116 THE STRUGGLE FOR EXISTENCE 



situation it is difficult to draw any reliable conclusions concerning 

 the causes of these oscillations. However, quite recently Lotka 

 (1920) and Volterra (1926) have noted on the basis of a purely mathe- 

 matical investigation that the properties of a biological system con- 

 sisting of two species one of which devours the other are such that 

 they lead to periodic oscillations in numbers (see Chapter III). 

 These oscillations should exist when all the external factors are in- 

 variable, because they are due to the properties of the biological 

 system itself. The periods of these oscillations are determined by 

 certain initial conditions and coefficients of multiplication of the 

 species. Mathematicians arrived at this conclusion by studying the 

 properties of the differential equation for the predator-prey relations 

 which has already been discussed in detail in Chapter III (equa- 

 tion 21a). Let us now repeat in short this argument in a verbal 

 form. When in a limited microcosm we have a certain number of 

 prey (Ni), and if we introduce predators (N2), 1 there will begin a 

 decrease in the number of prey and an increase in that of the preda- 

 tors. But as the concentration of the prey diminishes the increase 

 of the predators slows down, and later there even begins a certain 

 dying off of the latter resulting from a lack of food. As a result of 

 this diminution in the number of predators the conditions for the 

 growth of the surviving prey are getting more and more favorable, 

 and their population increases, but then again predators begin to 

 multiply. Such periodic oscillations can continue for a long time. 

 The analysis of the properties of the corresponding differential equation 

 shows that one species will never be capable of completely destroying 

 another: the diminished prey will not be entirely devoured by the 

 predators, and the starving predators will not die out completely, 

 because when their density is low the prey multiply intensely and in a 

 certain time favorable conditions for hunting them arise. Thus a 

 population consisting of homogeneous prey and homogeneous predators 

 in a limited microcosm, all the external factors being constant, must 

 according to the predictions of the mathematical theory possess periodic 

 oscillations in the numbers of both species. 2 These oscillations may be 



1 It is assumed that all individuals of prey and predator are identical in their 

 properties, in other words, we have to do with homogeneous populations. 



2 According to the theory, such oscillations must exist in the case of one 

 component depending on the state of another at the same moment of time, as 

 well as in the case of a certain delay in the responses of one species to the 

 changes of the other. 



