Our first model will be based on the following assumption:
The rate of change of the population is proportional to the population.
On the face of it, this seems pretty reasonable. When there is a relatively small number of people, there will be fewer births and deaths so the rate of change will be small. When there is a larger number of people, there will be more births and deaths so we expect a larger rate of change.
If is the population years after the year 2000, we may express this assumption as
where is a constant of proportionality.
Use the data in the table to estimate the derivative using a central difference. Assume that corresponds to the year 2000.
What is the population
Use your results from (a) and (b) to estimate the constant of proportionality in the differential equation.
Now that we know the value of we have the initial value problem
Find the solution to this initial value problem.
What does your solution predict for the population in the year 2010? Is this close to the actual population given in the table?
When does your solution predict that the population will reach 12 billion?
What does your solution predict for the population in the year 2500?
Do you think this is a reasonable model for the earth’s population? Why or why not? Explain your thinking using a couple of complete sentences.
Hint.
Small hints for each of the prompts above.
Answer.
or in the year 2056.
billion.
Solution.
We let be the population after year 2000 with where is a constant of proportionality.
Using the data in the table,
Since corresponds to the year
Using and the preceding values at we have so
The solution for the initial value problem is
The year 2010 corresponds to So The model predicts that the population in 2010 will be about 6.888 billion.
The population will be 12 billion when So we solve the equation for We see that so The population will reach 12 billion during the year 2056.
The year 2500 corresponds to So The model predicts that the population in 2500 will be about 3012 billion or about about 3 012 300 000 000. This is rather unreasonably large.