Note: Please explain your work carefully and in complete English sentences. dp Problem. Suppose that the population of fish in some lake is described by the logistic equation, kP(M-P). It is reasonable to assume that of a given level of effort, the rate at which fish are caught is proportional to the population, i.e., is given by EP, there is some constant, E > 0. Then the equation modeling the fishery becomes dP - P(M-P) - EP dt (a) Determine how does the number and character of equilibrium solutions depend on E. Which values of E give asymptotically stable equilibrium solutions P.! () A sustainable yield Y of the fishery is a rate at which fish can be caught indefinitely. It is the product of the effort E and the asymptotically stable population Ps. Find Y as the function of the effort E (the graph Y = Y(E) is called the yield-effort curve). (c) Determine E ghring the maximum sustainable yield Ym- Sketch the slope fields and some trajectories that illustrate your conclusions using technology. There are many free tools available, eg, dfield JAVA Applet. Let me know if you need any help with this. Note: Please explain your work carefully and in complete English sentences. dp Problem. Suppose that the population of fish in some lake is described by the logistic equation, kP(M-P). It is reasonable to assume that of a given level of effort, the rate at which fish are caught is proportional to the population, i.e., is given by EP, there is some constant, E > 0. Then the equation modeling the fishery becomes dP - P(M-P) - EP dt (a) Determine how does the number and character of equilibrium solutions depend on E. Which values of E give asymptotically stable equilibrium solutions P.! () A sustainable yield Y of the fishery is a rate at which fish can be caught indefinitely. It is the product of the effort E and the asymptotically stable population Ps. Find Y as the function of the effort E (the graph Y = Y(E) is called the yield-effort curve). (c) Determine E ghring the maximum sustainable yield Ym- Sketch the slope fields and some trajectories that illustrate your conclusions using technology. There are many free tools available, eg, dfield JAVA Applet. Let me know if you need any help with this