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Diff EQ Lab I - APPM 2360 Lab 1 Fish Population Modeling...

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dy/dt t dy/dt Figure : Change in Pop. Vs. Pop. y y Figure : Population vs. Time t t Figure : Vector Field dy/dt y Figure 5: Change in Pop. Vs pop. dy/dt y Figure 6: Change in Pop. Vs Pop. t y Figure 7: Vector-Field t y Figure 8: Population vs. Time dy/dt y Figure 9: Change in Pop. Vs. Pop y y t t Figure 11: Population vs. Time Figure 10: Vector-Field dy/dt Change in Carp Population; 0 < y < 2 y dy/dt Change in Carp Population: 13< y < 15 y y Carp: Population Slope-Field t y t Carp: Sample Solutions dy/dt Change in Catfish Population: 0 < y < 2 y dy/dt y Change in Catfish Population: 5 < y < 6 y t Catfish: Slope-Field y t Catfish: Sample Solutions dy/dt Bass: Change in Population vs. Population y y t Bass: Slope-Field y t Bass: Sample Solutions APPM 2360 – Lab 1 Fish Population Modeling Spring 2008 Filip Maksimovic (810689087) Theodoros Horikis Recitation 34: John Vilavert Michaela Cui (810585604) Theodoros Horikis Recitation 036: Sean Nixon Brandon Parks (810689666) Theodoros Horikis
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Recitation 034: John Vilavert 1.0 Introduction The goal of this lab is to determine the growth of different species of fish given a model equation. The given model for population growth takes into account the death rate, birth rate, and rate of harvest. In order to determine the behavior of the different species’ models, the equation had to be classified, solved, and thoroughly analyzed. This was accomplished through numerical analysis as well as with the help from a computer-aided numeric solver, Mathematica. The resulting conclusion was that an environment based solely on carp would result in overpopulation, while one based on catfish would result in a final population slightly large or average, and bass would result in an average population no matter what the initial condition. It was concluded that bass would be the most desirable fish to stock a pond with. 2.0 Model The model for the entirety of the lab is as follows; = - - dydt r1 yLy H (0) and can also be written as; = - - dydt ay cy2 H (1) where = a r and = c rL . This equation can be easily classified order to easily obtain a solution. Upon inspection, the equation can be classified as first order, nonlinear, homogeneous, and autonomous. There are no instances of the independent variable in the equation which leads to the conclusion of autonomy. All coefficients are constants relating to statistics of the population growth of the fish.
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The first term, ay, is the rate of birth of the fish, while cy2 describes the rate of death of the fish. H, the harvesting parameter, also deducts from the fish population, hence the negative sign. Units of ay and cy2 must be fish per day, as this is the rate of change of the population of fish, y, with respect to t, which is measured in days. Therefore, the units of a must be - t 1 , and the units of c must be ( )- ty 1 , where a and c are both constants relating the fish being studied. For the lab’s purposes, a will always be 0.65.
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