In this video I go over an example on population growth and this time try to model the world population during the 20th century (that is from the year 1900 to 2000). The first model involves estimating the relative growth rate from the 1910 population, but the accuracy becomes an issue in the more recent years. Similarly, with the second model, which uses the 1950 population to estimate the relative growth rate, despite this model being more accurate than the first model, it is still not that accurate in the recent years. But regardless of the accuracy of these models, this is a good video in understanding how to go about applying the differential equations concepts to creating a model for population growth. In later videos, I will go over more accurate models so stay tuned for those!
NOTE: I made a mistake at 7:00: should be 0.6% and NOT 6%.
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Example
Assuming that the growth rate is proportional to population size, use the data in the table below to model the population of the world in the 20th century.
What is the relative growth rate?
How well does the model fit the data?
One way of estimating the relative growth rate, k, is using the year 1910 which has a population of 1750 million.
In other words, the relative growth rate is about 0.6% per year.
The model fits the data well for about the first 30 years but then becomes less and less accurate in later years. In fact the model is off by over a factor of 2 in the year 2000.
Can we make a better model?
Well, another possibility for estimating k is to use the given population for 1950 instead of the year 1910.
Thus the relative growth rate is about 0.88% per year.
Thus, although the new model is more accurate over a longer time period than the first model, it is still lags behind in recent years.
And the model is off the mark in the year 2000 by just over 2 billion people!