APR. 19, 1923 LOTKA: CONTRIBUTION 
Consider the case 
TO QUANTITATIVE 
PARASITOLOGY 155 
rm 0 (24) 
dy 
i >0 (25) 
At time T = 0 let 
X = Xo (26) 
Y ie 
~ (XoYo) = Ko 
Then at time T + dT ; 
2G SO (27) 
p are 
dy 
 % (&yY1) = eK Yo) +4, dT (28) 
= K, (29) 
ae. (30) 
But the curve ¢ = K,is wholly enclosed within the curve ¢g = K.. 
Therefore the point x, y always moves from any given curve ¢ 
to a neighboring one g = K’ lying wholly outside the former. 
having once left the area enclosed by ¢ 
It follows that when R > 0 the origin is a point of 
again enter it. 
K 
Hence, 
K, the point can never 
unstable equilibrium. Similarly it is readily shown that if R < 0 
the origin is a point of stable equilibrium, and the point x, y continually 
approaches the origin. 
(See Fig. 1.) 
This approach, however, is asymptotic, the origin is never quite 
reached within finite time. 
simply, by (16), (17) 
dT 3 
xdx + ydy 
x? + y? 
% 
y, 
ll 
r constant 
r cos T 
r sin T 
For, when, x y are very small, we have 
(31) 
(32) 
(33) 
Thus in its final stages the process is nearly periodic, with a period 
T, = 27, or tp = 2raBo, the rise and fall of the parasite popula- 
