Catch (thousands of tons)

(a) To get a sense of the issues, make plots of CPUE vs year (remembering that CPUE is an index of abundance), catch vs year, and cumulative catch vs year.

(b) You are going to use a Schaefer model without process uncertainty but with observation error to analyze the data. That is, we assume that the biological dynamics are given by Eq. (6.10). Ray Hilborn and I treat the case in which both r and K are unknown, but here we will assume that r is known from other sources and is r = 0.39. However, carrying capacity K is unknown. Assume that the index of abundance is CPUE and is proportional to biomass; the predicted index of abundance is /pre(t) = qN(t), where q is the catchability coefficient. As with r, Hilborn and I consider the case in which q also has to be determined. To make life easier for you, assume that q = 0.000 45. However, the index /pre(t) is not observed. Rather, the observed CPUE is CPUE(t) = Jpre(t)eZ(t) where Z(t) is normally distributed with mean 0 and standard deviation a. (c) Show that Z(t) = log{CPUE(t) —log(/pre(t))} so that the log-likelihood of a single deviation Z(t) is L(t) = —log(a) — (1/2)log(2p) — (Z(i)2/2a2). The total log-likelihood for a particular value of K is the sum of the single year log-likelihoods LT(K|data) = i=1965L(t), where I have emphasized the dependence of the likelihood for K on the data. (d) Compute the total log-likelihood associated with different values of carrying capacity K, asK ranges from 2650 to 2850 in steps of 10. To do this, use Eq. (6.10) to determine N(t) for each year, assuming that the population started at K in 1965. Find the value of K that makes the total log-likelihood the largest. Denote this value by K* and the associated total log-likelihood by Lt; it is the best point estimate. Make a plot of LT (x-axis) vs K (y-axis) and show K* and Lt. (e) From Chapter 3, we know that the 95% confidence interval for the carrying capacity are the values of K for which the total log-likelihood LT = Lt — 1-96. Use your plot from part (d) to find these confidence intervals. (Note: if you look in Hilborn and Mangel (1997), you will see that the confidence intervals are much broader. This is caused by admitting uncertainty in r and q, and having to determine a as part of the solution. But don't let that worry you. We also used the negative log-likelihood, which is minimized, rather than the likelihood, which is maximized.) (f) At this point, you should have estimates for the 95% confidence interval for carrying capacity. Now suppose that the management objective is to keep the population within the optimal sustainable region, in which N(t) > 0.6K from 1988 to 2000 (assume that you were doing this work in 1988). Determine the catch limit that you would apply to achieve this goal. Hint: How do you determine the population size in 1987?

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