The Logistic Differential Equation: An Introduction

What the Logistic Differential Equation Is For

The exponential growth model assumes that a population can keep growing at the same proportional rate forever. That's useful for short-term growth, but it eventually becomes unrealistic. Real populations face limited food, space, nutrients, and other resources.

The logistic differential equation modifies exponential growth by adding a term that slows the growth rate as the population approaches a maximum sustainable level called the carrying capacity.

Here, represents the population at time , while and are positive constants. The constant is the carrying capacity: the population level that the model approaches over time.

Understanding the Logistic Model

The most important part of the logistic equation is the factor . It determines how strongly the carrying capacity affects the growth rate.

• When , the population is below carrying capacity and is positive.
• When , the population is exactly at carrying capacity and .
• When , the population is above carrying capacity and is negative.

So the logistic model automatically captures three different behaviors: growth below the carrying capacity, no change at the carrying capacity, and decline above it.

Why the Growth Starts Fast and Then Slows Down

Suppose the population is very small compared with the carrying capacity. Then is close to 1, so the logistic equation behaves approximately like exponential growth:

As the population gets closer to , the factor gets closer to zero. Therefore, the growth rate decreases.

At exactly , the factor becomes zero, so the population stops changing:

This is the central idea of the logistic model: growth is self-limiting.

Equilibrium Solutions

An equilibrium solution is a constant solution whose derivative is zero. For the logistic differential equation, there are two equilibrium populations.

Equilibrium 1: Zero Population

If , then:

So is an equilibrium solution. If the population starts at zero, the model keeps it at zero.

Equilibrium 2: Carrying Capacity

If , then:

Thus is also an equilibrium solution. A population that begins exactly at carrying capacity remains there.

Determining Whether a Population Is Increasing or Decreasing

AP Calculus questions frequently ask whether a population is increasing or decreasing at a particular population value. You do not need to solve the differential equation to answer this.

Simply substitute the given population into the differential equation and determine the sign of :

• If , the population is increasing.
• If , the population is decreasing.
• If , the population is constant at that instant.

The Maximum Growth Rate

One of the most important features of the logistic model is that the population does not grow fastest when it is smallest. Nor does it grow fastest near carrying capacity.

The growth rate is largest when the population is halfway to the carrying capacity:

Why? The logistic growth rate is proportional to . This product is largest when the two factors balance each other, which occurs at .

Finding the Logistic Growth Rate at a Given Population

Sometimes the question gives the population and asks for its instantaneous rate of change. This is a direct substitution problem.

Logistic Growth vs. Exponential Growth

The two models look similar at first, but their long-term behavior is very different.

For exponential growth, the growth rate remains proportional to the population, so the model predicts unlimited growth. For logistic growth, the additional factor progressively reduces the growth rate as the population approaches .

This makes the logistic model much more realistic for populations living in an environment with finite resources.

Reading a Logistic Differential Equation on the AP Exam

When you see a logistic differential equation, immediately identify these three quantities:

1. Population:
2. Growth constant:
3. Carrying capacity:

Then ask what the problem is actually requesting. If it asks whether the population is increasing or decreasing, find the sign of . If it asks when growth is fastest, use . If it asks for a rate of change, substitute the given population directly into the differential equation.

Practice Problems