SECT. 4] THEORY OF FOOD-CHAIN RELATIONS IN THE OCEAN 439 



animals. Wangersky and Cunningham (1957) have examined this problem. 

 With variations in the time lag in relation to certain other parameters, they 

 were able to demonstrate a variety of results including damped oscillations, 

 monotonic approach to equilibrium and an expanding oscillation which prob- 

 ably ultimately approaches a limit cycle. True Volterra oscillations were 

 possible with a finite time lag, but only within a narrow range. 



Bartlett (1957) studied the effects of stochastic fluctuations on several kinds 

 of population models with results that cast doubt on the accuracy of deter- 

 ministic solutions. In the case of the prey-predator equations, his stochastic 

 model showed three distinct peaks of prey and predators, then a sudden 

 change to a smaller and poorly defined cycle, and finally extinction of the 

 predators. 



Ecological models present more problems than the classical population 

 models. In all that thus far have been devised, the prey and predators have 

 been plankton populations containing an assemblage of species. An attempt 

 must be made to frame physiological coefficients in average terms. The fact 

 that there are species differences in behavior and a variable species composi- 

 tion inevitably increases the margin of error. Furthermore, ecological models 

 generally are devised to include effects of variations in physical and chemical 

 properties as well as biological relationships. This is clearly a more difficult 

 problem than the laboratory models which postulate a constant environ- 

 ment. The effects of stochastic fluctuations have not been thoroughly in- 

 vestigated in ecological models, but they must be accepted as a possible cause 

 of error. 



For these reasons an attack on ecological problems on a purely theoretical 

 basis would inspire little confidence. It is necessary to apply the equations to a 

 particular situation in nature and compare theoretical results with field observa- 

 tions. In some cases it will be seen that the comparison is quite accurate, 

 particularly when random errors are minimized by using average values for 

 environmental factors and plankton which have been obtained at a series of 

 closely spaced stations. Useful conclusions have already been obtained by 

 means of the theoretical approach, crude as it is in its present state, and it 

 seems to be a promising tool for future development. 



2. Mathematical Models of Plankton Populations 



Harvey, Cooper, LeBour and Russell (1935) described the spring diatom 

 flowering in the English Channel and proposed that its termination was pri- 

 marily due to grazing by the zooplankton population, which increased during, 

 and as a result of, the flowering. Fleming (1939) postulated a relationship based 

 on equation (1), taking the form 



dP 



_ = p [a -(b + ct)], (3) 



where P is the total phytoplankton population underlying a unit area of sea 



