(d) We say that a system is in equilibrium when the rate of change is equal to 0. For our model, that is equivalent to socing when L - 0.071 (1- 100, 000 - h - 0. Clearly, if L - 0 then we can have no harvesting, so h is also 0, thus we will have dL/dt - 0. This simply tells us if the Lake Sturgeon population is zero, then there will be no growth in the population. This makes sense. However, to better understand the relationship between L and h we can use Matlab. i. Use Matlab to plot the graph of I' when h - 1, 000 (use the interval (0, 100, 000] for your graph). On the same graph, plot L' - 0 (recall, the "hold on' command will let you place a second graph on the same axes). ii. Are the any equilibrium values for L? If so, what are their approximate values and interpret their meaning? (Use your previous plot to answer these questions). iii. Use your plot (and perhaps your Logistic m-file) to determine what would happen to the Lake Sturgeon population if we started this harvest rate when L(0) - 15,000 fish. iv. Use your plot (and perhaps your Logistic m-file) to determine what would happen to the Lake Sturgeon population if we started this harvest rate when L(0) - 20,000 fish. v. Use your plot (and perhaps your Logistic m-file) to determine what would happen to the Lake Sturgeon population if we started this harvest rate when L(0) - 90,000 fish. vi. Discuss how you can use plots of ', the logistic model, and harvest rates to help manage fish populations at sustainable levels. 4. Instead of a constant harvest rate h, suppose now the harvest is proportional to the number of Lake Sturgeon. (a) Modify your IVP in (1.) to include proportional harvesting (b) Now what are the units for h? (c) What are reasonable values for h? (d) For all harvesting proportionality constants, will the population of Lake Sturgeon be sustainable? Supply graphs to support your answer! (e) Use a strategy similar to (3d.), and h - 0.5 to discuss equilibrium values, harvest rates, and sustainability. 5. What do you feel is a better method of management of the fish population - constant harvesting or proportional harvesting?MP 3: Gone Fishing One of the earliest applications of using differential equations to effectively manage the population of a species was with fisheries. A goal of managing fish populations is to determine the rate that fish can be harvested that will still result in a sustainable fish population. This project explores this some simple fishery models. For all of the questions below, we consider a large lake of vol- ume approximately 100 cubic miles (similar to Lake Erie). The fish of interest will be Lake Sturgeon. Lake sturgeons can be huge, topping six feet (two meters) long and weighing nearly 200 pounds (90 kilograms). They are also extremely long-lived. Males may live some 55 years, and females can reach 150. Unfortunately, populations of Lake Sturgeon in North America have been decimated. In 1885, 8.6 million pounds of sturgeon were taken from the Great Lakes. By 1928 the catch totaled 2,000 pounds. The fish that had abounded in drainages of Hudson Bay, the Great Lakes and the Mississippi River as far south as Alabama suddenly faced extinction. Today commercial fishing for lake sturgeon is banned in the U.S. and strictly limited in Canada and recovery work is underway throughout most of the historic range. There are many ongoing efforts to restore the Lake Stur- goon populations, but it is a slow struggle. Hampering efforts is that Lake Sturgeon don't reach reproductive age until 15 years. Thus the rebound won't be measured in years, but rather decades. Suppose it is estimated that in our lake, the carrying capacity of Lake Sturgeon is thought to be approximately 1000 Lake Sturgeons per cubic mile of water. 1. In many lakes with no harvesting, we can use the logistic growth differential equation to model a fish population. Suppose L represents the total number of Lake Sturgeon in our lake, it is currently estimated the lake contains approximately 20,000 Lake Sturgeons, and the lake contains 100 cubic miles of water (similar to estimated totals in Lake Erie). Discuss why it is reasonable to use the logistic growth mathematical model to model the Lake Sturgeon population. Assuming logistic growth, what is the initial value problem (IVP) for the situation described? (User - 0.07/yr in your IVP). 2. Use Matlab and Euler's method to create a graphical solution of your IVP in (1.). Use time steps of a half-year for 100 years (You may simply modify Logistic.m).. (a) Approximately how many years will it take for the population to attain 60% of the carrying capacity? (b) Suppose the population of Lake Sturgeon in the lake is extremely depleted (which it was in most of the North American lakes) and the entire lake contained only 1000 Lake Sturgeon. Run your logistic m-file again using this new initial condition. Now how long will it take for the population to reach 60% of the carrying capacity? (c) Given that serious conservation efforts for Lake Sturgeons started in North America approximately 45 years ago, are the plots in (a.) and (b.) consistent? Discuss. 3. Suppose it is believed that when the population has rebounded to 60% of the carrying capacity we can allow harvesting of the Lake Sturgeon. (a) Modify your IVP in (1.) to include a constant harvest rate of h Lake Sturgeon per year. (b) If h is set at 1000 fish/year, will the population be sustainable? Use Matlab m-files to explore this question. (c) If h is set at 2000 fish/year, will the population be sustainable? Use Matlab m-files to explore this