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Chapter 4: Problem 61

Describe a difference between exponential growth and logistic growth.

Short Answer

Expert verified
The core difference between exponential and logistic growth lies in the concept of carrying capacity. Exponential growth happens without any constriction and it increases over time. On the other hand, logistic growth considers resource limitations and thus begins to slow down as resources dwindle, eventually reaching a plateau when the carrying capacity, the maximum population an environment can support, is reached.

Step by step solution

01

Definition of Exponential Growth

Exponential growth is a type of growth where a quantity increases at a rate proportional to its current value. In other words, the larger the quantity, the faster the rate of growth. From a graphical perspective, exponential growth is illustrated by a J-shaped curve, where growth accelerates over time without reaching an upper limit.
02

Definition of Logistic Growth

Logistic growth, on the other hand, starts off similarly to exponential growth, with a rapid increase in the quantity. However, over time, as resources become increasingly scarce, the rate of growth slows down and eventually stops, flattening out into a S-shaped curve. This indicates that the population/quantity has reached its carrying capacity, which is the maximum number the environment can support.
03

Describing the Difference

The main difference involves the presence or absence of a carrying capacity. Exponential growth assumes unlimited resources, thus the growth is unrestricted and can continue indefinitely. Meanwhile, logistic growth recognizes that resources are limited; the growth slows, and eventually stops, as the capacity limit (carrying capacity) is reached.

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Most popular questions from this chapter

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=112.5 e^{0.012 y}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(50 \% ?\)

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{5}\left(\frac{\sqrt{x}}{25}\right)\)

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{6} 17\)

Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.
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