APR. 19, 1923 LOTKA: CONTRIBUTION TO QUANTITATIVE PARASITOLOGY 157 
observation: “‘With all very injurious Lepidopterous larvae we con- 
stantly see a great fluctuation in numbers, the parasite rapidly 
increasing immediately after the increase of the host species, over- 
taking it numerically, and reducing it to the bottom of another 
ascending period of development.® 
Fig. 2. Special case. Integral curves are spirals winding about the origin and about a 
limiting cycle, after the pattern of the eye of a cyclone. 
In the limiting case that R = 0 we must establish, in place of the 
function y, a similar function ¢’ 
OS ey eee ee beaks Gir sat rey (35) 
and the stability of the system at the origin of x, y then depends on the 
first ¢; of even degree for which the condition 
M4 NO = (36) 
cannot be satisfied.® 
5L.O. Howard. A Study in Insect Parasitism. U.S. Department of Agriculture, 
Bureau of Entomology, Technical Series No. 5, 1897, p. 48. 
6 See Poincaré, Jl. de Mathématique 1885 ser. 4, vol. 1, p. 178; Picard, Traité 
d’Analyse vol. 3, p. 217. 
