Problem 1: Fisheries Population Dynamics [3.5 points] Consider a tuna stock at the beginning of time t, denoted by the variable Xt, measured in metric tons. Suppose that in each period, the stock of tuna grows according to the difference equation X1\" = X; + PUG] Ht, where 17(th is a function representing the net growth of tuna and H; is the harvest of tuna at time t. Suppose that net growth of tuna depends on the size of the sh stock and is given by the function For.) = X,{1 xt). LA. [11.5 points] Explain the concept of a steady state. What are the biological steady states for the tuna stock if there is no harvesting? Are these steady states stable? LB. [0.5 points] Is there some level of X at which the net growth of tuna is maximized? If so, what is it? LC. [t].5 points] Now suppose that we have the following simple rule for determining the harvest of tuna: H, = text. Determine the steady state stock and harvest levels of tuna. (Note: Actually, there are two steady states, one when X = . i'm only interested in the second one.) II). [11.5 points] Suppose your goal is to maximize sustainable harvests of tuna. What level of or would you choose? LE. Now suppose that prots from the shery are given by 11', = pH: (chKHEth), where p is the perunit price for sh and o is a cost parameter. For the following questions, let p = o and e = 4. i} [0.5 points] Using Excel\" Solver tool, nd the or that maximizes steadystate prots. Is this the same as the n that maximizes sustainable yields? Why or why not? What is the steady-state stock level X\"? ii} [0.5 points] Now suppose that we are interested in the present value of profits over let} periods, that is H = 23:3] 1311 + djis't, where d is the discount rate. Let (1' = [1115 and suppose that we are initially at X (D) = X\