56 THE STRUGGLE FOR EXISTENCE 



was justified in the case of the competition for common food dis- 

 cussed before, we now take a further step and express a non-linear 

 relation of the increase per predator with the concentration of the 

 prey. All this will be easier to understand when in Chapter VI we 

 pass on the analysis of the experimental material. We shall then 

 explain also the meaning of the coefficient X in the equation (23). 



(3) The question now arises as to how to express the natural in- 

 crease of the prey. As noticed already, the growth of the prey in a 

 limited microcosm in the absence of predators can be expressed in the 

 form of a potential geometric increase biNi, which at every moment 

 of time is realized in dependence on the unutilized opportunity for 



growth — ^ ? — -. Therefore, the natural increase in the number of 



prey per unit of time can be expressed thus : 



_ = 6 1 Ar 1 _^__. 



It is easy to see that in the presence of the predator devouring the 

 prey the expression of the unutilized opportunity for growth of the 

 latter will take a more complex form. The unutilized opportunity 

 for growth will, as before, be expressed by the difference (taken in a 

 relative form) between the maximal number of places which is pos- 

 sible under given conditions (K{), and the number of places already 

 occupied. But the number of prey (A r i) which in the presence of the 

 predator exist at the given moment, does not reflect the number of 

 the "already occupied places." In fact the prey which have been 

 devoured by the predator have together with the actually existing 

 prey participated in the utilization of the environment, i.e., consumed 

 the food and excreted waste products. Therefore, the degree of 

 utilization of the environment is determined by the total of the pres- 

 ent population (JVi) and that which has been devoured (n). 8 The 

 expression of the unutilized opportunity for growth of the prey will 

 have then the following form: 



8 These calculations are true only in the case of the competition for a certain 

 limited amount of energy (see Chapter V). In other words it is assumed that 

 the nutritive medium is not changed in the course of the experiment. With 

 change of the medium at short intervals (as discussed in Chapter V) the term 

 n disappears, and the situation becomes more simple. 



