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)
Transcribed Image Text: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.
Transcribed Image Text:Project: Modeling fish populations for Eco Fisheries, Inc.
Eco Fisheries, Inc. operates a hugely successful network of fish farms that are scattered over the
northern region of West Virginia. Our fish products offer a necessary and environmentally
sound food supply to thousands of happy clients in the western Pennsylvania region. Though our
hallmark has been the freshness of our fish, we have unfortunately not been able to expand our
distribution to include the eastern region of Pennsylvania.
However, we have recently acquired a large lake in Strasburg, not too far from Lancaster. This
lake would permit the establishment of a fish farm in that location, allowing our company to sell
our fresh fish products in eastern Pennsylvania. Needless to say, it is essential that if we
approach such an undertaking, it be from a position of absolute assurance that it will be able to
succeed, and it is for the analysis of a model of the farm that we are approaching you.
It is our experience that the reproduction rate of the fish is both proportional to the size of the
fish population and limited by the number of fish that the farm can support. Additionally,
especially in such a location as Strasburg, we expect predation to be significant. While it should
be possible to restrict this to a reasonable level, predation will produce a measurable effect on the
fish population whenever there are significant numbers of fish present.
To model the fish population, an outside consulting company proposed the following model.
dN
dt
N - RV (1-A)- P(1-2 =)
(1)
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.
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