Math 347, Project 2. DUE April 25, 2017 (Assignment turned after the due date will be penalized at the rate of 5% per day) Work in teams of size 1-3. and hand in one report per group. You may not have any of the same partners as you had for the first project. Every student is expected to be actively involved in the solving the problem as well as in the write-up. Your report should have introductory and concluding paragraphs and it should be nicely written in a format similar to an expository text in a textbook or in a research paper. It should be be written as one integrated paper rather than as an outline. In particular, you should not assume that your reader knows about this assignment. If it takes some eort to understand the situation described below, give details which make it easier to understand. Write with complete sentences, even in your mathematical derivations. Make sure to address the questions in all six parts below, and to provide as much analysis and explanation as necessary. The paper will be graded for both mathematical correctness and expository quality. It must be neat and well-organized and it should be typed, although you may fill in some formulas by hand. Problem. Suppose that size the population of deer in a forest is governed by the logistic growth equation below, where D represents the number of deer in the forest. Suppose further that the agency charged with managing the deer population grants a limited number of permits to deer hunters. If we assume that the rate at which deer are killed by hunters is proportional to both the deer population and the number of hunting permits issued, we obtain the dierential equation ( ) D dD = rD 1 ED, dt K where r (the \"natural growth rate\" of the deer population) and K (the environmental capacity) are constants and E is a parameter with units of time1 that measures the amount of hunting eort and is taken to be proportional to the number of permits issued (so E = f P , for some constant f , where P is the number of permits issued). This model is called the Schaefer model, named after the biologist M. B. Schaefer, who applied it to fishery management. Give an intuitive explanation of what the dierential equation above is saying, and why it is a reasonable model for the change in the deer population. Logistic equations are discussed on pages 36 and 37 of the course notes. You will want to use the formula on the bottom half of page 37 (with the variables and constants replaced with the ones used here). You should derive this formula, using the fact that the dierential equation at the top of page 37 is separable. (1) The model is easier to analyze if it is scaled by a suitable change of variables. Define new variables y and and a new parameter h by D E , = rt, h = . K r Note that y is the size of the deer population relative to the environmental capacity and h is the hunting eort relative to the natural growth rate. Changing variables (show how this works), we arrive at the following dierential equation y= dy = y(1 y) hy. d (2) Suppose that E r. What would happen to the deer population over time? (3) Suppose instead that E < r. Find the asymptotic value yc = 0 (in terms of h) and note that this gives a constant solution. Sketch several solution curves including the constant solution (using MATLAB, using a fixed value of h between 0 and 1). 2 (4) Notice that the value of yc can be manipulated by controlling the number of hunting permits. Sketch a graph of h versus yc (using MATLAB or by hand). Explain how the graph could be used to determine the appropriate number of permits for any desired equilibrium deer population. (5) Suppose the goal of the management program is to allow a maximum amount of sustainable deer hunting. The hunting is represented by the term hy in the dierential equation so the goal is to maximize the function Y (h) = hyc . Sketch the graph of function Y (h). (6) Determine the value of h that maximizes Y