Logistic Models with Differential Equations
Abigail Young
6 min read
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Study Guide Overview
This study guide covers logistic models with differential equations, specifically for AP Calculus BC. It explains the concept of carrying capacity and how to determine it from a given differential equation. It also covers how to find the population size at the fastest growth rate, which is half the carrying capacity. Example problems and practice questions are included.
#7.9 Logistic Models with Differential Equations
Besides exponential models, differential equations can also be used with logistic models. In this topic, we’ll cover what logistic models are and how differential equations are relevant to them.
#📈 Logistic Models with Differential Equations
The logistic model or logistic growth model is a differential equation that describes how a population grows over time—it grows proportionally to its size but stops growing when it reaches a certain size. Specifically, the model states that the rate of change of a population is jointly proportional to the size of the population and the difference between the population and the carrying capacity. The carrying capacity is the maximum number of individuals that the environment can sustain. 🌵
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Image of two graphs, one displaying exponential growth and the other displaying logistic growth with a carrying capacity.
Image Courtesy of Biology LibreTexts
Mathematically, we have the following differential equation, where is the population, is a positive constant representing the growth rate, and is the carrying capacity.
\frac{dy}{dt} = ky(M - y)
$...
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