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                                                Growth Rates in Populations

            A simple approach to populations considers the changes through births, deaths, immigration and emigration to arrive at the changing population.  Through census data the population can be measured.  Countries that are facing rapid increases (India, Africa) or that have large populations (China) need to constantly keep track of their population numbers and rates at which they change.

            Thanks to our mathematics friends many scientific fields like physics, and other fields such as business and psychology can keep track of important numbers and the trends they represent.  In this activity we will use a few equations that have been developed and have proven to be useful to population demographers and those studying environmental issues.  We will not worry about the derivations (Mr. Dye and others can) but will use these equations to learn about the changes in populations.

 

Rate of change-  How fast populations change their rate is essential if one is to be able to plan ahead for roads, jobs, schools, etc.   To determine the rate of change in a population one useful equation is:

                                    R = (lnP2 – lnP1) / t

                        Where:  R = the intrinsic rate of increase

                                      lnP1 = the population at a first time interval

                                      lnP2 = the population at a second time interval

                                      t = time in years (any time actually)

 

This formula allows us to calculate the rate of increase.  If a percentage increase is desired you only need to multiply the answer by 100 !

 

Doubling Time-  A frequently used standard for population changes is the doubling time of a population.  This is actually analogous to the half-life in a nuclear reaction.  To determine the time for a population to double the useful equation is:

 

                                    t = 0.69.3 / r   ( if you use the r as a % then use t = 69.3 / r )

                        Where:  t = time

                                      R = the intrinsic rate

                                      0.693 or 69.3 is a constant

Future population numbers can be calculated using a formula that is often used for investors who wish to know how much their money will be worth.  Here we are concerned with future populations. 

 

 The future population equation is:

                                    P =  Po ert          

Where:  P is the new population

             Po = the original population

             e = natural log

             r  = rate

             t  =  time

Part A:  Doubling time calculations.  The table below has estimates for the size of the human population at various stages of history.  Use your calculator to determine the growth rate and doubling time.

 

Period

Population

Rate

R = (lnP2 – lnP1) / t

 

Doubling Time

t = 0.69.3 / r  

300,000 BC

1 million

 

 

10,000 BC

3 million

 

 

1 AD

200 million

 

 

1650 AD

0.5 billion

 

 

1900 AD

1.6 billion

 

 

1950 AD

2.4 billion

 

 

2000 AD

6 billion

 

 

 

 

Part B:  Comparison of initial population and rate of increase.  Use the table below to develop population growth histories at different rates and different initial population sizes.  Use the following equation: P =  Po ert          Graph the results using a linear scale.  USE GRAPH PAPER !

 

Year

0

R=1, Po=100

R=1, Po=200

R=1, Po=1000

R=2, Po=100

R=10, Po=100

1

 

 

 

 

 

2

 

 

 

 

 

3

 

 

 

 

 

4

 

 

 

 

 

5

 

 

 

 

 

10

 

 

 

 

 

20

 

 

 

 

 

30

 

 

 

 

 

40

 

 

 

 

 

50

 

 

 

 

 

 

Part C:  Questions/Conclusions:

1.                  Based on your doubling time calculations what conclusions can you make about the future world population numbers? 

2.                  Based on your comparisons in part B, what is the only option for a country such as China with a present world population of 2+ billion persons?  What about Africa with a growth rate as high as 7%?

3.                  Which seems to have the greatest impact on long term populations, rate of increase or the initial size of the population?