Explanation Given: The population community levels off at 2000. That is, carrying capacity K = 2000 , the initial population growth P ( 0 ) = 200 and the final population increases 600 in first year. P ( 1 ) = 600 . Result used: The differential form of a logistic equation is P ′ ( t ) = r P ( 1 − P K ) , where r Natural growth, carrying capacity K and population P . Calculation: Use the above result mentioned to write the logistic differential equation with K = 2000 . That is, P ′ ( t ) = r P ( 1 − P 2000 ) Rewrite the above logistic equation as follows, P ′ ( t ) P ( 1 − P 2000 ) = r Integrate on both sides of the above equation with respect to t . That is, ∫ P ′ ( t ) P ( 1 − P 2000 ) d t = ∫ r d t ∫ ( P ′ ( t ) P ( t ) + P ′ ( t ) 2000 − P ( t ) ) d t = ∫ r d t Take P ( t ) = z . P ′ ( t ) = d z d t P ′ ( t ) d t = d z Then, ∫ d z z + ∫ P ′ ( t ) 2000 − P ( t ) d t = ∫ r d t ln z − ln ( 2000 − P ( t ) ) = r t + C ln ( z 2000 − P ( t ) ) = r t + C Substitute the value of z = P ( t ) in the above equation. That is, ln ( P ( t ) 2000 − P ( t ) ) = r t + C Where C is constant. Take exponential on both sides of the above solution as follows, e ln ( P ( t ) 2000 − P ( t ) ) = e r t + C P ( t ) 2000 − P ( t ) = c 1 e r t Reciprocate the above equation as follows, 2000 − P ( t ) P ( t ) = A e − r t The general solution of the logistic equation is 2000 − P ( t ) P ( t ) = A e − r t . The initial growth of population is P ( 0 ) = 200 . Substitute the initial value P ( 0 ) = 200 in the above general solution

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

See similar textbooks

Concept explainers

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 …

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experime…

Videos

Question

Chapter 9.5, Problem 17E

To determine

To find: The logistic function of the given population problem and graph the population function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.5, Problem 16E

Chapter 9.5, Problem 18E

Chapter 9 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 9.1 - What are the orders of the equations in Example 2?...Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - In Example 7, if the height function were given by...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - The solution to the initial value problem y(t) = 2...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The logistic function of the given population problem and graph the population function.

Solution Summary: The author explains the logistic function of the given population problem and graphs the population function.

Concept explainers

Videos

To find: The logistic function of the given population problem and graph the population function.

Want to see the full answer?

Chapter 9 Solutions