124 THE STRUGGLE FOR EXISTENCE 



Or else (2) as shown in Figure 31a certain number of prey might have 

 in the refuge been entirely out of reach of the predators, and the 

 latter will perish finally from lack of food. (3) Lastly, prey may 

 from time to time leave the refuge and be taken by the predators; as a 

 result a mixed population consisting of prey and predators will 

 continue to exist for a certain time. All this depends on the circum- 

 stance that in our experiments the absolute numbers of individuals 

 were not large, and the amplitude of fluctuations connected with 

 multiplicity of causes proved to be wider than these numbers. 



(3) Let us consider the corresponding data. In one of the experi- 

 ments 30 microcosms were taken (tubes with 0.5 cm 3 of oaten medium 

 with sediment), in each of them five Paramecium and three Didinium 

 were placed , and two days after the population was counted . It turned 

 out that in four microcosms the predators had entirely destroyed 

 the prey whilst in the other 26 there were predators as well as prey. 

 The number of prey fluctuated from two to thirty-eight. In another 

 experiment 25 microcosms were examined after six days; in eight of 

 them the predators had died out entirely and prey alone remained. 

 Therefore, in the initial stage for every individual microcosm we can 

 only affirm with a probability of -£ 5 that it will develop in the direction 

 indicated in Figure 31. Certain data on the variability of popula- 

 tions in individual microcosms are to be found in Table 7 (Appendix). 

 Further experimental investigations are here necessary. First of all 

 we had to do with too complicated conditions in the microcosms owing 

 to variability of refuges themselves. It is not difficult to standardize 

 this factor and to analyze its role more closely. 



In concluding let us make the following general remarks. When 

 the microcosm approaches the natural conditions (variable refuges) 

 in its properties, the struggle for existence begins to be controlled by 

 such a multiplicity of causes that we are unable to predict exactly 

 the course of development of each individual microcosm. 3 From the 

 language of rational differential equations we are compelled to pass 

 on to the language of probabilities, and there is no doubt that the 

 corresponding mathematical theory of the struggle for existence may 

 be developed in these terms. The physicists have already had to 



3 This means only that the development of each individual microcosm is in- 

 fluenced by a multiplicity of causes, and it would be totally fallacious to con- 

 clude that it is not definitely "caused." All our data have of course no relation 

 to the concept of phenomenal indeterminism. 



