136 



THE STRUGGLE FOR EXISTENCE 



A mathematical analysis of the properties of the complicated equa- 

 tion of relaxation given in Fig. 38 shows that there is a point on the 

 map of the curves of interaction which is usually called "singular 

 point." The powers of mortality, natality and interaction of prey 

 and predators are so balanced that the "classical" oscillations in 

 numbers are theoretically possible around it. But in the case of 

 Paramecium and Didinium the coordinates of this singular point are 

 exceedingly small. In other terms the zone of possible classical oscil- 

 lations is displaced here to such small densities that these oscillations 



Line of Aortgo/rda/ tangents 

 1/ 



A/ z (Didinium) -^ 



Fig. 38. The solution of the complicated equation of relaxation (21c). 

 ah represents the threshold concentration of the prey. 



are completely annihilated by the statistical factors which are much 

 more powerful in this zone. 



(4) In conclusion let us consider the appearance of periodic varia- 

 tions in numbers under the influence of immigrations (a slight and 

 synchronous inflow of Ni and N2 after intervals of time t). In other 

 terms we have to deal here with the problem of the influence of small 

 impulses. At the origin (a, Fig. 37) they lead to a return of the curve 

 to the ordinate. Relaxations arise when the concentration of Ni rises 

 above the threshold. From Figure 38 it is easy to calculate how a 



