Mathematical Model for Fish Population Dynamics: The initial model equation, denoted as Equation (1), is provided as: N = RN (1-X) -P (1- e-A) In this expression, N represents the fish population, with R, K, P, and A as constants, and & as a very small parameter. A revised model, known as Equation (2), has been derived from the initial one: =ru (1- )(1- e-cau) du dt where time and population have been rescaled as t = ax and N = Bu, respectively, allowing for the selection of a and 3 to simplify the original model. The constants r and q are related to the growth rate and the adjusted carrying capacity, and & remains a very small positive value. The constant r is within the range 1 ≤ r ≤ 30.
Mathematical Model for Fish Population Dynamics: The initial model equation, denoted as Equation (1), is provided as: N = RN (1-X) -P (1- e-A) In this expression, N represents the fish population, with R, K, P, and A as constants, and & as a very small parameter. A revised model, known as Equation (2), has been derived from the initial one: =ru (1- )(1- e-cau) du dt where time and population have been rescaled as t = ax and N = Bu, respectively, allowing for the selection of a and 3 to simplify the original model. The constants r and q are related to the growth rate and the adjusted carrying capacity, and & remains a very small positive value. The constant r is within the range 1 ≤ r ≤ 30.
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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the derivation of (2) from (1)
![Mathematical Model for Fish Population Dynamics:
The initial model equation, denoted as Equation (1), is provided as:
\[
\frac{dN}{dt} = RN \left(1 - \frac{N}{K}\right) - P \left(1 - e^{-εA}\right)
\]
In this expression, \( N \) represents the fish population, with \( R, K, P, \) and \( A \) as constants, and \( ε \) as a very small parameter.
A revised model, known as Equation (2), has been derived from the initial one:
\[
\frac{du}{dt} = ru \left(1 - \frac{u}{q}\right) \left(1 - e^{-εαu}\right)
\]
where time and population have been rescaled as \( t = αx \) and \( N = βu \), respectively, allowing for the selection of \( α \) and \( β \) to simplify the original model. The constants \( r \) and \( q \) are related to the growth rate and the adjusted carrying capacity, and \( ε \) remains a very small positive value. The constant \( r \) is within the range \( 1 \leq r \leq 30 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5504642f-0145-4427-b5f2-66a9310e79bc%2F471d1676-908c-4bba-b691-e4bc51de1df1%2Fus04xy8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Mathematical Model for Fish Population Dynamics:
The initial model equation, denoted as Equation (1), is provided as:
\[
\frac{dN}{dt} = RN \left(1 - \frac{N}{K}\right) - P \left(1 - e^{-εA}\right)
\]
In this expression, \( N \) represents the fish population, with \( R, K, P, \) and \( A \) as constants, and \( ε \) as a very small parameter.
A revised model, known as Equation (2), has been derived from the initial one:
\[
\frac{du}{dt} = ru \left(1 - \frac{u}{q}\right) \left(1 - e^{-εαu}\right)
\]
where time and population have been rescaled as \( t = αx \) and \( N = βu \), respectively, allowing for the selection of \( α \) and \( β \) to simplify the original model. The constants \( r \) and \( q \) are related to the growth rate and the adjusted carrying capacity, and \( ε \) remains a very small positive value. The constant \( r \) is within the range \( 1 \leq r \leq 30 \).
![**Project: Modeling Fish Populations for Eco Fisheries, Inc.**
Eco Fisheries, Inc. operates a successful network of fish farms scattered over the northern region of West Virginia. Our fish products offer a necessary, environmentally sound food supply to thousands of satisfied clients in the western Pennsylvania region. However, despite our hallmark being the freshness of our fish, we have not expanded our distribution to include eastern Pennsylvania.
We recently acquired a large lake in Strasburg, near Lancaster. This lake will enable the establishment of a fish farm in that location, allowing us to sell our fresh fish products in eastern Pennsylvania. It is essential to approach this undertaking from a position of assurance that it will succeed. Hence, we are analyzing a model of the farm.
Our experience indicates that the reproduction rate of the fish is proportional to the size of the fish population and limited by the farm's capacity. Additionally, in Strasburg, significant predation is expected. While this should be restricted to a reasonable level, predation will measurably impact the fish population when there are many fish.
To model the fish population, an outside consulting company proposed the following model:
\[
\frac{dN}{dt} = RN \left(1- \frac{N}{K} \right) - P \left(1 - e^{-N^{2}/\varepsilon K^{2}} \right)
\]
The report from the consultant company was partially destroyed when coffee was spilled on it. Part of the explanation concerning this model became illegible. We understand that \( N \) is the number of fish and \( R, K, P, \) and \( \varepsilon \) are constants. The significance of \(\varepsilon\) is unclear, as the consulting company is no longer available for communication due to a liquidation of assets.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5504642f-0145-4427-b5f2-66a9310e79bc%2F471d1676-908c-4bba-b691-e4bc51de1df1%2F0797ynb_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 scattered over the northern region of West Virginia. Our fish products offer a necessary, environmentally sound food supply to thousands of satisfied clients in the western Pennsylvania region. However, despite our hallmark being the freshness of our fish, we have not expanded our distribution to include eastern Pennsylvania.
We recently acquired a large lake in Strasburg, near Lancaster. This lake will enable the establishment of a fish farm in that location, allowing us to sell our fresh fish products in eastern Pennsylvania. It is essential to approach this undertaking from a position of assurance that it will succeed. Hence, we are analyzing a model of the farm.
Our experience indicates that the reproduction rate of the fish is proportional to the size of the fish population and limited by the farm's capacity. Additionally, in Strasburg, significant predation is expected. While this should be restricted to a reasonable level, predation will measurably impact the fish population when there are many fish.
To model the fish population, an outside consulting company proposed the following model:
\[
\frac{dN}{dt} = RN \left(1- \frac{N}{K} \right) - P \left(1 - e^{-N^{2}/\varepsilon K^{2}} \right)
\]
The report from the consultant company was partially destroyed when coffee was spilled on it. Part of the explanation concerning this model became illegible. We understand that \( N \) is the number of fish and \( R, K, P, \) and \( \varepsilon \) are constants. The significance of \(\varepsilon\) is unclear, as the consulting company is no longer available for communication due to a liquidation of assets.
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