Limited Population Growth

Part 1: Background Information

You may have studied recently the "natural" growth of biological populations, starting with assumptions of growth rate proportional to the population and no restrictions on growth. As you know, these assumptions lead to a model formula that is exponential. In this module we model a population for which there is a limit on the population size. Rather than start from data on a real population, we start from a theoretical assumption about the growth rate, and we study what this assumption predicts about the population.

Our Theoretical Assumption: Let M be the maximum population that the environment will support. We assume (as biologists often do) that the rate of change of the population is proportional to the product of

• the population and
• M minus the population.

• In your worksheet, write this Theoretical Assumption as a formula, using c to denote the proportionality constant and P to denote the population.

Here are two reasons for assuming such a relationship:

• When the population P is small relative to M, the difference M - P is essentially the same as M (a constant), and then dP/dt is essentially proportional to just the population P. This is the "natural growth" situation you have studied already. Hence, for small populations, our new model should result in population predictions that will look a lot like "natural population growth." This is what a biologist would expect to happen with most populations.
• At the other extreme, when the population gets close to M, the factor M - P is close to zero, and this effectively shuts down further growth. Thus, the factor M - P captures in a natural way the effect of a limited environment.

In the next Part we consider a way to generate numerical and graphical solutions to any problem of the form

Find a function P = P(t) such that

• dP/dt = some formula in P, and
• P has some known value at the starting time.

This problem has two pieces of given information:

• an equation involving dP/dt, called a differential equation, and
• a starting value P₀ for P at the starting time t = 0; P₀ is called an initial value.

Such a problem is called a differential-equation-with-initial-value-problem, which is usually abbreviated to initial value problem.

In Part 3, we will apply our solution procedure to a simulated population of fruit flies in a laboratory environment.

Last modified: September 20, 1997