Homework: Spring 2009, Math 509  (Finotti)



Math509 Home

Math509 Calendar

Guest Speaker Calendar

Office Hours

Math509 Miscellaneous










****  Don't forget to write up your homework according to the guidelines in the syllabus! ****

General Guidelines 
** Remember - confusion is the first step to wisdom! It's not only normal to not immediately understand something, it's often necessary for learning and very necessary for curiosity.

**The purpose of the homework sets are for you to work through the understanding of the material and see what it is you do and do not understand so that you can fine tune your understanding from there -- please keep this viewpoint in mind when working through problems. 


**Educational research shows that studying and learning in peer groups often leads to better comprehension of the material.  Get to know your classmates and form study groups. 

**The other side of the coin is that just as beneficial to your learning is time spent grappling with the material alone.  Always do your final homework writeups on your own, and feel free to reach for help when you need it.  Doing it first on your own helps you to know what it is that you really do or don't understand

**If you are suspected of plagerism (think solution manual), you will
get a zero for the homework set, and will be reported to University authorities.


*Please look at the following interesting graph of
material retention vs. time lapse from first exposure until review (retention curve) and the subsequent tips for textbook reading

Keep this in mind when writing up your homework solutions -- it IS as important for learning math to practice explaining your solutions well as it is finding them!  Mathematics is a language.

Homework Set #

Due Date

Homework/Boardwork Assigments



****  Don't forget to write up your homework according to the guidelines in the syllabus! ****

 1


1/14

Read handout article by Alfio Quarteroni, Chapter 2 in Taubes, and the HIV article at the end of Chapter 2 for next time
Download MATLAB (http://oit.utk.edu/software) (contact me with any questions/problems)

Written Assignment:  Ch 2 Taubes - Problems  1, 2ab, 3, 5

 2


1/21

Written Assignment: 
1.  Recall the equation from the first lecture
d^2(Theta)/dt^2 + (g/l)* sin(Theta) = 0
that describes the dynamics of the angle Theta of an oscillating pendulum over time.  If we let the length of the pendulum be 1 meter then g/l is approximately 9.8 m/s, rename Theta as f,  and move the second term to the right hand side, we have
f '' = -9.8*sin(f)
 a.  Find a 1st degree Taylor Series approximation for the right hand side of this ode, assuming f(0) = C1 and f '(0) = C2
 b.  Find a 2nd degree Taylor Series approximation for the right hand side, assuming f(0) = C1 and f ' (0) = C2
 **Computer assignment: 
 a. Use ode45 in MATLAB to compare the true solution, the 1st order approximation, and the 2nd order approximation assuming C1=0.5 and C2 = 2
 b.  do the same for C1=1 and C2=2
 [note:  in order to do this, because this is a second order ode [i.e. has two derivatives in time] you'll need to set this up as a system, with y1 = f  and y2 = y1 ' = f ', so you get the system:
y2 ' = -9.8*sin(y1)
y1 '  = y2

y1(0) = C1
y2(0) = C2  ]


2.  A lake is eutrophic if it is high in nutrient input and plant production, has murky water, and is toxic.  The primary cause of eutrophication in the US is usually excessive phosphorus, due mainly to agrucultural and urban runoff.   A very simple model of the phosphorus concentration in a lake is given by

dp/dt = L - sp + r*( p^q / (m^q + p^q) )

where q can range from 20 for a shallow warm lake to 2 for a deep cold lake ( q is alway >= 2),  r is called the maximum recycling rate of phosphorus and m is the concentration of phosphorus at shich recycling is half the maximum rate, L is the rate of input of phosphorus from the watershed, and s is the rate of loss of phosphorus from sedimentation, outflow, and absorption by consumers or plants.

For Lake Mendota, near Univ. of Wisc. at Madison, s= 0.817/year, r= 731,000 kg/year, m=116,000 kg and q = 7.88 (though these are approximate, averaged and have alot of uncertainty).

a.  Find a first and second degree Taylor Series approximation to the right hand side of the equation, assuming p(0) = C and q = 7.88.

**Computer assignment
in parts a,b,c assume that the minimum feasible phosphorus input rate  per year is L = 3800 kg/year.  On all parts, be sure to take a long time span (like [0 50])

a. Use ode45 to compare the true solution, first order approximation, and second order approximation for the values given in part a above, with C = 80,000.
b.  Use ode45 to compare the TRUE solutions (only) for the cases where q = 2, 7.88, and 20 and C=4,000; 80,000; or 90,000.  What does this say qualitatively about the state of the lake over time in these different instances?
c.  Use ode45 to compare the true solution for the case with q=7.88 and C=4,640; 80,000; 90,000; and 800,000.   Again, what does this say about the state of the lake over time in each case?
d.  What changes if there is a new development which raises the minimum phosphorus input L to 8000 kg/year?  Assume here that C = 4,640; 80,000; and 800,000. 


Project Progress:  Today you need to turn in who you will be working with, if anyone, and have narrowed it down to three project topics (that you have agreed on with your partner(s))


3
1/28



Written Homework:  Do problems 3, 5, and 17 from Chapter 4 in Parkhurst.  (for part b of number 3, give a "closed form" solution... meaning give a formula for the solution function)
On problem 17, in paragraph 3, he's trying to tell you that H = CT + (possibly some constant), so heat and temperature are related in much the same was as concentration and mass.  Be careful to note that throughout the description he's talking about how the level of heat H in the house changes, but the final question asks for an ode describing how the temperature T changes.
***Let me know if you have spent more than half an hour working out any of these written problems.

Computer Homework: 
1.  Enter the system you derive for problem 3 into MATLAB.  You'll need values for q_a, q_b (note q_c = q_a + q_b), c_A, V, f, and the inital mass of Arsenic (As) in the water.   Start with
q_a = 2000 m^3/day
q_b = 3200 m^3/day
c_A = 0.18 g/m^3
V = 2.9 km^3 = 2.9 x 10^9 m^3
f = 0.1 (10%)
m(0) = intial mass of Arsenic = (.02 g/m^2) * (2.9 x10^9 m^3) = 5.8x10^7 g = 58,000,000 g = 58000 kg

A toxic concentration of As is considered to be around 0.05 g/m^2.  Does the lake become toxic?  If so, does it take long?
Try a couple of different values of f to see how strongly that effects the toxicity over time.
Try a couple of different values of c_A to see how strongly that effects the toxicity over time.
Try a couple of different values of m(0) to see it's effect on the toxicity over time.
 
***If you are having difficulty with this, please let me know ASAP.  If this takes you more than 2 hours to do the first problem, also let me know that as well, and ignore the second problem.  I don't want these MATLAB assignments to be a huge time sink, but as I said - after you are accustomed to using MATLAB, they should go pretty quickly.

2.  Choose reasonable values of V, q, and C_max for problem number 5 from PH and enter it into MATLAB.  Be sure to let me know what values you've chosen in one way or another (having them listed in the code is fine).  In order to determine the value of your first solution on the seventh day (to use it as the initial condition for the second solution), you can do the following:
if you have output your solution to your ode as
[T,F] = ode45(.....etc....);
typing
length(T)
will return the length of the vector  T.  let's pretend MATLAB gives you a value of length(T) = 45.
then F(45,1) will give you the final value of your solution function... so in general F(length(T),1) is the value of your solution at the final time.  Let me know if that doesn't make sense.


4

2/4

Written:
Taubes - Problems 1,3,4

Project Progress:  One page summary on your project including a brief description of the subject, two problems that you are interested in exploring and what you hope to accomplish, including proposed uses of MATLAB to explore your model.  Also include a reference list with at least two books that you will use to learn about your project.

5

2/11




Written: Taubes - Problems 1,3,4
Additional: A population of sandhill cranes has been modeled by a logistic equation with constant harvesting
dx/dt = rx(1 - x/K) - H
with carrying capacity of  K =194,600 members and a growth rate of r =0.0987 [1/yr]. Find the critical harvest rate for which constant harvesting will drive the population to extinction, and find the equillibrium population size under constant harvesting of 3000 birds per year.

Computer:
1. Find and classify the equillibria of the ode from the Lake Medota problem of homework #2 using MATLAB and the "fzero" function (hint there are three! one of which is unstable). What does this say about how you would develop policy to protect the lake from becoming toxic?   (Give a graph with several plots to illustrate the behaviorof the phosphorus levels for different inital  values that clearly indicate  where the equillibria are)

2.Use MATLAB to analyze the sandhill crane model. Show what happens to the population if the harvesting rate is above and below the critical rate (give a graph in matlab with two plots on it, one from each case). Also, assume that the habitat of the sandhill cranes is disrupted by development and so the carrying capacity is decreased to half the former value. How does this effect the evolution of the population over time when subjected to the same harvesting that you used in the first part of this problem (here give a graph from matlab with two plots - one for each carrying capacity)?

3. Imagine a small herd of cows in a field of modest size. The following example shows how the inital condition might affect the final outcome. May developed a model to describe the dynamics of the amount of vegetation V in the field (1974):
dV/dt = [rV(1-V/K)] - (HbV^2)/(V_0^2 + V^2)
where r = 1/3, K = 25, b = 0.1, V_0 = 3, H = 10.
Find all possible equillibria and determine their stability (notice you can use the "roots" function for this problem by setting the rhs equal to zero and multiplying through by V_0^2 + V^2 which gives you a degree 4 polynomial in V). Illustrate these equillibria  by solving the ode in MATLAB with different inital conditions and plotting on the same graph. How much does this change if H=20? H=30?

** Exam 1 is next week!
5.5
not to
be turned
in!
Find the nullclines, equillibria, and full phase plane diagram for the following model:
dS/dt = -bSI/(S+I) + gI
dI/dt = bSI/(S+I) - gI
(here S = number of susceptibles, I = number of infecteds and S+I = total population (which is constant in this case.. do you see why?)
You should have two cases here, one for b>g and one for b< g. 

What does b represent?
What does g represent?

Solution


6


3/4



Written Homework:  Do a full phase-plane analysis for the following two predator-prey models (note these a slightly more complex than the ones in class, but still very similar)
1.  dx/dt = x(1- x/30 - y/(x+10));  dy/dt = y(x/(x+10) - 1/3)
2.  dx/dt = x(1- x/30 - y/(x+10));  dy/dt = y(x/(x+10) - 3/5)

Computer work:
Write a MATLAB code to solve each system from the written homework, and for each one, choose a point from each region in your phase plane as your inital values and run to see the behavior of the populations over time.  Make sure to take a long enough time interval to see what the equillibrium behavior is. 

So, for system from 1, you should have 4 distinct regions in your phase plane (only do for x >=0 and y >=0).  You should run your code four times, taking a different intial value each time by choosing a point from each region in the phase plane.  Graph the evolution of the two populations over time on the same graph.  You should hand in your graphs (4) and your code, but  just give the code for one set of inital values (i don't need a seperate copy for each inital value you used).    Repeat for the system from problem 2. 
Project Progress:  Bring to class today  a summary of the work you have done so far on the project along with questions that still need to be addressed and any computer issues that have arisen that you would like help with.  (just 3 weeks till your first draft is due!  remember, it can take some time to do the writeup even if the work is done, so make sure you get the math done early enough to leave time to work on writing things up)


7

3/11
Choose one of the following readings and write a one to two page summary including the questions they are trying to answer, the model used and what the models say (meaning of terms, coefficients, etc), assumptions made, conclusions drawn.:  Reading 7.2, 8.1, 10.1, or 10.2

Look at the following model for the dynamics of smokers/ potential smokers/ and permanent quitters in a high school:
dP/dt = m - mP -bSP
dS/dt = bSP - (m + g)S
dQ/dt = gS - mQ

-Since the average length of time a person spends in the system is 4 years, let m = 1/4 [ year^-1]
-Assume that P,S and Q are the fraction of the population that is in each class (so 0 <= P,S,Q <=1).

1.  Show that the model is consistant with P+S+Q being constant.
2.  We don't really need the final equation for Q to know how the population Q changes.  Why?
3.  Explain what each of the coefficients and terms in the model represent.
4.  Do an analysis of the dynamics like we did in class on 3/4.  Meaning:  find a criterion for having dP/dt > 0 or < 0 that has the form S < ... or S > ....,  and similarily find a criterion for dS/dt to be > 0 or <0 that has the form P > ... or P < ... .  Give the four possible scenarios that result and describe what would happen to the dynamics of the population in each subcase. 
5.  Is there a possibilty for an "equillibrium cycle" here?  Why or why not?
6.  Suppose now that the total population is always 1000 students.  Can you come up with values of b and g that will guarantee that there is no one in the P class?  Can you come up with b and g so that no one will be a smoker?  (Try to not just guess and check - analyse the criteria for the signs of dP/dt and dS/dt  to force one or the other to always be positive or always be negative)
7.  Use Matlab to check the values you came up with in 6.  Turn in your graphs and a sample code for one set of values.


8
3/25
Taubes, pg 213-215: 1, 2, 3, 7, 8, 9,11, 12, 14, 15, 16,17
(i'm sure this looks like a lot, but these should go very quickly!  if they aren't let me know so i can help you)


9


4/1
 Taubes, pg 225:  6,7,8,9,10ace
 Reading 15.2 - pgs 237-243

Project Progress: First Draft of Project Writeup Due Today!

10
not
collected
Homework:  Taubes - 1,2,4acd
Parkhurst, Chapter 11, problem 12 (notice that they are assuming that the  insect distribution is independent of y if y measure the position along the road (so the only thing that matters is distance into the forest x)

Solutions

midterm 2


4/8

Second Midterm!  Will cover all material since first midterm (remainder of phase plane stuff, advection, diffusion and corresponding MATLAB code)

4/15
  tests back, discuss, wrap up any "loose ends", discuss projects

4/22
Project Presentations in Class today

4/29
 Final Written Project Due!