Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 6000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation dP P(1-2), dt determine the constant k, and then solve the equation to find an expression for the size of the population after t years. k= P(t) = = (b) How long will it take for the population to increase to 3000 (half of the carrying capacity)? It will take years.
Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 6000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation dP P(1-2), dt determine the constant k, and then solve the equation to find an expression for the size of the population after t years. k= P(t) = = (b) How long will it take for the population to increase to 3000 (half of the carrying capacity)? It will take years.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 6000. The number of fish tripled in the first year.
(a) Assuming that the size of the fish population satisfies the logistic equation
\[
\frac{dP}{dt} = kP \left(1 - \frac{P}{K}\right),
\]
determine the constant \( k \), and then solve the equation to find an expression for the size of the population after \( t \) years.
\( k = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \),
\( P(t) = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \).
(b) How long will it take for the population to increase to 3000 (half of the carrying capacity)?
It will take \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ years.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30a42a85-c58f-45ac-a4af-faeed1a599e1%2F04619f28-7cf3-4695-a2f7-c4283d1d6003%2Faxg9ltc_processed.png&w=3840&q=75)
Transcribed Image Text:Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 6000. The number of fish tripled in the first year.
(a) Assuming that the size of the fish population satisfies the logistic equation
\[
\frac{dP}{dt} = kP \left(1 - \frac{P}{K}\right),
\]
determine the constant \( k \), and then solve the equation to find an expression for the size of the population after \( t \) years.
\( k = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \),
\( P(t) = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \).
(b) How long will it take for the population to increase to 3000 (half of the carrying capacity)?
It will take \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ years.
Expert Solution
Step 1
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
