Justify and analyze the model proposed in equation (2), covering in particular the following issues: an analysis of the validity of equation (2) as a model for the fish population in a fish farm; 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, 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). **Project: Modeling Fish Populations for Eco Fisheries, Inc.****Introduction:**Eco Fisheries, Inc. runs a successful network of fish farms across northern West Virginia. Our products supply thousands of customers in western Pennsylvania. Despite our success, expanding to eastern Pennsylvania has been challenging.**New Opportunity:**Recently, we acquired a lake in Strasburg, near Lancaster. This location could support a new fish farm, enabling expansion into eastern Pennsylvania. Before proceeding, a thorough analysis of a farm model is essential.**Reproduction and Predation:**The fish reproduction rate is linked to population size and farm capacity. Predation, particularly in Strasburg, is an expected challenge but should be manageable. However, predation can significantly impact fish numbers.**Proposed Model by Consultants:**The model provided by consultants is:\[ \frac{dN}{dt} = RN \left(1 - \frac{N}{K}\right) - P \left[ 1 - e^{\frac{-N^2}{A}} \right] \]- \( N \): Number of fish- \( R, K, P, A \): Constants- \( \varepsilon \): A small parameter (\(< 1\))Unfortunately, the consultants' report was damaged. The firm has since closed down. Yet, the analysis shows substituting variables leads to a simplified model:**Equation (2):**\[ \frac{du}{d\tau} = ru \left[ 1 - \frac{u}{q} \right] - \left[ 1 - e^{\frac{-u^2}{\varepsilon}} \right] \]Where:- \( r \) and \( q \) are constants- \( q \) ≈ 1- \( r \) relates to the fish production rate (expected \( 1 \leq r \leq 30 \))**Analysis and Justification:**Evaluate the model in Equation (2) for:- Derivation from Equation (1)- Validity as a fish population model in the proposed farm- Potential for achieving a stable fish population suitable for harvesting- Size of the initial population needed for stability and the time required to achieve it (numerical methods recommended).

Given that : where are constants and is a parameter very much less than 1.We have to choose so that equation(1) becomes,Let Then, and Then, equation(1) becomes, Set and , Also, let Then,Then, equation(3) can be rewritten as Thus, equation(1) can be written as . (Proved)Clearly solution is explained above.

Math

Advanced Math

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 2²-xx (₁-4)-0(₁-²²) = 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 B to simplify it to the form du = = ru (1- ² ) - (1-e =) dr 9 where r 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: ● r is related to the production rate of the fish, which we can control through a feeding policy. We expect that1