STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 43 



"The position of sciences in which the methods of experimentation 

 and the theory develop hand in hand is thus more favorable than the 

 position of those fields of knowledge where experimental methods far 

 outstrip the theory, as is, e.g., observed in certain domains of experi- 

 mental biology, and then the development of the theory becomes more 

 difficult and complicated" (p. 6). 



Therefore we must not be afraid of the simplicity of the logistic 

 curve for the population of unicellular organisms and criticize it from 

 this point of view. At the present stage of our knowledge it is just 

 sufficient for the rational construction of a theory of the struggle for 

 existence, and the secondary accompanying circumstances investiga- 

 tors will discover in their later work. 4 



(7) The application of quantitative methods to experimental 

 biology presents such difficulties, and has more than once led to such 

 erroneous results that the reader would have the right to consider 

 very sceptically the material of this chapter. It is very well known 

 that the differential equations derived from the curves observed in 

 an experiment can be only regarded as empirical expressions and they 

 do not throw any real light on the underlying factors which control 

 the growth of the population. The only right way to go about the 

 investigation is, as Professor Gray ('29) says, a direct study of factors 

 which control the growth rate of the population and the expression of 

 these factors in a quantitative form. In this way real differential 

 equations will be obtained and in their integrated form they will 

 harmonize with the results obtained by observation. In our experi- 

 mental work described in the next chapter special attention has been 

 given to a direct study of the factors controlling growth in the sim- 

 plest populations of yeast cells. It has become evident that the value 



4 The usual objection to the differential logistic equation is that it is too 

 simple and does not reflect all the "complexity" of growth of a population of 

 lower organisms. In Chapter V we shall see that this remark is to a certain 

 extent true for some populations of Protozoa. The realization of the biotic 

 potential in certain cases actually does not gradually diminish with the de- 

 crease in the unutilized opportunity for growth. But this ought not to 

 frighten any one acquainted with the methodology of modern physics: it is 



K — N 



evident that the expression — — — is but a first approximation to what actually 



K. 



exists, and, if necessary, it can be easily generalized by introducing before N a 



certain coefficient, which would change with the growth of the culture (for 



further discussion see Chapter V). 



