2. You've been hired by the New York City Office of Public Sanitation to analyze the pests in the sewers. The Chief of Public Sanitation expects you to justify your

conclusions using both graphical and numerical data. You should aim to be as precise as possible in your analysis. Please upload screenshots of any graphical material as well as any Excel files (not screenshots) you use. The following system of differential equations describes the populations of cock-roaches and rats living in the sewers. x^{\prime}(t)=.05 x-.001 x y y^{\prime}(t)=-.01 y^{2}-.1 y+.02 x y Both species are measured in millions and time is measured in months. • In the "PPLANE Phase Plane" window menu bar, select Options → Delay Time Per Point → 10 Milliseconds Options → Solution Direction → Forward ● In the “PPLANE Equation” window, you can change the values in the"Display Window." Your minimum values should be zero since populations can't be negative. You will need to decide what you want the maximum values to be. • Remember that the command Graph → Both x-t & y-t allows you to seea graph of the solution functions! (a) Which of x or y is the population of cockroaches? Which of x or y is the population of rats? Completely justify your answer. (b) In the absence of one of these two species, describe the growth of the other. (c) Starting with a population of 6 million rats and 4 million cockroaches,when does the number of cockroaches overtake the number of rats? Then,how long does it take for the number of rats to catch back up and overtake the number of cockroaches? What is the minimum rat population during this period of time? (d) Find and classify the equilibrium points of the system. What does this tell you about the long term behavior of the two species? The Chief of Public Sanitation wants to know whether it is more effectiveto poison the cockroaches or rats. What recommendation do you make?Completely justify your answer.