SECT. 4] FISHERY DYNAMICS THEIR ANALYSIS AND INTERPRETATION 475 



accentuates the changes in the latter with F or t c , and hence in the growth rate, 

 which in turn affects the spawning potential of the adult stock. 



The problems involved in the treatment of recruitment as a factor in the 

 analytical population model have been discussed at some length because, 

 generally speaking, the characteristics of the recruitment to marine fish 

 populations — its degree of fluctuation, its relation to stock size and the in- 

 fluence on it of changing environmental conditions — are the key to the inter- 

 pretation and prediction of the long-term dynamics of a fishery, whether by 

 the Beverton-Holt or by the Schaefer approach, which will now be described. 



3. The Schaefer Approach 



If, in equations (1) and (2), we combine the rates of recruitment, growth and 

 natural mortality into a single function of P, f{P), called "the coefficient of 

 natural increase", 1 we obtain for average environmental conditions: 



1 (IP 



-—=f {P )-F(X) (14) 



and, for steady-state equilibrium, 



F(X)=f(P), (15) 



Y = PF(X) = Pf(P), (16) 



where Y is the total catch. f(P) is taken to be a single-valued function of P. 

 At the environmental-limited upper value of the average population magni- 

 tude, L, in the absence of any fishery, it must be equal to zero, since, in this 

 circumstance, dP/dt = and F = Q. It must increase with decreasing P. This is 

 a "negative feedback" term which is necessary to describe the self-regulating 

 property by which the rate of natural increase is regulated appropriately to 

 bring the population into balance again with the imposition of a fishery. 



The utility of this approach in fishery dynamics, and in other branches of 

 population dynamics as well, depends on being able to apply a fairly simple 

 approximation to f(P). f(P) may be expressed as a power series, f(P) = 

 k(L — P — bP 2 — cP s ■ • ■). Now, the simplest approximation that we can make is 

 to assume that it is linear with P (Moran, 1954), that is, to take only the first 

 two terms of the series, so that 



f(P) = k(L-P). (17) 



This, of course, corresponds to the Verhulst-Pearl "logistic" or "sigmoid" law 

 of population growth, which provides a useful approximation for a wide variety 

 of organisms, and seems to be also a useful first approximation to the growth 

 in biomass of at least some fish populations. 



1 Schaefer (1954) has called the equivalent of Pf(P) the "rate of natural increase", 

 which was designated /(P) in that paper. 



