Development of the Model 



In fishery biology, it is generally assumed that after some 

 early life history stages, the mortality rate is constant. 

 That is, the proportion reaching age t is given by 



§ =e-* (1) 



where Z is the constant instantaneous rate of mortali- 

 ty, N, is the number surviving to age t, and N is the 

 initial number present so that N/N is the proportion 

 surviving to age t. 



Suppose the longevity of a stock is defined as the 

 age, (,, to which a proportion, k, of the animals sur- 

 vive, where k is some arbitrarily small constant (e.g., 

 0.01). Then 



and 



k = e- /J <- 



\n(k) = -Zt,, (2) 



Equation (2) describes a hyperbola which can be 

 linearized by plotting the mortality rate against \/t L 

 or by plotting log (Z) against log {t L ). 



In Equation (2), t, is a quantile that is determined by 

 aging the fish in the upper tail of a length- frequency 

 sample. However, it is considerably easier to find the 

 maximum age, t mkx , in a sample (by aging just the 

 largest few fish) than it is to estimate a quantile. 

 Thus, it is of interest to know if Equation (2) will hold, 

 at least approximately when t max is substituted for t L . 



Tanaka (1960) plotted the mortality rate versus 1/ 

 t m „ for five fish species and suggested that the ap- 

 parently linear relationship deserves further 

 investigation. Beverton (1963) and Bayliff (1967) 

 made the same kind of plot for fishes in the families 

 Clupeidae and Engraulidae, and Ohsumi (1979) in- 

 vestigated the situation within the Cetacea. 



In this paper, plots of log (mortality) versus log (t m J 

 were investigated for three taxonomic groups com- 

 prising 134 stocks. 



Data and Results 



Data on the total mortality rates and the corre- 

 sponding maximum observed ages were taken mainly 

 from the compendia by Beverton and Holt (1959), 

 Ohsumi (1979), and McBride and Brown (1980). 

 Most of the data pertain to unexploited or lightly ex- 

 ploited stocks. All of the data are shown in Figure 1 

 and their sources are listed in Hoenig (1982). The data 

 for the mollusks are shown separately in Figure 2. 



Results of calculating ordinary least squares linear 

 regressions on the log transformed data are given in 

 the following table: 



The predictive equations are of the form 



ln(Z) = a + b In (t m J. 



The four regression lines are very similar. The com- 

 bined regression equation makes use of data over the 

 widest possible range of ages (1-123 yr) and has the 

 highest coefficient of determination (r). It is suggest- 

 ed that the combined regression equation be used for 

 predictive purposes for all three groups. 



Discussion 



The high values of the coefficients of determination 

 in the above regressions indicate that the equations 

 have considerable predictive power. The relation- 

 ship between mortality rate and maximum age ap- 

 pears to hold within a species as well. This is demon- 

 strated by the data for 1 stocks of Pacific razor clam, 

 Siliqua pa tula, and 6 stocks of NuttalFs cockle, ( 'lino- 

 cardium nuttallii, shown in Figure 2. 



In deriving the regression approach, it was assumed 

 that the mortality rate does not vary with age. How- 

 ever, it is well known that in at least some groups of 

 fish (e.g., sturgeons, Ricker 1975: ch 2; clupeids and 

 engraulids, Beverton 1963; and salmonids, Gerking 

 1957) the mortality rate appears to increase with age. 

 Concave catch curves, suggestive of decreasing mor- 

 tality rate with age, have sometimes been reported 

 but these have usually been given other interpre- 

 tations (Ricker 1975: ch. 2). In general, not much is 

 known about the mortality rates among the oldest 

 animals of most species (and how mortality might 

 vary among taxa). 



The regressions presented here are based largely on 

 data from unexploited stocks. Since the scatter plots 

 and regression statistics indicate a strong linear re- 

 lationship between the maximum age and the mor- 

 tality rate, the method works well for predicting 

 mortality rates in unexploited stocks. If age trunca- 

 tion is a common phenomenon among the stocks for 

 which data were available, then the application of this 

 technique to heavily exploited stocks may result in an 

 underestimate of the mortality rate. 



Applications 



The regression technique can be used in several dis- 

 tinct applications: 



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