Logistic Growth

In a population showing exponential growth the individuals are not limited by food or disease. However, in most real populations both food and disease become important as conditions become crowded. There is an upper limit to the number of individuals the environment can support. Ecologists refer to this as the "carrying capacity" of the environment. Populations in this kind of environment show what is known as logistic growth.

Click the following button to run an applet you can use to experiment with logistic growth. If you are accessing this lesson over a slower network connection it may take several seconds for the applet to appear.

You must have Java installed and enabled to run this.

This applet is similar to the one you used with exponential growth. We assume you have have already worked with the Exponential Growth applet and are familiar with its controls and their functions. The only new field present is the carrying capacity field which is initialized at 1000.

While in the Habitat view, step the population for 25 generations. The population becomes quite crowded, but it levels off near 1000 individuals. Switch to the Graph view and you will see the classic "S" shaped curve that is characteristic of logistic growth. While the population is small it shows exponential growth. However, as the population approaches the carrying capacity the growth slows down and the curve levels off.

One way to think about the carrying capacity is that it modifies the "effective" birth rate. As the population reaches the environment's carrying capacity the effective birth rate declines until it is 1.0 and each individual is just replacing itself in the next generation.

Experiment 1:

In this experiment and those following you should leave the applet in the Graph view to help interpret the results.

Click Reset All to clear any past data. Make sure the birth rate is set to 1.5. Now do a series of simulations using carrying capacities of 200, 400, 600, 800, and 1000. Step or Run the population for about 30 generations with each value of carrying capacity. Be sure to click the Reset button between each simulation in order to start over but leave the past results in place. Do this now.

Notice that all the curves look similar for the first 10 or 12 generations. It is not until the population begins to near its carrying capacity that the curve deviates from and exponential growth curve.

Experiment 2:

Click Reset All to clear any past data. Reset the carrying capacity to 1000. Now do a series of simulations using birth rates of 1.2, 1.4, 1.6, 1.8, and 2.0. Step or Run the population for about 30 to 40 generations with each value of birth rate. Be sure to click the Reset button between each simulation in order to start over but leave the past results in place. Do this now.

Changing the birth rate affects how quickly the population reaches its carrying capacity but has no effect on to final population size achieved.

Experiment 3:

Click Reset All to clear any past data. Make sure the carrying capacity is set to 1000. Set the birth rate to 3.0 and Step or Run the population for 40 generations.

Here we see something new! The population appears to be oscillating around the carrying capacity. What's going on? With birth rates greater than 2.0 the population will overshoot the carrying capacity and have more offspring than the environment can support. The following generation, because of this overcrowding, the effective birth rate will be much less than 1.0 and the population will decline below the carrying capacity. Then it will overshoot again and the cycle will be repeated. You will notice that with time the overshoots get less and less and the population still apears to converge on its carrying capacity.

Experiment 4:

There is a wide range of interesting behaviors this simple model can show. For birth rates greater than 3.0 the population will appear to oscillate between two or more points. Unlike the situation at 3.0 and below, the population will never converge to one stable population size. Experiment with different birth rates greater than 3.0 and see what happens

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