50 lotka: discontinuous evolution 



the other. Among the various possible cases of this kind we can 

 distinguish the following: 



1. A group Ai feeds on living portions of a group Ah, which 

 are either 



(a) expressly killed in the act of feeding, or 



(b) continue to live, at least for a time, while A\ is parasitic 

 upon them (parasites, disease germs). 



2. The waste products A n of a group Ah. serve as food for A\. 



3. Several groups, such as A, and A] feed competitively on 

 the same group Ah. 



Schematically we may represent these three cases by the 

 following diagram 



(1) Ah-^A^A k 



(2) A h (->A H )->A,->Ak 



(3) A 





In the case of type (1) Ai will as a rule tend to exert a prejudi- 

 cial influence upon the growth of Ah. An exception occurs where 

 there is symbiosis with mutual benefit, the term being here used 

 in a general sense, to include for example the relation between 

 man and farm plants and animals. 



In the case of type (2) the influence of Ai upon Ah will be rather 

 beneficial than otherwise, since waste products are by the very 

 nature of things more or less harmful to the group from which they 

 originate. So long, however, as there is no undue crowding, so 

 that the waste products are sufficiently spread out in space ("di- 

 luted"), this effect will as a rule be small, and may often be negli- 

 gible. 



Case (3) may be regarded as a special form of cases (1) and (2). 



If we denote the mass of any given group by the letter M with 

 the proper subscript, we may express in analytical form the state- 

 ments made in the last three paragraphs. Arranging the facts 

 in a table we have: 



