# Population Growth

Applications of Differential Equations

Population Growth

Populations tend to have a growth rate proportional to the present popluation. This is true whether we’re talking about a the population of a country or a colony of bacteria or a herd of deer.

If we call the population, P, we can write this as a differential equation:

dP/dt = kP

where k is the constant of proportionality. This constant k is a function of societial factors for the growth of a human population, but we can determine it if we have information about previous growth rates. Let’s work a problem.

Problem:    The population of a certain country is 2 million and has doubled in the last 20 years. Find the expected population in 80 years.

First summarize the given information:

When          t  = 0 yrs             P = 1 x 106

t   = 20 yrs         P = 2 x 106

t = 100 yrs        P = ?

And             dP/dt = kP

Now we separate the variables of our differential equation:

dP/P = k dt

Then integrate and solve for P:

ln P = kt + C

P = e kt + C

P = C e kt

Refer back to previous example if you are unclear on this step.

Now use the given information to solve for two constants, k and C.

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