AMATH 383 Winter 2014 Homework 7 Due: Thursday, February 27, 2014 Show work for full credit! 1. Beverton-Holt discrete population model A goal of fishery scientists has been to determine the relationship between the size of a fish population and the number of offspring from it that survive to enter the fishery. Fishery scientists refer to a population of a fish as a stock, and the process of entering the fishery as recruitment. The relationship, called the stock-recruitment relationship is often obtained by empirical fitting of the data to an analytic function. The Beverton-Holt stock recruitment is governed by the difference equation Nn+1 = RNn , R>1 1 + R−1 K Nn (a) Find the equilibria and determine their stability. (b) Find an exact, closed form solution in terms of the first population N1 . (Hint: Use the substitution Xn = 1/Nn . The equation for Xn turns out to be linear.) This is one of the rare cases where a nonlinear difference equation can be solved exactly. 2. The grandparent effect We wish to model a species of animal that has a distinct breeding season, and whose life span is long enough that a significant portion of individuals live to see their grandchildren born and are able to help the parents in raising the newest generation. Create a new model that incorporates this effect into the discrete Logistic population model discussed in class: Nn Nn+1 = Nn + r∆tNn 1 − K Assume the influence of the grandparents on the rate of new births is proportional to the number of living members of the grandparents’ generation. Write the difference equation for your new model, and find the equilibrium points. Note that the same model could describe a negative influence from the grandparents if the proportionality constant is negative, i.e., the middle-aged generation must care for both their children and their parents, reducing the rate of new births. Comment on the difference between the equilibrium points found in this model, and those found in the original discrete Logistic population model. 1 3. The diet problem We seek to decide the number of units of different foods to consume every day so we meet the minimum daily requirement (MDR) of different nutrients at minimum cost. carbs/unit protein/unit vitamins/unit cost/unit barley 2 5 3 0.7 maize 3 2 5 0.8 MDR 9 16 10 Let x be the number of units of barley and y be the number of units of maize to consume every day. We formulate this problem as a linear program: min 0.7x + 0.8y subject to 2x + 3y ≥ 9 5x + 2y ≥ 16 3x + 5y ≥ 10 x≥0 y≥0 (a) Make a sketch of the feasibility region. Are any of the constraints redundant? (b) On the same sketch, make a contour plot of the total cost by drawing lines of constant objective function value. Then, graphically solve for the optimum (x, y) that minimizes the objective function. (c) Let’s assume the cost of barley is still 0.7 per unit, but the cost of maize is more volatile. What range of cost per unit for maize will result in the same optimum solution found in part (b)? What would the optimum solution be if the cost of maize went below this range? Above? (d) Take the cost per unit of maize to be 0.8 again. Research has found that we actually need more vitamins than previously thought. To what value could the vitamin MDR be raised before we have to start worrying about changing our diet to accomodate this new information? (Hint: Since this is a 2-variable problem, you should be able to answer these graphically, without resorting to the simplex method.) 2 4. Transportation problem In order to produce a certain product, raw material must be transported from the mine to a plant where it is refined. Say a company has control over 2 mines in Colorado and Virginia, and 3 plants in Alabama, Minnesota, and New Mexico. We denote the mines C and V, and plants A, M, and N, respectively. Below is a table of the estimated shipping costs between each plant and mine, in thousands of dollars per ton of material. C V A 22 14 M 18 20 N 7 24 For a certain week, the Colorado mine is expected to output 150 tons of ore, and the Virginia mine 130. To fulfill demand in the same week, the Alabama plant requires 88 tons of ore, the Minnesota plant 125 tons, and the New Mexico plant 55 tons. (a) Formulate the problem as a linear program. Denote xCA as the amount of ore shipped (in tons) from Colorado to Alabama, etc. (b) Introduce slack and/or surplus variables s1 , s2 , ... to make the inequality constraints (other than nonnegativity constraints) equality constraints. (c) Write the LP in matrix form. (d) A summer engineering intern from Rival State University suggests that it would be easiest if the Colorado mine exclusively supplied New Mexico and Alabama, and the Virginia mine exclusively supplied Minnesota (only enough to meet demand.) Check that not only is this option feasible, it is a nondegenerate basic feasible solution. Find the total shipping cost (in thousands of dollars) in this case. (e) Starting at the BFS mentioned in part (d), use the simplex method to determine the optimum values for the six shipment routes. What is the total shipping cost (in thousands of dollars) in this case? How much did we save as compared to part (d)? 3