242: ESM Assignment #1 Due in class Tuesday April 17, 2007 1 Recreational Abalone Diving
Here we consider recreational abalone diving off the coast of Northern California. In this question we'll develop and analyze a simple stylized model to assess the impact of various policy changes in the sport abalone fishery. Consider the following population growth model: Nt+1 = Nt + rNt 1 - Nt K - Ht (1)
where Nt is the biomass of abalone, r = 0.2 is the intrinsic growth rate, K = 1000 is the carrying capacity of the abalone population, and Ht is the harvested biomass in year t. 1. Identify the variable(s) and the parameter(s) of this model. 2. Suppose the abalone stock is severely depressed; say it is at 10% of the carrying capacity. Write a program in Excel to simulate the abalone population size over a 25 year period without harvest (i.e. if the fishery is shut down). Graph the recovery over time. Suppose we say that a population is "recovered" when it reaches 90% of carrying capacity. In what year did this population "recover"? Graph the "recovery" time as a function of the intrinsic growth rate, r. 3. Here, fishing "effort" is measured in diver-days. One diver-day is the equivalent of one person diving for the entire day. The average catch per diver-day (Ct ) depends on the daily limit and on the population in the following way: Ct = min(L, Nt ) (2)
where L is the daily bag limit, Nt is the population, and = .006 is called the "catchability coefficient". Use Excel to produce several plots of C vs. N (with varying limits, L). Describe in words how C depends on N . 4. The number of diving-days depends on the expected catch: Et = a ln(b Ct ), where a = 20 and b = 10. Generate a few plots of E vs. C, and describe in words how effort depends on expected catch. 5. Finally, total harvest of abalone in a year is given by the effort (number of diving-days) multiplied by the average catch per day: Ht = Et Ct . Use Excel to develop a model of this stylized abalone fishery. Assume the limit is L = 4. Your model should simulate N , C, E, and H over a 25 year period. Describe the time path of these variables over the 25 year period an for abalone population that starts at carrying capacity (N = K). 6. Do the same (as the above question) for an abalone population that starts at 10% of the carrying capacity (N = 100). 7. Now suppose an oceanographic regime shift causes r to decrease from 0.2 to 0.1 and K to decrease to 850. Again, starting the population at 100, how does this regime shift affect the above variables through time? 8. Returning to the original parameter values (and starting at N = 1000), how does the population trajectory depend on the limit? (NOTE: you can allow for non-integer limits). Try this for L between .5 and 4. Describe, interpret, and explain your results. 1
9. Mathematically, how would you incorporate illegal fishing (i.e. harvesting over the legal limit) into this model?
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Exploring Abalone Policy
Use Solver to explore different harvest policies for abalone: 1. Use the original parameter values and start with N = K. Suppose you want to maximize the total number of diving-days over a 25 year period. What harvest limit (L) would you choose? What is the tension here (i.e. why might you want L to be smaller? Why might you want L to be bigger)? 2. Now suppose you want to maximize the present value of consumer surplus of divers over a 25 year period. If each dive-day is worth $30 and the discount rate is 5%, what harvest limit (L) should be chosen? Explain how/why your answer differs from above. 3. Suppose instead you want to maximize the average catch per dive-day over the 25 year period. What is your choice of L? 4. Start with N = K/2 and determine the limit that maximizes consumer surplus with a 5% interest rate. Now add a constraint that N25 700. How does this recovery constraint impact your choice of L?
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Term Paper Proposal
Write a proposal not to exceed two pages that identifies the question you will address with your term paper research. Be as specific as possible about your research question. Discuss the relevance of your question. Briefly outline the methods you will use to answer this question. Identify what data and experts you will pursue. Conclude with a time-line of your proposed research. [Recall, this counts for 10% of your final term paper grade.]
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