56 



lotka: discontinuous evolution 



No attempt was made to investigate the convergence of the 



series, but the solution was tested by substituting it in the right 



hand member of (I) and comparing the values thus obtained with 



dx 

 the corresponding values of — obtained directly from the solution 



at 



as tabulated above. These latter are shown as a continuous 



curve x in the accompanying diagram while the crosses indicate 



corresponding values obtained by substitution. Beyond the 



last cross shown the agreement was complete within the limits 



of plotting error. 



While the curves shown in the diagram refer specifically to the 

 reaction mentioned above, in their general appearance they are 

 typical of the solution (25) of the general case, and we may here 

 discuss them as if they related to the groups A u and A { which 

 we have been considering. 



We note, then, that at the start there is an abundance of the 

 food material x, and accordingly the feeding group y increases 

 rapidly, with the result that soon the increased consumption causes 

 the total food supply to dimin sh, tho it is sti 1 for a time quite 

 plentiful, and y accordingly continues to increase. After a cer- 

 tain time, however, when x, the food supply, has fallen below 

 a certain value, 3 the feeding group y now also begins to diminish. 

 This alternate rise and fall of the two curves, with a certain phase 

 difference, would go on indefinitely if we extended our curves to 

 infinity, for it can be seen by inspection of (24) (25), that for 

 very large values of t the solution reduces simply to the form of 

 damped harmonic oscillation. Actually the curve for x has been 



3 That this value happens to be zero is peculiar to the special case here consid- 



dv 

 ere'd, x being a factor of — (see Equation II) : in general this is not the case. To 



at, 



be precise, x represents not the food sujply, but the excess of this over its equi- 

 librium value. See Equation (16). 



