Harvesting Due to overharvesting, the population of western Atlantic cod collapsed in the early 19905. The government has since declared a moratorium on the Northern Cod shery, which drastically changed the shing industry in Atlantic Canada. Suppose that the cod population off the eastern coast of Canada recovers enough to support shing. The Department of Fisheries and Oceans will want to decide on a. safe xed proportion of sh, It, to authorize for harvmt each year, following the reproduction season. This means that h. is a nonnegative real number and the size of the catch is determined by multiplying h times the cod population at the end of the previous year. The modied discrete-time dynamical system will be: r = 4 N N "I (Home) ' h ' Use this for problem 3 and onward. N will still represent sh in tens of millions and t will still be in years. We will work to nd the maximal number of sh that can be harvested, while maintaining a stable population. 3. Again suppose that for Atlantic cod '3' = 2.51 and nd the cquilibria points for this new system algebraically (the expression for one of the equilibrium points will contain the parameter h}. 4. 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 problem 3 is stable. The annual yield of the cod population at equilibrium will be the product of two terms: 0 the positive equilibrium value and o the safe proportion value h. Write down the formula for the annual yield. YUZ), a function of h. 5. The Department of Fisheries and Oceans wants to maximize the annual yield. Find the value for h that will give you a Iilaximal annual yield, i.c. find the local Inaxirnulri value of the function you found in problem 4. and show that it is indeed maximal using the let derivative test - we will see that this is actually a global max. (remember that h is non-negative, so the domain of Y(h) is [0, oo)}. Please include the rst two decimal places and round the answer you get from the calculator. 6. Using; the expressions for the positive equilibrium population from problem 3 and the annual yield from problem 4, nd the specic values by substituting the value It you found in problem 5. Please include the rst two decimal places and round the answer you get from the calculator. (Give the answer also in terms of numbers of sh.) 7. Use the slope criterion from the derivative test to determine whether this equilibrium popu- lation [the one referenced in problem 6] is indeed stable. You will want to rst substitute the value for h from problem 5 before differentiating. The zero population represents a collapse of the shery. Determine if the zero solution is stable using the slope criterion. Problem 3. The equilibrium points are I = Problem 4. The annual yield Y (h) = Problem 5. The value for h that gives a maximal annual yield h = Problem 6. With the maximal h from problem 5, the equilibrium population is z = ( fish), the annual yield Y (h) = ( fish). Problem 7. The equilibrium point * = is (stable or unstable). The equilibrium point _ = is (stable or unstable)