Homework D2: Single-Variable Dynamical Systems First, do this problem: #1: Consider the following simulated data on bacteria growing in a petri dish. Population is measured in milligrams. Hour Population 0 1.00 1 2.06 2 4.10 3 7.57 4 12.13 5 15.57 BIG HINT: do NOT use excel's auto-fill feature to extend the data series. That is, if you got the numbers 17.72, 20.76, etc. you're doing it wrong. a) Figure out the carrying capacity and the appropriate multiplicative factor. Hint: look at the graph of delta a_n versus a_n and eyeball or compute a curve for it. A little bit of algebra might be required. You could do it by hand or use wolframalpha.com for it, or do both! b) Using your results in part (a) and the population at hour 4 of 12.13, predict the population at hour 5. You should get 15.57, obviously, because the data is artifical and perfectly follows a model. c) Now predict the population at hour 6. Then, choose 2 of these following questions to do. Note that some of the data sets don't actually fit the models we might hope that they fit, and that's okay-- it doesn't mean you get a bad grade on that question. If you would like to substitute a problem of your own choice on this homework, just let me know. The Variable-Rate question: Which grows faster: * a population that grows at a steady 3% each year, or * a population that grows at 0% one year, 6% the next, 0% the next, 6% the next, etc--thus, "averaging" 3% ? * can you find a better way to average 0% growth and 6% growth to get a single number that is the long-term growth rate? The US Population Question: Use the US Census population values from 1790 to now and fit a limited-population-growth dynamical system, where the carrying capacity and growth constant are the decision variables. Comment on the resulting carrying capacity. The Turkey Question: Pick one of the turkey-cooking data sets given in the powerpoint slides and fit a dynamical system to it. Comment on how the fitted ambient temperature matches (or not) the oven setting. If you have time to fit both data sets, comment on how the fitted parameters compare across the data sets. The Very, Very Sad question: A recent news story about parents leaving their children in hot cars included a graph of how the temperature in the car changes with time: http://www.indystar.com/article/20120709/LOCAL/207100318/Inside-hot-car-can-become-unbearable-just-10-15-minutes Fit a dynamical system model to it. It will need to be a little more complicated than Newton's law of heating/cooling. The Harvesting Question: Consider a limited-population-growth model with a carrying capacity of 100 and a growth constant that gives a reasonable growth curve. What is the maximum harvest amount (average amount per year) that you can sustainably capture? Explain how this works. Optional: What if you try something fancy, like alternating between two different harvest amounts? The Finance Question: a) Consider a mortgage for $100,000 with a monthly payment of $536.82 for 30 years. What must the yearly interest rate be? b) Here is an ad for Rentway.com, a rent-to-own retailer. I will give you the entire text of the ad; some parts may be irrelevant. Of course, the real ad (I can show it to you) has some things in BIG PRINT and others in much smaller print :-) "Suite" Dreams Sale: Get a beautiful 6-pc. bedroom...including a queen mattress set! SALE! 6-pc. bedroom including mattress, 1 low price $19.99 a week. 104 weeks for a total cost of $2,086.96. cash price $1039.48 The fine print mentions miscellaneous fees which you can ignore. So my question is: what is the effective annual percentage rate? You may have to guess various rates and try to narrow it down, rather than compute it in one big step. You could use Goal Seek or Solver in Excel to help. c) Consider the following strange kind of loan: borrow $400, pay $40 every 2 weeks, and it's paid off in 1 year (26 payments). What must the yearly interest rate be? This is essentially the way "payday loans" work. You borrow $400 for a "fee" of $40, and pay it back in 2 weeks. If you don't have the $400 to pay back, you can "roll it over" and have another 2 weeks to pay it off, for another $40 fee of course. Many people do this over and over. If you did it for a whole year, it would essentially be the scenario described above. The Scaling Question: a) Suppose you have a population-growth model with a growth rate of 1% per year. Now you want your time steps to be months instead of years. How do you change the growth rate to get the same behavior? b) Suppose you have a carrying-capacity model whose time steps are in decades, with a growth constant of 0.005. Now you want to change the time steps to years. How do you change the growth constant to get the same behavior from the model? Is it exactly the same, or just close? c) Suppose you have a carrying-capacity model whose carrying capacity is 100, with a growth constant of 0.005. You have been saying to yourself that the population is in units of "thousand animals", so that the actual environment has 100,000 animals. Now you want to change from units of thousand-animals to single-animals, so the new carrying capacity in the model is 100,000. How do you change the growth constant to get the same behavior? Is the behavior exactly the same, or just close? d) (optional) Starting with the S-I-R model we did in class, change the time steps from weeks to days by changing the various parameters. Do you get exactly the same behavior, or just close? The Logistic Question: The carrying-capacity model generates data that look a lot like a Logistic curve. Are they the same? Explore. The Facebook Question: (thanks to a previous class member for finding this data!) Here are some statistics on Facebook membership. Do some math modeling with them, using dynamical systems and perhaps more. Ask good questions and answer them. Year Members(millions) 2004 1 2005 5.5 2006 12 2007 50 2008 100 2009 300 2010 550 2011 800 Ben Phoster. 12 April 2012. How Many Users on Facebook? Available from http://www.benphoster.com/facebook-user-growth-chart-2004-2010/