SYST 201
SYST 220: Dynamical Systems
HW 1
1. Consider the population of the United States. Suppose that (aside from immigration) the population increases by 4 percent each year – that is, for every 100 people who start the year living in the U.S., there are 104 people living in the U.S. a year later due to births and deaths. In addition to this intrinsic growth rate, people also immigrate to the U.S. Suppose that 1 million people immigrate to the U.S. every year. Suppose that the U.S. population at the start of year 2005 is 260 million.
a. Give a dynamic systems model for the population of the U.S. every year.
b. Using a spreadsheet/calculator, what will be the population of the U.S. in the year 2010?
2. You have a mutual fund that earns 8% interest annually. At the beginning of each year, you add $1,200 to the account.
a. Create a dynamic systems model for the amount of money you have in the account at the end of each year. Define all variables.
b. Suppose you start saving when you are 20. Using a spreadsheet/calculator, determine how much money you will have when you are 30.
c. Suppose instead, you save $4,000 each year, starting when you are 25. How much money will you have when you are 30? Which is a better savings plan?
d. What is your current age? Determine how much money you would need to save each year to have $100,000 at 30.
3. A certain lake contains a population of fish. The fish reproduce every year in the spring. Each fish gives birth to (on average) 3 new fish every year. In the summer, people come to fish at the lake. Each year, they catch a total of 220 fish. The lake starts with 100 fish.
a. Create a dynamic systems model for the number of fish in the lake at the end of year n. Define all variables. Specify any assumptions you are making.
b. Using a spreadsheet/calculator, determine how many fish are in the lake after 10 years.
c. Suppose that each year people catch a total of 340 fish (instead of 220). How many fish are in the lake after 10 years?
d. Suppose you are in charge of setting the maximum number of fish that people can catch each year so that the system stays in equilibrium. How high can this maximum be to maintain the fish population?
4. Each day in the Washington metro area, cars release 10,000 tons of nitrogen oxides (NOx) into the air. Also during the day, ambient winds reduce the quantity of nitrogen oxides in the air by 40%.
a. Create a dynamic system to model NOx in the atmosphere over time.
b. Using a spreadsheet/calculator, determine how many tons of NOx will be in the air after 20 days.
c. To meet air quality standards, the metro area must have an average of 8,000 tons of NOx in the air each day. What must the total daily NOx car emissions be to achieve this goal? (hint: Force the equilibrium point to be at 8000).
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