To model the fish population, an outside consulting company proposed the following model. -N² = RN (1-X) - P(1- ²+²) dN dt The report issued by the consultant company was partially destroyed when a coffee was spilled on it. Owing to this error, much of the explanation associated with this particular model is illegible, though we understand that N is the number of fish, R, K, P, and A are constants, and & is a parameter very much less than 1. The original consultant company liquidated its assets after a bankruptcy and no longer available for communication. In a legible portion below the above equation, the consultant concludes that "by substituting t = at and N = Bu into this equation, it is possible to choose a and ß to simplify it to the form du dr =ru 1- 9 (1) |- (₁1-e=²) where r and q are constants". In this equation, (2)
To model the fish population, an outside consulting company proposed the following model. -N² = RN (1-X) - P(1- ²+²) dN dt The report issued by the consultant company was partially destroyed when a coffee was spilled on it. Owing to this error, much of the explanation associated with this particular model is illegible, though we understand that N is the number of fish, R, K, P, and A are constants, and & is a parameter very much less than 1. The original consultant company liquidated its assets after a bankruptcy and no longer available for communication. In a legible portion below the above equation, the consultant concludes that "by substituting t = at and N = Bu into this equation, it is possible to choose a and ß to simplify it to the form du dr =ru 1- 9 (1) |- (₁1-e=²) where r and q are constants". In this equation, (2)
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
the derivation of (2) from (1)
![**Project: Modeling Fish Populations for Eco Fisheries, Inc.**
Eco Fisheries, Inc. operates a successful network of fish farms in northern West Virginia. These farms provide fresh and environmentally friendly fish to clients in western Pennsylvania. Although the company excels in fish freshness, it hasn't expanded to eastern Pennsylvania yet.
Recently, Eco Fisheries acquired a lake in Strasburg, near Lancaster, allowing the potential establishment of a fish farm in this area. This development could enable the company to distribute fresh fish in eastern Pennsylvania.
**Reproduction and Predation in Fish Populations**
The reproductive rate of fish is proportional to their population size and the capacity of the farm. Predation is expected, especially in Strasburg, which could affect population size. Controlling predation is essential for maintaining fish numbers.
**Proposed Fish Population Model**
A consulting company proposed a model to analyze fish population dynamics:
\[ \frac{dN}{dt} = RN \left(1 - \frac{N}{K}\right) - P \left[1 - e^{-\frac{N^2}{A}}\right] \]
Where:
- \(N\): Number of fish
- \(R, K, P, A\): Constants
- \(\varepsilon\): A small parameter
The consultant company faced issues after an original report was damaged, leading to partial information. They suggested a transformation using:
\[
t = \alpha \tau \quad \text{and} \quad N = \beta u
\]
This leads to simplified equation:
\[ \frac{du}{d\tau} = ru \left[1 - \frac{u}{q}\right] - \left[1 - e^{-\frac{x^2}{\varepsilon}}\right] \]
Where:
- \(r\) and \(q\) are constants
- \(\varepsilon\) is very small
- \(q\) is close to 1
- \(r\) relates to fish production rate, typically between 1 and 30
---
**Analysis Requirements**
Analyze the proposed model focusing on:
- Derivation of this model
- Validity assessment for fish farm populations
- Possibility and conditions for a stable fish population for harvesting
- Initial population size and time required to reach stability (numerical solver suggested).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5504642f-0145-4427-b5f2-66a9310e79bc%2F06fc76c7-f710-422d-bead-768439e071bc%2F2jhid3g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Project: Modeling Fish Populations for Eco Fisheries, Inc.**
Eco Fisheries, Inc. operates a successful network of fish farms in northern West Virginia. These farms provide fresh and environmentally friendly fish to clients in western Pennsylvania. Although the company excels in fish freshness, it hasn't expanded to eastern Pennsylvania yet.
Recently, Eco Fisheries acquired a lake in Strasburg, near Lancaster, allowing the potential establishment of a fish farm in this area. This development could enable the company to distribute fresh fish in eastern Pennsylvania.
**Reproduction and Predation in Fish Populations**
The reproductive rate of fish is proportional to their population size and the capacity of the farm. Predation is expected, especially in Strasburg, which could affect population size. Controlling predation is essential for maintaining fish numbers.
**Proposed Fish Population Model**
A consulting company proposed a model to analyze fish population dynamics:
\[ \frac{dN}{dt} = RN \left(1 - \frac{N}{K}\right) - P \left[1 - e^{-\frac{N^2}{A}}\right] \]
Where:
- \(N\): Number of fish
- \(R, K, P, A\): Constants
- \(\varepsilon\): A small parameter
The consultant company faced issues after an original report was damaged, leading to partial information. They suggested a transformation using:
\[
t = \alpha \tau \quad \text{and} \quad N = \beta u
\]
This leads to simplified equation:
\[ \frac{du}{d\tau} = ru \left[1 - \frac{u}{q}\right] - \left[1 - e^{-\frac{x^2}{\varepsilon}}\right] \]
Where:
- \(r\) and \(q\) are constants
- \(\varepsilon\) is very small
- \(q\) is close to 1
- \(r\) relates to fish production rate, typically between 1 and 30
---
**Analysis Requirements**
Analyze the proposed model focusing on:
- Derivation of this model
- Validity assessment for fish farm populations
- Possibility and conditions for a stable fish population for harvesting
- Initial population size and time required to reach stability (numerical solver suggested).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 17 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,
