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HSEV 101- CHEM 317
Homework No. 3 – Due October 16, 2013
Exercises in Population Dynamics
The following exercises are based on the data represented in Fig. 8-1 in Wright and Boorse’s Environmental Science. According to these data, the size of the human population has reached a value of N = 6 billion in 1999. According to the Figure the population size between 1980 and 2100 agrees very well with the logistic model using a carrying capacity, K, of 11.5 billion, and an initial value of ro = 0.036/year (or 3.6% per year) for the population growth factor. (Of course, in 1999 the current growth factor, r, is already significantly smaller because of the approach to the carrying capacity.) Note that the good agreement between the model and the data does not extend before to the period before 1980, because improvements in public health in the second part of the 20th century resulted in a decline in mortality rates, thus causing a rise in the value of r.
1. Plot the population size of the world as a function of time between the years 1900 and 2100 according to the exponential (Malthusian) model of Equation (2). Start with a population size of No = 0.17 billion in 1900. Use a constant value of r = 0.036/year.
2. Plot the rate of change in population size of the world as a function of time between the years 1900 and 2100, dN/dt, according to the exponential (Malthusian) model of Equation (1) and the data of Problem No. 1.
3. Calculate the doubling time of the population corresponding to the value of r in Problem No. 1 according to Equation (2a).
4. Plot the population size of the world as a function of time between the years 1900 and 2100 according to the differential form of the exponential (Malthusian) model (Equation (1)) and the data of Problem No. 1. [To use Equation (1), first calculate dN/dt for the year 1900, combine it with the population size for that year No to obtain N for the year 1901. (Since each step of the calculation is performed for a period of 1 year, the population increase dN has the same numerical value as dN/dt.) Use N for 1901 to calculate dN/dt or dN for that year, and add it to N for 1901 to obtain a value of N for 1902. Repeat the calculation on a yearly basis until you reach the year 2100.]
5. Plot the population size of the world as a function of time between the years 1999 and 3000 according to the integrated form of the logistic model (Equation (16).) Use values of No = 0.35 billion for the population in 1900, an initial value of ro = 0.036/year for the population growth factor, and a carrying capacity of K = 11.5 billion.
6. Plot the rate of change in population size of the world as a function of time between the years 1900 and 3000, dN/dt, according to the differential form of the logistic model (Equation (14)) and the data of Problem No. 5.
7. Plot the population size of the world as a function of time between the years 1900 and 3000 according to the differential form of the logistic model (Equation (14)) and the data of Problem No. 5. [To use Equation (14), first calculate dN/dt for the year 1900, combine it with the population size for that year No to obtain N for the year 1901. (Since each step of the calculation is performed for a period of 1 year, the population increase dN has the same numerical value as dN/dt.) Use N for 1901 to calculate dN/dt or dN for that year, and add it to N for 1901 to obtain a value of N for 1902. Repeat the calculation on a yearly basis until you reach the year 2100.]
8. Compare the plot of population size as a function time obtained according to the differential form of the logistic model (Problem No. 7) with the corresponding plot obtained according to the integrated form of the logistic model (Problem No. 5).
9. Plot the population size as a function of time between the years 1900 and 3000 according to the delayed regulation model of Equation (18). Use the data for No (in the year 1900), ro, and K as those given in Problem No. 5. Obtain separate plots for each of the following values of the time lag, T: 0, 20, 30, 40, and 50 years. [For a value of T = 30 years, for instance, the calculation of N as a function of t is identical with the calculation with T = 0 for the first 30 years, between 1900 and 1930. However, in 1930 the value of N(t-T) in Equation (18) is N in 1900, for the year 1931 N(t-T) in Equation (18) is N in 1901, etc.]
10. Based on the results of Problem No. 9, describe the effect of the lag time, T, on the population-vs.-time dependence.
11. Plot the population size of the world as a function of time between the years 1900 and 3000 according to the delayed regulation model of Equation (18), but that instead of the initial value of the population growth parameter of the world used in the previous problems, ro = 0.036/year, use a value larger by 50%, i.e. ro = 0.054/year. Use a lag time of T = 30 years. In this calculation, use a value of No = 0.06 billion for the year 1900. As in previous problems, use a level of K = 11.5 billion for the carrying capacity. [The higher value of ro = 0.054/year is typical of many countries in Africa and South Asia.]
12. Based on the results of Problem No. 11 for the case of T = 30 years, describe the effect of the initial value, ro, of the population growth parameter, on the population-vs.-time dependence.
13. What are the general conclusions regarding population trends that you can obtain from the previous problems?
14. A certain island has a constant and rapidly renewable supply of grass A = 120 tons. Initially, X = 30 sheep and Y = 30 wolves live on the island. The rate constant of grass consumption and sheep reproduction, ka, is 0.001 per month. The rate constant of sheep consumption and wolf reproduction, kb, is 0.002 per month. The death rate constant of the wolves, kc, is 0.06 per month.
Plot the population sizes of sheep and wolves as a function of time over a period of 500 months using Equations (7) and (10) at constant intervals of 1 month. To use these equations, first calculate dX (which has the same numerical value as dX/dt for an interval of 1 month) and dY (which has the same numerical value as dY/dt for an interval of 1 month) based on the initial values of A, X, and Y. Then add dX to the existing value of X and dY to the existing value of Y to obtain new values of X and Y. (A remains constant). Repeat these procedure 500 times and plot X and Y as a function of t.
Problem No. 1 – Partial solution
1. Open Excel, press “File” tab, then “New” and select “Home” tab.
2. Enter “Year” in cell A1, “Year since 1900” in C1, and “Pop,billion” in B1.
3. Enter “1900” in A2, “=A2-1900” in C2, and “=0.17*(EXP(0.036*C2))” in B2.
4. Enter “1901” in A3.
5. Highlight the two cells A2 and A3 simultaneously. A “drag handle” (+) appears at the bottom right corner of A3. Drag the handle all the way to cell A202 to fill in all the years from 1900 to 2100.
6. Highlight B2 and C2 simultaneously. Press “Copy”.
7. Highlight the block of cells from B3 to C202. Press “Paste” and “Paste”.
8. Select “Insert” tab.
9. Highlight the block of cells from A1 to B202.
10. Press “Scatter” and the data-point diagram without data lines directly under “Scatter”.
11. Select “Chart Tools – Layout” tab.
12. Select “Chart Title”, then “Above chart” and enter the title “Integrated Malthus Model”.
13. Select “Axis Titles”, “Primary Horizontal Axis Title” and enter the title “Time, years”.
14. Select “Axis Titles”, “Primary Vertical Axis Title”, “Rotated title”, and enter the title “Population, billions”.
15. Start a file labeled “Population Exercises” and save the chart “Integrated Malthus Model” in this file.
Problem No. 4 – Partial solution
1. Open Excel, press “File” tab, then “New” and select “Home” tab.
2. Enter “Year” in cell A1, “Year since 1900” in D1, “Pop,billion” in B1, and “Pop change” in C1.
3. Enter “1900” in A2, “=A2-1900” in D2, “0.17” in B2, and “=0.036*B2” in C2.
4. Enter “1901” in A3, “=A3-1900” in D3, “=B2+C2” in B3, and “=0.036*B3” in C3.
5. Highlight the two cells A2 and A3 simultaneously. A “drag handle” (+) appears at the bottom right corner of A3. Drag the handle all the way to cell A202 to fill in all the years from 1900 to 2100.
6. Highlight B2, C2 and D2 simultaneously. Press “Copy”.
7. Highlight the block of cells from B3 to D202. Press “Paste” and “Paste”.
8. Select “Insert” tab.9. Highlight the block of cells from A1 to B202.10. Press “Scatter” and the data-point diagram without data lines directly under “Scatter”.11. Select “Chart Tools – Layout” tab.
12. Select “Chart Title”, then “Above chart” and enter the title “Differential Malthus Model”.
13. Select “Axis Titles”, “Primary Horizontal Axis Title” and enter the title “Time, years”.14. Select “Axis Titles”, “Primary Vertical Axis Title”, “Rotated title”, and enter the title “Population, billions”.
15. Save the chart “Differential Malthus Model” in the “Population Exercises” file.
