r and the positive constant K is referred to as the environmental carrying

R and the positive constant k is referred to as the

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r , and the positive constant K is referred to as the environmental carrying capacity or saturation level. 2
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Rewriting (4) and assuming a linear harvesting function, i.e. h(t) = y , we have (5) ) 1 1 ( x K rx y x A more general version of (5) can be written in discrete form as (6) ) 1 1 ( ) ( 1 1 1 t t t t t x K rx y x x which collapses to the logistic curve in (4) when =1 . 2. Data The data used here is for the Norwegian spring-spawning herring from 1950 to 1995. During this period the size of the stock varied greatly. It peaked at 11.5 million tons in 1951 but in the 16 years following 1956 the stock showed an almost uninterrupted decline, finally shrinking to 4.2 thousand tons in 1972, at which time the Norwegian spring-spawning herring was at the brink of extinction. The spawning stock remained very low until the mid 1980’s when the herring stock began to increase again and measured 5 million tons in 1995, almost half of it’s size in 1951. The bulk of the spring-spawning herring was traditionally caught by Norwegian boats, but the 1950’s witnessed a drastic increase in the landings by Russian boats. Iceland and the Faroe Islands have also been engaged in the fishing, as well as EU- countries. 3
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Figure 1. Development of the spawning stock and total landings of the Norwegian spring-spawning herring 1950-1995 (thousand tons). 0 2000 4000 6000 8000 10000 12000 14000 1950 1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 spawning stock landings 3. Results The growth function described in (6) was estimated using both linear and non-linear techniques. The estimated equation was (7) 1 2 1 1 1 ) ( t t t t t x x y x x where 1 equals r in (6) and 2 equals K r . Consequently, K can be expressed as 2 1 . Equation (7) was first estimated using ordinary least squares (OLS) but as tests indicated the presence of hetereoskedasticity - the White test statistic was 18.18 - equation (7) was re- estimated using the White heteroskedasticity consistent covariance matrix estimator (OLS in Table 1). This estimator provides correct estimates of the coefficient covariances in the presence of heteroskedasticity of unknown form. The estimated intrinsic value was -0.00005 and estimated carrying capacity was 9.2 million tons. Using OLS may lead to simultaneity bias, since we have lagged values of the herring stock on both sides of equation (7). Further, the estimates of herring stock may be subject to measurement error. Thus, (7) was also estimated using the method of instrumental variables (IV) with two different sets of instruments, lagged values of the annual catch (IV-1 in Table 1) and second lags of the herring stock (IV-2).
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