156 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 8 
tion, as measured by y, lagging by one quarter-period behind that of 
the host population as measured by x. 
More generally, if we write (16), (17) in the form 
=e (34) 
then the integral curves of (34), when x and y are plotted in rectangu- 
lar codrdinates, are spirals‘ closing in toward the origin and becoming 
Fig. 1. Illustration of the argument: The curves ¢g = K forma family, curves of lesser K 
d 
being wholly enclosed by those of greater K. When R < O then a < O, and the 
ih 
integral curves are therefore traversed in inward direction (see arrows). The zones 
between the curves ¢ = K thus form a trap, as it were, for the point x, y, permitting 
only one-way traffic, toward the origin. This latter is therefore a point of stable 
equilibrium. 
more and more nearly circular as they do so; the process is no longer 
strictly periodic for larger amplitudes, though it remains oscillatory. 
It is interesting to compare this conclusion with Dr. L. O. Howard’s 
d ‘ : 
4For, according to (9), (10), + and 4 are always of opposite sign so long as 
x 
p q 
1 Eo me 
x Fi ov B 
