Previously, we looked at the exponential growth model for a population P. The constant k represents a reproduction parameter.

In this model the population growth continues unabated.

Although such a model may be accurate during the early stages of growth, as time goes on, other factors, such as food supply, will begin to affect the real growth of the population.

A better description of population growth is given by the logistic equation, which includes an inhibiting factor P/K.

Suppose the value of P is much smaller than the constant K.

If the population starts out at a value much less than K, the beginning of logistic growth is similar to exponential growth.

Now, let's look at the slope field for a logistic differential equation.

A particular solution curve corresponding to a given initial population is shown on the slope field. Move the slider to change the initial population value. The value of P/K is shown for each curve.

The constant K is known as the carrying capacity. It is the maximum population an environment can sustain in the long term.

Notice that if the population starts out less than the carrying capacity, the population increases at first, and then levels off at the carrying capacity.

if the population starts out greater than the carrying capacity, the population decreases at first, and then levels off at the carrying capacity.

If the initial population is equal to the carrying capacity, then the rate of change of the population is zero.

Therefore, the population remains constant over time.

The logistic equation is separable, so we can rearrange the differentials.

To solve the equation, we take the integral of both sides.

The solution of the integral on the right-hand side is trivial.

The integral on the left-hand side can be solved using the method of partial fractions.

After determining the general solution, we can impose the initial condition that the population is P0 at time t equals zero.

We then are able to express the population P in terms of P0.

We see that as the time approaches infinity, the population P approaches the carrying capacity K.

Now, consider this logistic equation that describes the growth of a certain population of animals.

The initial population of the animals is 200.

Making a graph of our results, we can see how close the animal population is to its carrying capacity at the given time.

How does the logistic growth model compare to the exponential growth model in this case? Click the "Submit" button after selecting your answer.

What does the constant K appear to be? Click the "Submit" button after selecting your answer.

If the population is initially equal to the carrying capacity, what happens to the population over time? Click the "Submit" button after selecting your answer.

What is the solution of the integral on the left-hand side? Click the "Submit" button after selecting your answer.

What is the population size at time t equals forty? Click the "Submit" button after selecting your answer.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education