\f\f(b) Constant Harvest Model: A Specific Case In the constant-harvest model fisher- men are allowed to harvest a set amount of fish during a fixed time. The parameter h indicates the constant harvesting rate. dP =TP dt -h h >0 . (3) Our goal is to understand what happens to the equilibrium values as h increases. Use r = 1 and K = 200 to find the equilibrium values. P. and P_ of equation (3) for each of the values h = 18, 32, 48, and 50. ii. For each of the values of h above draw a phase diagram and classify the stability of the equilibrium values P+ and P_. iii. What happens to P. and P_ as h increases? iv. A. For which value of h is there only one critical point? Draw a phase line for this case . B. What does this mean in terms of the sustainability of the fishery? C. Explain what happens to fishery if h - 50.1 and if h - 80. D. What fraction of the carrying capacity is the h-value that yields a single critical point? (c) Bifurcations The model dealing with constant harvesting provides an example of bifurcation. A bifurcation is essentially a dramatic change in the qualitative structure of the phase line, such as the appearance or disappearance of equilibrium solutions (critical points) A bifurcation diagram for a family of DEs is a graph that shows the location and stability of the critical points for each parameter, in our case the parameter is h. the harvesting rate. Consider our example r = 1 and K - 200. The critical points satisfy ap - TP (1 - ) - n -0. ( 4 ) and these can be solved using the quadratic formula. i. Plot the graph of P vs h from (5) for 0 S h s 200. fi. What is the bifurcation point and what does it tell you about the limits on h if the goal is a sustainable fishery? C N O 37OF 2/ DELL F6 F F8 F9 F10 F11 F12 PrtScr Insert %2. TREATING INDUSTRIAL POLLUTION USING A RETENTION POND Often the result of an industrial operation is a waste fluid that contains toxic chemicals. There are different ways of processing waste fluid, but one way is to store the waste fluid in a retention pond, especially if water is used in the industrial operation. This method allows any particles that are suspended in the waste fluid to settle to the bottom of the pond and remain trapped there. The remaining contaminated water can then be pumped out of the pond, treated and recycled. Suppose that an industrial operation stores its waste fluid in a pond that has an initial volume of 40,000 cubic meters. When the operation begins, the pond is completely filled with clean water. The pond has polluted water flowing into it while some of the liquid in the pond is also pumped out. Suppose that 2400 cubic meters per day of polluted liquid flows into the pond and 2400 cubic meters of liquid are pumped from the pond each day to be processed and recycled. Thus, the water level of the pond remains constant. At time t = 0, polluted input liquid, which is contaminated with a toxic chemical at a rate of 8 kilograms per 1200 cubic meters, flows into the pond. We will assume that liquid in the retention pond is well mixed so that the concentration of the toxic chemical through out the pond liquid is fairly uniform. In addition, any particulate matter that flows into the pond in the input stream settles to the bottom of the pond at a rate of 80 cubic meters per day, and contains no toxic chemical. Thus, the volume of the retention pond is reduced by 80 cubic meters each day, and will eventually become full. We shall assume that the particulate matter and the toxic chemical pollutant are included in the 2400 cubic meters that flow into the pond from the input stream each day. Questions: (a) Find a differential equation that will model the amount of chemicals in the retention pond at any particular time. Use z(f) to represent the amount of the toxic chemical in the pond at time, t where t is measured in days. Show the steps by which you arrived at your model. (b) Solve the differential equation you found. Show all of your work. c) Graph z(t) vs t. (d) On what day will a(t) be a maximum? Show your work (e) On what day will the retention pond be full of sediment? Explain your answer (f) On what day will r(t) = 07 n 99+ O N O 37OF DELL F5 F6 F F8 F9 F10 F11 F12 PrtScr Ins % &