20. THEORY OF FOOD-CHAIN RELATIONS IN THE OCEAN 



G. A. Riley 



1. Introduction 



Descriptive oceanography reveals a variety and complexity of species 

 relationships and environmental influences that defy quantitative analysis 

 except by recourse to simplifying procedures of one sort or another. The 

 obvious solution to many of these problems is laboratory experimentation under 

 simplified and controlled conditions. Such work leads to a better quantitative 

 understanding of ecological relations than mere observation of nature ; however, 

 it is desirable to take the further step of recombining experimental results into 

 a quantitative synthesis of the interaction of processes in nature. This is the 

 only way we can demonstrate to our satisfaction that the experiments are 

 realistic. In order to effect such a synthesis, the experiments must be combined 

 within a logical framework which duplicates, as closely as our knowledge 

 permits, the relationships of organisms and environmental factors that we find 

 in nature. 



All of the work that has been done on the theory of food-chain relations has 

 stemmed from the prey-predator relationship postulated by Volterra (1928). 

 Defining N\ as the number of prey and No as the number of predators, Volterra 's 

 equation for the rate of change of prey with respect to time was 



^i = N 1 (b 1 -k 1 N 2 ), (1) 



where b\ is the growth coefficient of the prey, and lc\ is a predation coefficient 

 having the sense that the predator "hunts" a constant amount of space in unit 

 time and consumes all of the prey therein. The equation for predators is 



^ = N^Nr-d*), (2) 



where lc% is a coefficient of food assimilation which in the first approximation 

 may be synonymous with k\ or may be some constant fraction of it. The 

 coefficient of death, e^, is also a constant. 



According to these equations, an increase in prey leads to an increase in the 

 number of predators, which in turn reduce the net rate of growth of the prey 

 until the latter begins to decline. Continuation of the sequence leads to the well 

 known prey-predator oscillation. 



In this ideally simple form neither the equations nor their integrated results 

 are biologically realistic. Smith (1952) has discussed some of the defects. How- 

 ever, the equations provide a logical framework, and changes that make them 

 more realistic do not alter the basic structure. The constants are not likely to 

 remain the same through the whole cycle, but this can be corrected on the basis 

 of known facts about the organisms in question. There is likely to be a time lag 

 between feeding and reproduction, which will vary considerably in different 



[MS received Auyust, 1960 J 438 



