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error). For example, if we assumed that biological production, but not catch, were subject to process uncertainty and that this uncertainty had a log-normal distribution, then the Schaefer model, Eq. (6.10), would be modified to

where Zp is a normally distributed random variable with mean 0 and standard deviation cp. Our index of abundance is still catch per unit effort, but this is now observed with error, so that we have an index of abundance

where Zobs is a normally distributed random variable with mean 0 and standard deviation o-obs.

Exercise 6.12 (E)

Referring to Chapter 3 and the properties of the log-normal distribution, explain why Eq. (6.30) produces a biased index of abundance, in the sense that E{I(t)} > qN(t). Explain why a better choice in Eq. (6.30) is that Zobs is a normally distributed random variable with mean —(1/2)(aobs)2 and standard deviation aobs. Would this cause you to change the form of Eq. (6.29)?

One of the great quantitative challenges in fishery management is to figure out practicable means of analysis of models such as Eqs. (6.29) and (6.30) (or their extensions; see Connections). The following exercise, which is a simplification of the analysis in Hilborn and Mangel (1997, chapter 10) will give you a flavor of how the thinking goes. Modern Bayesian methods allow us to treat process uncertainty and observation error simultaneously, but that is the subject for a different book (see, for example, Gelman et al. (1995), West and Harrison (1997)).

Exercise 6.13 (M)

The Namibian fishery for two species of hake (Merluccius capensis and M. paradoxus) was managed by the International Commission for Southeast Atlantic Fisheries (ICSEAF) from the mid 1960s until about 1990. Your analysis will be concerned with the data from the period up to and including ICSEAF management. Hake were fished by large ocean-going trawlers primarily from Spain, South Africa, and the (former) Soviet Union. Adults are found in large schools, primarily in mid-water. While both species are captured in the fishery, the fishermen are unable to distinguish between them and they are treated as a single stock for management purposes. The fishery developed essentially without any regulation or conservation. Catch per unit effort (CPUE), measured in tons of fish caught per hour, declined dramatically until concern was expressed by all the nations fishing this stock. The concern about the dropping CPUE led to the formation of ICSEAF and subsequent reductions in catch. After catches were reduced, the CPUE began to increase. In the data used in this analysis, CPUE is the catch per hour of a standardized class of Spanish trawlers. Such standardized analysis is used to avoid bias due to increasing gear efficiency or differences in fishing pattern by different classes or nationalities of vessels. The data are as follows.

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