Mathematical Modeling Task 1

Below are the instructions for the assignment. I have included my first draft of the assignment as a microsoft word doc along with notes from my professor on the assignment about what needs to be adjusted.
A lake with a fixed carrying capacity contains a certain fish population. The fish population in the lake has a growth rate that is proportional to its size when the population is very small relative to the carrying capacity. However, when the fish population exceeds the carrying capacity, the growth rate is negative.
A. Write a differential equation that models the population of fish described in the Scenario section, defining all parameters and variables.
1. Explain how the differential equation models both conditions in the Scenario section.
B. Adjust the differential equation from part A to account for the following modification to the Scenario section: the fish are continually harvested at a rate proportional to the square root of the number of fish in the lake.
1. Explain why the adjustment to the differential equation from part B models the modification to the Scenario section.
C. Acknowledge sources, using in-text citations and references, for content that is quoted, paraphrased, or summarized.
Professors notes on assignment submission:
“In part A, you have correctly analyze P/K and 1-P/K for both scenario. You would need to continue your analytic to kP(1-P/K) to finishing your discussion on how to reach the conclusion of the scenarios.
In part B. Note that the given scenario does not specific the proportionality constant for the harvest is the same as the growth proportionality constant. So you should be using the same variable for these two constants.”
Notes on A1. The work correctly noted that when P/K is close to one, then (1 – P/K) will be close to 1, and dP/dt is close to k*P. Also, it is correct that if P/K > 1, then (1 – P/K) will be negative. It is unclear how the two given scenarios (the population is very small relative to the carrying capacity, and the fish population exceeds the carrying capacity) are used in the discussion.
Notes on B. Subtracting an expression from the differential equation in part A has merit in adjusting the model. The work subtracted k√P from the original equation. Using the same constant k as the proportionality constant used in part A is inaccurate.
Notes on B1
It is correctly noted that a quantity that is proportional to √P is subtracted because when fish are harvested, those fish are taken out of the total fish population. The given explanation will be evaluated once the work provides an appropriately adjusted differential equation in part B.


Approximately 250 words