**Project: Modeling Fish Populations for Eco Fisheries, Inc.**Eco Fisheries, Inc. operates a network of fish farms in northern West Virginia, supplying fresh fish to western Pennsylvania. However, they aim to expand their distribution to eastern Pennsylvania through a new fish farm on an acquired lake near Strasburg, Lancaster.The reproduction rate of fish depends on population size and the carrying capacity of the farm. Predation is also a concern, especially in Strasburg, affecting fish population significantly.**Proposed Model:**The model for fish population dynamics is given by:\[ \frac{dN}{dt} = RN \left( 1 - \frac{N}{K} \right) - P \left( 1 - e^{-\frac{N}{A}} \right) \]Where:- \( N \): Number of fish- \( R, K, P, A \): Constants- \( \epsilon \): Parameter very much less than 1**Simplified Form:**By substituting \( t = \alpha \tau \) and \( N = \beta u \), the equation becomes:\[ \frac{du}{d\tau} = ru \left( 1 - \frac{u}{q} \right) - \left( 1 - e^{-\frac{u}{\epsilon}} \right) \]Where:- \( r \) is related to the fish production rate.- \( \epsilon \) is small.- \( q \) is close to 1.- \( 1 \leq r \leq 30 \).**Analysis and Justification:**Tasks include:- Derivation of the simplified equation from the original model.- Evaluation of the model's validity for the fish farm.- Determination of stable population levels for harvesting.- Calculation of initial population size and time required to reach stability, where numerical solvers are recommended.

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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. -N² dN * - ** (1 - - (1---*) = RN 1-- dt K 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=ar and N = Bu into this equation, it is possible to choose a and ß to simplify it to the form du dr =-=- ru (1-² )-( 1-0 =²) -=ru wherer and q are constants". In this equation, ● ● ● ● (1) & is very small (positive value closer to zero); qis close to 1 and ● Justify and analyze the model proposed in equation (2), covering in particular the following issues: (2) r is related to the production rate of the fish, which we can control through a feeding policy. We expect that1 < r ≤ 30. an analysis of whether, based on model equation (2), we may expect a stable fish population from which harvesting could take place; and, if so, the derivation of (2) from (1); an analysis of the validity of equation (2) as a model for the fish population in a fish farm; an analysis of how large an initial population of fish will be required to obtain this stable population and the length of time required for the stable population to be established (numerical solver recommended).