Suppose the equation P' = 0.5P (1000 1.6P) models the growth of a population. If P(0) = 700, what happens to the population over a long period of time? [Hint: You do not need to actually solve the equation to answer this.]
Suppose the equation P' = 0.5P (1000 1.6P) models the growth of a population. If P(0) = 700, what happens to the population over a long period of time? [Hint: You do not need to actually solve the equation to answer this.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4. Suppose the equation P' = 0.4P(100 – 0.08P) models the growth of a population. If P(0) = 600, what
happens to the population over a long period of time? [Hint: You do not need to actually solve the equation to
answer this.]
Option 1:
Solving P' = 0 yields critical numbers at 0.4P = 0 ⇒ P = 0 and 100 – 0.08P = 0 ⇒ P = 1250.
Testing the interval (0,1250) into P' shows that P'>0 over that interval. So, the solution curve is increasing when P =
600 and is asymptotic to P = 1250.
So, as t→ ∞o, P→ 1250
Option 2:
Note that the equation is logistic, but not in the standard form.
P' = 0.4P (100 – 0.08P) ⇒ 0.4P[0.08](1250 – P) ⇒ 0.032P (1250 – P)
So, the limiting population of the model is 1250.
Page 2 of 3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e9c030-c4e6-49da-b6c6-4b7d6c7f6fcd%2Fe708cfd0-8842-4685-b1af-6fc60b28b3b9%2F4g4f3fo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Suppose the equation P' = 0.4P(100 – 0.08P) models the growth of a population. If P(0) = 600, what
happens to the population over a long period of time? [Hint: You do not need to actually solve the equation to
answer this.]
Option 1:
Solving P' = 0 yields critical numbers at 0.4P = 0 ⇒ P = 0 and 100 – 0.08P = 0 ⇒ P = 1250.
Testing the interval (0,1250) into P' shows that P'>0 over that interval. So, the solution curve is increasing when P =
600 and is asymptotic to P = 1250.
So, as t→ ∞o, P→ 1250
Option 2:
Note that the equation is logistic, but not in the standard form.
P' = 0.4P (100 – 0.08P) ⇒ 0.4P[0.08](1250 – P) ⇒ 0.032P (1250 – P)
So, the limiting population of the model is 1250.
Page 2 of 3
![4. Suppose the equation P' = 0.5P(1000 – 1.6P) models the growth of a population. If P(0) = 700, what
happens to the population over a long period of time? [Hint: You do not need to actually solve the equation to
answer this.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e9c030-c4e6-49da-b6c6-4b7d6c7f6fcd%2Fe708cfd0-8842-4685-b1af-6fc60b28b3b9%2F6bcnqc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Suppose the equation P' = 0.5P(1000 – 1.6P) models the growth of a population. If P(0) = 700, what
happens to the population over a long period of time? [Hint: You do not need to actually solve the equation to
answer this.]
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