THE PROBLEM 7 



are as yet in the stage of nothing but a general analysis of the proc- 

 esses of ontogenesis. 



In past years several eminent men were deeply conscious of the 

 need for a mathematical theory of the struggle for existence and took 

 definite steps in this domain. It often happened that one investi- 

 gator was ignorant of the work of another but came to the same con- 

 clusions as his predecessor. Apparently every serious thought on 

 the process of competition obliges one to consider it as a whole, and 

 this leads inevitably to mathematics. A simple discussion or even 

 a quantitative expression of data often do not suffice to obtain a 

 clear idea of the relationships between the competing components 

 in the process of their growth. 



(9) About thirty years ago mathematical investigations of the 

 struggle for existence would have been premature, or in any case sub- 

 ject to great difficulties, due to the absence of the needed preliminary 

 data. Of late years, owing to the publication of a number of investi- 

 gations, these difficulties have disappeared of themselves. What is 

 it that these indispensable preliminary researches represent? 



There is no doubt that a rational study of the struggle for existence 

 among animals can be begun only after the questions of the multipli- 

 cation of organisms have undergone a thoroughly exact quantitative 

 analysis. We have mentioned that the struggle for existence is a 

 problem of the relationships between species in mixed growing groups 

 of individuals. We must therefore begin by analyzing the laws of 

 growth of homogeneous groups consisting of individuals of one and 

 the same species, and the competition between individuals in such 

 homogeneous groups. During the second half of the last century and 

 the beginning of the present much has been said about multiplica- 

 tion, and "equations of multiplication" have even been proposed of 

 the following type: the coefficient of reproduction — the coefficient 

 of destruction = number of adults. (Vermehrungsziffer — Vernich- 

 tungsziffer = Adultenziffer; see Plate ('13) p. 246.) Usually, how- 

 ever, things did not go any further, and no attempts were made to 

 formulate exactly all these correlations. Recently the Russian geo- 

 chemist, Prof. Vernadsky, has thus characterized from a very wide 

 viewpoint the phenomena of multiplication of organisms ('26, p. 37 

 and foil.): "The phenomena of multiplication attracted but little 

 the attention of biologists. But in it, partly unnoticed by the natu- 



