Questions
1. Water Reservoir System
Consider the water reservoir system, illustrated in Fig. 1, which is desired to supply water
to a city. The reservoir is fed by the streams q1, q2 and q3 where qi = q1 + q2 + q3 given in
units of m3/hour. The outlet water flow, qo (m3/hour), is controlled in such a way that the
flow to the city is rendered proportional to the quantity of water in the reservoir, w (m3
).
Proportionality constant is K = qoγ/wγ where qoγ = qo,max − qo,min and wγ = wmax − wmin.
(a) Find the relationship between qo and qi
.
(b) Find and plot qo when qi
is a unit step and initial conditions are zero.
(c) Find wγ, the difference between specified maximum and minimum storage of water
(m3
), such that we do not have to drain water via the bypass pipe before 5 days if
a sudden rise of magnitude B × K × wmax (m3/hour) in qi occurs due to excessive
rainfall (winitial = wmin).
1
Figure 1: Schematic of the Reservoir System
2. An Ecological System
You have already been introduced with ecological system modeling during lectures. In
this question, you are basically going to simulate a prey-predator system under different
assumptions.
this model, you are required realize the following set of equations.
y˙1 = (1 − y2/η2) y1 (1)
y˙2 = −(1 − y1/η1) y2 (2)
Notice that y1 and y2 stand for the number of prey and predator species, respectively.
Besides, η1 and η2 are constants determining the relation between two populations.
(b) Simulate the model for the following set of parameters. In your report, include a
figure exhibiting y1 and y2 over time. The figure should have a title, labeled axes, and
reasonable axis limits.
(η1 = 300, η2 = 200, y1,initial = 400, y2,initial = 100)
(c) Comment on the pattern of populations. Is there anything remarkable about observed
responses? Can you give an example of a nonlinear system having similar properties?
(d) Commit the current version of the model with a plausible commit message and push
the local repo to the remote repository on GitHub.
(e) Find the nonzero equilibrium point of the system, i.e., (y1,initial 6= 0, y2initial 6= 0).
When the initial value of populations is close to the equilibrium point, how does the
period and characteristics of the oscillations change?
2
(f) In this step, modify the model according the equations given below. Note that the
modification is to introduce a growth limiting term for the prey population.
y˙1 =

1 −
y1
2 ∗ y1,initial 
(1 − y2/η2)y1 (3)
y˙2 = −(1 − y1/η1)y2 (4)
(g) Simulate the model with the previous parameter set. In your report, include a figure
exhibiting y1 and y2 over time. The figure should have a title, labeled axes, and
reasonable axis limits. Comment on the shape of the solution curves? Are they still
periodic, if so how does the length of a period change?
(h) Commit this version of the model and push it to the remote repository on GitHub.
3. We will evaluate your response considering the files included in your GitHub repository.