Population Increase


(Adapted from Consider a Spherical Cow by John Harte.)

Below is a problem which is partially worked out. Finish the problem, answering the questions as you go along. Don't panic if you get stuck; we will discuss this in the next class and then you will turn in a corrected solution the following class period (email is good but send it so it gets to me before the next class). Since probably everyone will have the same nearly the same numerical answers, points will be awarded on the thoroughness of your discussion and analysis.

Problem: Suppose the human population grows exponentially. (This is not exactly true but is approximately true for humans and some other populations.) In other words the number of people is N(t) = N(0) ert where t is time N(0) is the number at t = 0 and r is the rate parameter. Given the table of population data below, how long will it take for the human population density on the land surface of the Earth to equal the present density in typical urban areas of the world?

Solution:

If we take the natural (base e) logarithm (ln) of both sides of the exponential equation above we have:

ln [N(t)] = ln [N(0)]  + rt 

which would be a straight line graph if t is plotted horizontally and ln [N(t)] vertically. The y-intercept is ln [N(0)] and r is the slope. Make a third column and fill it in with ln(population).

Question 1: plot the year on the x-axis and ln(population) on the y-axis (you will turn this graph in).

Table:

year         population (billions = 109)    
1650         0.5
1850         1.1
1900         1.6
1910         1.7
1930         2.0
1950         2.5
1960         3.0
1970         3.6
1980         4.5
1990         5.3
2000         6.0

Question 2: Is the graph close enough to a straight line to justify the assumption of exponential growth? Justify your answer.

Question 3: What is the rate of increase (do a least square fit on the data to get the slope, r, of the graph- this is the growth rate)? What are its units? Multiply this number by 100 to get the percentage growth rate.

Very large urban areas are something like 104 people per square kilometer. The total land mass of the Earth is around 1.5x108 km2. 

Question 4: Verify these two numbers. You can do this by finding them on the Internet (or in the library!) or making an approximation using some simple estimates.

If we use the above total land mass and the data for 1980 in the table we have that the population density was 4.5x109 /1.5x108 km2 = 30 people per km2. If we assume the available land mass does not change then the density increases at the same rate the population does.

Question  5: In order to find when the density reaches 104 people per km2 solve 104 = 30 e rT for T using the r value from your graph. Hint: Divide both sides by 30 and take ln of both sides. How many years away is this?

Question  6: Assume the average mass of a human is 70 kg. When will the mass of people living on the earth equal the mass of the earth?

Question  7: List each of the assumptions made in this problem and discuss the effects each one has on the result (For example, suppose the average human mass is only 60 kg. How does that effect your calculations?)