STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 31 



investigate the change with time (t) of the number of persons infected 

 with malaria (z). 



The equations of Ross were submitted to a detailed analysis by 

 Lotka ('23, '25) who in his interesting book Elements of Physical 

 Biology gave examples of some other analogous equations. As Lotka 

 remarks, a close agreement of the Ross equations with reality is not 

 to be expected., as this equation deals with a rather idealized case: 

 that of a constant population both of human beings and mosquitoes. 

 "There is room here for further analysis along more realistic lines. 

 It must be admitted that this may lead to considerable mathematical 

 difficulties" (Lotka ('25, p. 83)). 



(3) The equations of Ross point to the important fact that a mathe- 

 matical formulation of the struggle for existence is a natural conse- 

 quence of simple reasoning about this process, and that it is organi- 

 cally connected with it. The conditions here are more favorable than 

 in other fields of experimental biology. In fact if we are engaged in 

 a study of the influence of temperature or toxic substances on the life 

 processes, or if we are carrying on investigations on the ionic theory of 

 excitation, the quantitative method enables us to establish in most 

 cases only purely empirical relations and the elaboration of a rational 

 quantitative theory presents considerable difficulties owing to the 

 great complexity of the material. Often we cannot isolate certain 

 factors as we should like, and we are constantly compelled to take 

 into account the existence of complicated and insufficiently known 

 systems. This produces the well known difficulties in applying 

 mathematics to the problems of general physiology if we wish to go 

 further than to establish purely empirical relations. As far as the 

 rational mathematical theory of the struggle for existence is con- 

 cerned, the situation is more favorable, because we can analyze the 

 properties of our species grown separately in laboratory conditions, 

 then make various combinations and in this way can formulate 

 correctly the corresponding theoretical equations of the struggle for 

 life. 



(4) Besides the interest of the equations of Ross as the first attempt 

 to formulate mathematically the struggle for existence, they allow 

 us to answer a very common objection of biologists to such equations 

 in general. It is frequently pointed out that there is no sense in 

 searching for exact equations of competition as this process is very 

 inconstant, and as the slightest change in the environmental condi- 



