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 The Mathematics of Population Increase

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(Most recent edit 06/12/2013)

The equations below can be used to calculate population growth rates in a simple, easily understood way. There are better mathematical treatments for real-world applications, like the Logistic Function to describe a system with limited resources, or the many cases where the birth and death rates aren't proportional to each other, but this page should establish an accessible grounding in the basic ideas.

Basic Relationship

We will use the equation below to derive all the other forms:

(1) $ \displaystyle \log\left(\frac{N}{N_{0}}\right) = r t $
Where:

Source: Wolfram Math World: Population Growth

Derived Forms

Here are the forms of equation (1) in terms of each of its variables:

(2) $ \displaystyle N = N_{0} e^{\left(r t\right)} $ (future population)

(3) $ \displaystyle N_0 = N e^{\left(-r t\right)} $ (present population)

(4) $ \displaystyle t = \frac{\log\left(\frac{N}{N_{0}}\right)}{r} $ (time)

(5) $ \displaystyle r = \frac{\log\left(\frac{N}{N_{0}}\right)}{t} $ (rate)

Note that e in the above equations is the base of natural logarithms, and log(x) refers to the natural logarithm of x.

Doubling time

At the present world population growth rate of 1.7% per year, how long will it take to double the world's population?

The appropriate equation for this case is (4) above, with the following arguments:

$ \displaystyle t = \frac{\log\left(\frac{N}{N_{0}}\right)}{r} = \frac{\log\left(\frac{2}{1}\right)}{0.017} = 40.773$ (years)

This equation shows that it will take about 41 years to double the world's population. If this prediction is borne out, there will be 11 billion people on Earth in 2034 (based on the 1993 population of 5.5 billion).

Rate of Increase

How many more people join us each day?

The base equation for this case is (2) above:

$ \displaystyle N = N_{0} e^{\left(r t\right)} $

But because we want the increase per day, we will use this form:

$ \displaystyle N_d = N_{0} e^{\left(r t\right)} - N_0 $

For this problem we will use 5.5 billion (1993) for the initial population N0, 0.017 for the rate of increase r, and 1/365 for the argument of time t, since we want the rate of population increase per day (Nd). The result:

$ \displaystyle N_d = N_{0} e^{\left(r t\right)} - N_0 = 256170.349$ (people per day)

Every day, about a quarter million people join us on planet Earth.

Comments

So, given this fantastic population increase, one might ask "What are they doing about it?" There are two answers to this question: (1) nothing, and (2) We are the "they" in the question. It makes no sense to blame third-world countries for uncontrolled population growth. America has 5% of the world's population but consumes 25% of the world's resources. So, in terms of resources used, each new American born equals five world citizens.

The outcome of the math isn't surprising. What is surprising is the fact that many people think we can solve this problem by conserving resources, recycling and so forth. Conservation programs are worthwhile and should be pursued, but they only treat the symptoms of the disease. Addressing resource issues without also confronting the population explosion is what I call "placebo environmentalism."

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