Longevity and Mortality Rates of Fish in Nature 159 



(1) as a means of representing mathematically the general 

 growth pattern of fish in terms of two parameters to provide 

 a simple means of relating size and growth to mortality and 

 lifespan. 



It is a property of equation (1) that it can be transformed 

 to a linear function relating length at age t to length at age 

 ^ + 1, namely: 



/,^i = L„(l-e-^) + /,e-^ (2) 



Fig. 3 shows the growth curves of Fig. 2 plotted in this way. 

 From equation (2) it will be seen that the slope of the line 

 drawn through the points provides an estimate of e~^, and 

 hence of K; and that the intersection of the line with the 

 bisector drawn through the origin (shown by broken lines in 

 Fig. 3) gives an estimate of the asymptotic length L^. Esti- 

 mates of L^ and K for all the species under consideration are 

 listed in Table I. 



Apart from providing a means of estimating the two para- 

 meters of the growth equation (1), plotting Z^ against Z^^^ in 

 this way is a valuable technique for the comparative analysis 

 of growth curves (Walford, 1946). For example, it can be seen 

 from Fig. 3 that male plaice not only have a lower L^ than do 

 females, but also grow towards it rather more quickly, i.e. 

 they have a higher K. In the case of Lahidesihes sicculus 

 (insert in lower part of Fig. 3) the lengths are at monthly 

 instead of yearly intervals, but when plotted one against the 

 next they nevertheless give a close approximation to a straight 

 line; in this case, however, the slope is e"^^^, and so in reality 

 is very much flatter than the other graphs of Fig. 3. The 

 method is also useful for detecting departures from the simple 

 growth pattern which sometimes arise because of special en- 

 vironmental conditions, of which lack of uniformity in the 

 supply of food to fish of difiPerent sizes is usually the most 

 important (see below and also papers by Aim, 1946, and 

 Decider, 1951). 



