STRUGGLE FROM VIEWPOINT OF MATHEMATICIANS 57 



K, - (Nr + n) 



K x 



(25) 



Here we arrive at a very interesting conclusion, namely, that the 

 development of a definite biological system is conditioned not only by its 

 state at a given moment, but that the past history of the system exerts a 

 powerful influence together with its present state. This fact is appar- 

 ently very widespread, and one can read about it in a general form 

 in almost every manual of ecology. Lotka and Volterra expressed 

 mathematically the role of this circumstance in the processes of the 

 struggle for existence. 



In order to complete the consideration we have to express the num- 

 ber of the devoured prey from the moment the predator is introduced 

 up to the present time. If in every infinitesimal time interval the 

 predator devours a certain definite number of prey then the total 

 number of the devoured prey will be equal to the sum of the elemen- 

 tary quantities devoured from the moment the predator is introduced 

 up to the given time (t) . This total can be apparently expressed by 

 a definite integral. 



(4) We cannot at present ignore the difficulties existing in the field 

 of the mathematical investigation of the struggle for life. We began 

 this chapter with comparatively simple equations dealing with an 

 idealized situation. Then we had to introduce one complication 

 after another, and finally arrived at rather complicated expressions. 

 But we had in view populations of unicellular organisms with an 

 immense number of individuals, a short duration of generations and a 

 practically uniform rate of natality. The phenomena of competition 

 are reduced here to their simplest. What enormous difficulties we 

 shall, therefore, encounter in attempts to find rational expressions for 

 the growth of more complicated systems. 9 Is it worth while, on 

 the whole, to follow this direction of investigation any further? 



There is but one answer to this question. We have at present no 

 other alternative than an analysis of the elementary processes of the 

 struggle for life under very simple conditions. Nothing but a very- 

 active investigation will be able to decide in the future the problem 

 of the behavior of the complicated systems. Now we can only point 



9 We can have an idea of this from the recent papers of Stanley ('32) and 

 Bailey ('33) who try to formulate the equations of the struggle for existence 

 for various insect populations. 



