How to Solve a Population Growth Linear Equation

Population increase is generally proportional to population size.

The term "linear equation" in this context refers to the differential model for population saying that the rate of growth of a population is proportional to the size of the population. The closed-form solution of this equation is exponential and so this linear model describes exponential population growth. Although sometimes more complex models are required to describe population growth, using two data points, this model can provide a good fit for many situations.

Things You'll Need

• Calculator

Instructions

1. Solve the differential equation dP/dt = rP where P is the population as a function of t (time), and r is a rate constant for growth. This describes how the instantaneous rate of change of population is proportional to the population at that time. The solution for this equation is P(t) = P0 * e^(r * (t - t0)), where P0 is the size of the population at a given reference point in time t0, and e is the natural number.

2. Pick a reference point in time and use the population at that time for P0. The point in time you pick is used as the reference time (t0) for all other times for which you want to find the population. For example, if you took the world population in 1980 as your reference, then when trying to figure out the world population for 1990, you would use t - t0 = 10 years.

3. Determine the rate r using a second point in time for which you know the population. Knowing the population (P1) at time t1, then you can determine the rate r = ln(P1/P0) / (t1 - t0) where ln is the natural log function. Given this value for r, you have now solved the linear equation for population growth and can use it to predict the size of a population at various times.

References

• Human Population Growth