DESTRUCTION OF ONE SPECIES BY ANOTHER 133 



tions of the classical equation (21a) and transform it into that of an 

 elementary relaxation? For all the technical details the reader is 

 referred to the original paper (Gause and Witt, '35), and we will dis- 

 cuss here only its essential ideas. 



In a first approximation to the actual state of affairs we can write 

 an elementary equation of relaxation. It can be admitted [on the 

 basis of the observations on Paramecium (Ni) and Didinium (Nz)] 

 that if iV2 is large the mortality of the predators is negligible when 

 Ni > 0. In addition, the increase of predators only slightly depends 

 on Ni (with an insufficiency of prey the predators continue to multi- 

 ply at the expense of a decrease in size of the individuals; in this con- 

 nection the consumption of prey but slightly depends on iVi). 



Introducing these assumptions into the equation (21) we write 



_ d#? = where N ^ and - ^r- = *#« where N t = 0.* To 



dt dt 



reduce the dependence upon N x of the members characterizing the 

 interaction of species, we substitute y/Ni to iVi.f Then 



dNj 

 dt 



dN 2 

 If 



= biNi - fciiV 2 VNl 



= b«Nt \/W x Ctfi * 0) 



= - (kN* (Ni = 0) J 



(21b) 



Figure 36 shows that the solution of the equation 21b (the integral 

 curves on the graph Ni,Ni) actually coincides with biological observa- 

 tions on Paramecium and Didinium. It is therefore safe to assume 

 that the general equation of the destruction of one species by another 

 (21) takes in our special case the form (21b) instead of the classical 

 expression of Lotka-Volterra (21a). 



(3) The equation of relaxation (21 b) represents but a first approxi- 

 mation to the actual state of things, and is true only if N x or iV 2 are 

 large. Looking at the trend of the experimental curves on the surface 

 Ni, N 2 with small densities (Fig. 37) we notice that they pass from 

 the right to the left and cross the ordinate (Fig. 37, a). This means 



* This condition is already sufficient for an exclusion of the "classical" 

 periodic fluctuations. 



f Special experiments show that this substitution is satisfactory (Gause 

 and Witt, '35). 



