Math 1215 Second Project - Due November 14, 2016 General Instructions This project must be typed or written neatly and legibly by hand. (Messy work can cost up to 5 points.) Projects consisting of more than one page need to be stapled! You need to give the question numbers with your answers. You may do this project in groups of one or two people from either section. Do not show your work to other students. If they copy it, you are responsible as well. Include a printout of all relevant spreadsheet work. We will not accept late submissions. Fish Population Growth The general discrete dynamical system model for a fish population assumes a per capita production rate given by r , p(N ) := 1 + 0.01N where N is the number of fish in thousands (i.e., N = 5 means 5000 fish). The value of r, the maximal possible reproduction rate, depends on the particular species of fish and environmental factors. 1. Give the corresponding discrete-time dynamical system to describe the development of a fish population Nt . (Assume that the unit for t is years. That is, t = 1 means one year.) 2. Suppose that for Atlantic cod, r = 2.6. What are the equilibrium points in this case? (Derive your answer algebraically.) 3. Determine the stability of each equilibrium point found in the previous part, using the derivative test. Harvesting Because of the over-harvesting, the population of western Atlantic cod collapsed in the early 1990s. The government has since declared a moratorium on the Northern Cod fishery, which drastically changed the fishing industry in Atlantic Canada. Suppose that the cod population off the eastern coast of Canada recovers enough to support fishing. The Department of Fisheries and Oceans will want to decide on a safe fixed proportion h to authorize for harvest each year, following the reproduction season. This means that h is a non-negative real number and the size of the catch is determined by multiplying h times the cod population at the end of the previous year. For the problems below, N will still represent fish in thousands and t will still be in years. 4. Give a discrete-time dynamical system that describes the development of the fish population with per capita production rate r p(N ) = 1 + 0.01N 1 and harvesting coefficient h. (For this problem, keep h and r as parameters; do not substitute in any values for them.) 5. Now substitute r = 2.6, and find the equilibrium points for this new system algebraically (your answers will still depend on h; i.e., the expression for the equilibrium points will contain the parameter h). 6. To prevent another collapse in the population of Atlantic cod, we need to harvest from a stable equilibrium population. Assume for the moment that the positive equilibrium point that you found in the previous question is stable. The annual yield of the cod population at equilibrium will be the product of this value and h. Give an algebraic formula for the yield Y (h) (as a function of h). 7. The Department of Fisheries and Oceans obviously wants to maximize the annual yield. Find a value for h that will give you a maximal yield and show that it is indeed maximal (remember that h is non-negative, so the domain of Y (h) is [0, )). 8. Find the equilibrium population and the annual yield corresponding to the value of h you found in #7. (Give the answer in numbers of fish.) 9. Use the slope criterion to determine whether this equilibrium population (the one referenced in #8) is indeed stable. The zero population represents a collapse of the fishery. Determine if the zero solution is stable using the slope criterion. 10. Write a statement for the Department of Fisheries and Oceans to give them advice about the proportion they should authorize for harvest, summarizing the above results and addressing the following issues: (a) Suppose that, right now, the Northern Cod fishery has a total of 30,000 fish; how many fish should they catch? (b) What can they expect for the future? i. What will the equilibrium populations be? ii. What will the yield be in that case? iii. Will this be a stable equilibrium (and interpret the practical meaning of 'stable equilibrium')? iv. With this policy, can the fish population recover from a serious setback? 2