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 10^{6}

t = 20 yrs P = 2 x 10^{6}

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.