SECT. 4] FISHERY DYNAMICS— THEIR ANALYSIS AND INTERPRETATION 465 



A very general model describing these dynamics may be formulated by 

 expressing the relative rate of change in biomass of the fished stock in terms of 

 the component factors influencing it ; 



1 dP 



Pit = r(P) + 9(P)-M{P)-F(X)+ri, (1) 



where P = biomass of the fishable part of the population, r, g and M = rates of 

 recruitment, growth and natural mortality respectively ; all of these are, in 

 general, functions of the biomass P of the population and its age composition. 

 i^ = rate of loss due to fishing, which is a function of the fishing effort, X. 

 77 = a (variable) rate of change of stock biomass, independent of both P and X, 

 due to variations in external environmental factors. In practice, these effects 

 are eliminated as far as possible by averaging over periods of time so that the 

 model can be regarded as describing events for average environmental 

 conditions. 



In the steady state, with the population in equilibrium under average 

 environmental conditions, dP/dt = Q and 17 = 0, so that 



F(X) = r(P) + g(P)-M(P). (2) 



In these circumstances, the average steady catch, Y = F(X) P, is equal to the 

 additions due to recruitment and growth, less the losses by natural death, i.e. 



Y = F(X) P = [r(P) + g(P)-M(P)] P. (3) 



Ideally, this general model could be applied by measuring the various ele- 

 mental rates and establishing from observation their dependence on the 

 biomass and age-composition of the population. In practice, however, a 

 complete description in such terms of the dynamics of exploited fish popula- 

 tions has not yet proved possible, and it is necessary to proceed by making 

 certain kinds of simplifications to the general model, depending on the kind 

 and amount of data available and the particular questions concerning the 

 population and its associated fishery which it is desired to answer. 



Two general types of approach have been developed. That most widely used 

 has been to retain the identity of the various elemental rates, to estimate their 

 magnitude and relations as far as is possible from the data available, and to 

 combine them in an appropriate special form of the general model (3), making 

 such simplifying assumptions as are required for practical application. This 

 analytical approach, with which are associated such investigations as those of 

 Baranov (1918), Thompson and Bell (1934), Ricker (1944) and others, has been 

 developed in detail by Beverton and Holt (1957) ; it is for convenience referred 

 to here as the "Beverton-Holt" approach. 



The alternative method does not attempt to distinguish between the ele- 

 mental rates of recruitment, growth and natural mortality, but considers only 

 their resultant effect as a single function of population size. This approach 

 stems from the concept of the logistic law of population growth (e.g. Pearl, 

 1930; Lotka, 1925 ; Gause, 1934) and was applied to the dynamics of a fishery 



16— s. TI 



