Analyzing the Logistic Model's Behavior

The Logistic Model

Exponential growth assumes that a population can continue growing at a rate proportional to its current size forever. Real populations usually cannot do that. Food, space, nutrients, and other resources eventually become limiting.

The logistic differential equation incorporates this limitation:

Here, represents the population at time , is the intrinsic growth rate, and is the carrying capacity — the population size that the environment can sustain in the long run.

The most important feature of the model is the factor . It changes the growth rate depending on how close the population is to its carrying capacity.

Understanding the Growth Rate

The differential equation tells us the instantaneous rate of change of the population:

To understand the behavior of the population, focus on the signs of the factors. Assuming and , the sign of depends entirely on the value of .

When the Population Is Below Carrying Capacity

If , then is positive. Therefore:

The population is increasing. The farther below the carrying capacity the population is, the larger the factor becomes.

When the Population Equals Carrying Capacity

If , then:

The population is not changing. The carrying capacity is therefore an equilibrium solution.

When the Population Is Above Carrying Capacity

If , then is negative. Therefore:

The population decreases toward the carrying capacity. This is an important feature of the logistic model: populations below tend to move upward toward it, while populations above tend to move downward toward it.

The Equilibrium Solutions

An equilibrium solution occurs when the population remains constant. In other words, we need:

For the logistic equation, there are two equilibrium populations:

The solution is usually the biologically relevant stable equilibrium. The solution is also an equilibrium mathematically, although a population of exactly zero remains zero forever in this idealized model.

Stability: Which Way Does the Population Move?

A particularly useful AP Calculus skill is determining whether an equilibrium is stable or unstable.

For the logistic model, consider three regions:

This means that is stable: populations on either side move toward it.

By contrast, is unstable in the mathematical model. A population slightly above zero grows away from zero, while a population slightly below zero decreases further away from zero.

Where Is the Population Growing Fastest?

The population grows fastest when is as large as possible.

Since:

we can view the growth rate as a quadratic function of . The maximum occurs at the vertex of that quadratic.

Therefore, the population has its maximum growth rate when it reaches half of the carrying capacity.

Substituting into the logistic equation gives the maximum growth rate:

This is one of the most important facts about logistic models to remember:

Why Growth Starts Slowly, Speeds Up, Then Slows Down

The logistic model produces the familiar S-shaped, or sigmoidal, population curve.

That shape makes sense if we think about the two factors in the differential equation:

When the population is very small, there are relatively few individuals reproducing, so the population grows slowly.

As the population increases, there are more individuals available to reproduce, while resources are still relatively abundant. Growth accelerates.

Once the population becomes large, however, the factor becomes smaller. Competition for limited resources slows the growth.

Eventually, the population approaches , where the net growth rate approaches zero.

Concavity and the Inflection Point

The logistic curve changes from concave up to concave down. The transition occurs at the same population size where the growth rate is greatest:

Thus:

The important connection is:

In words, the population grows fastest exactly when the logistic curve changes concavity.

Long-Term Behavior

Suppose a population begins at some positive value below the carrying capacity:

The population increases, but the growth rate gradually decreases as the population approaches .

In the long run:

The population approaches the carrying capacity but does not normally reach it at a finite time in the standard logistic model.

If the initial population is above the carrying capacity, the population instead decreases toward the same limiting value:

Therefore, for any positive initial population in the standard logistic model, the long-term behavior is generally:

Worked Examples

A Quick Logistic-Model Checklist

When an AP Calculus problem gives you a logistic differential equation, you can usually organize your analysis around a few key questions.

1. What is the carrying capacity? Look for .
2. Is the population below or above the carrying capacity? This determines the sign of .
3. Is the population increasing or decreasing? Use the sign of the derivative.
4. When is growth fastest? At .
5. Where is the inflection point? At .
6. What happens in the long run? A positive population approaches .

Common Mistakes

Practice Problems