468 SCHAEFEK AND BEVEHTON [CHAP. 21 



age, K is a constant which determines the rate (for given units of age) at which 

 the asymptotic weight W& is approached (i.e. the "curvature" of the growth 

 curve), and t is an arbitrary constant which is. in effect, the theoretical origin 

 of the age-scale, i.e. the age at which the weight would be zero, if growth 

 followed the von Bertalanffy equation at all ages. Actually, this equation does 

 not always hold for very young fish (well below the exploited ages), so that t 

 does not correspond to the real origin in time. Substituting in (8) this expression 

 for w t and that for N t from (7), and integrating over the whole life of the fish 

 after they first enter the fished stock at age t c , gives the following equation for 

 the equilibrium catch : 



Y = F RW<» e-MV'-tr) ^Q n e- nK «c-tv)l(F + M + nK). (10) 



In this equation the summation term is a shorthand way of writing the cubic 

 term of the growth equation (9), with 



Q = +1, Qi = -3, Qo = +3, Q 3 = -1. 



Beverton and Holt (1957) applied this equation to the plaice and haddock 

 fisheries of the North Sea, obtaining estimates of the parameters it contains 

 from the extensive data available for these two stocks. The methods of estima- 

 tion cannot be described in detail here, although some of the principles involved 

 may be outlined. The most favourable situation is one in which reliable age- 

 composition data of the catch are available over a period of years, together with 

 statistics of fishing effort in units which are proportional to the fishing mortality 

 rate generated. In a trawl fishery, for example, the total hours fished per year 

 multiplied by the average tonnage of the vessels is a suitable measure of total 

 fishing effort. It is then possible to express the age-composition for each year 

 in units of catch per unit effort, which can be regarded as indices of true 

 abundance, and hence to estimate the total mortality coefficient (F + M) from 

 the rate of decrease in the abundance of year-classes. In the absence of data of 

 fishing effort, only percentage age -compositions can be obtained, but, if these 

 are available over a sufficient period, reasonably good estimates of (F + M) can 

 usually be derived by combining all the data, a procedure which tends to 

 average out the effect of differences in year-class abundance. More difficult is to 

 separate the total mortality into its components, F and M. If statistics of 

 fishing effort are available, and if these cover periods during which the amount 

 of fishing has changed considerably, it may be possible to establish the relation 

 between total mortality coefficient (F + M) and fishing effort, which, if effort 

 is measured in proper units, should be a linear one. Extrapolation of the 

 regression of (F + M ) on effort to the origin where F = zero therefore provides 

 an estimate of the natural mortality coefficient, M . Alternatively, the fishing 

 mortality coefficient F may be estimated independently by tagging experi- 

 ments, although to obtain good estimates by this method it is necessary that 

 the population of tagged fish should be reasonably representative of that of the 



