APR. 19, 1923 LOTKA: CONTRIBUTION TO QUANTITATIVE PARASITOLOGY 153 
births per unit of time in the parasite species will be, evidently, 
ha,N,N2. If the deaths among parasites are d.N:, we have 
dNa fs ha,N; Nz = d, N; (2) 
dt 
Simplifying our notation we will write (1) and (2) in the form 
“ = uX — vXY = X(u — vY) (3) 
“ SS py ae oy xy (4) 
where the coefficients u, v, U, V, are in general functions of X and Y. 
A state of equilibrium (which may or may not be stable) ensues 
when 
U 
X = west (5) 
u 
Vimiiry=d (6) 
Introducing new variables 
X—-—p X-p 
xX = —_- = 7 
V pv a ”) 
Y-q. Y-4q | 
= —__ = — = 8 
y Vg B (8) 
we find 
dx a 
Roly +2 xy ) (9) 
dy _ ( B ) 
at ag aa xy (10) 
In these equations a and £ are functions of x and y, since they contain 
vand V. Wemay write 
a et se mm at es Hi (11) 
B=fot xt pyt. (12) 
Substituting (11), (12), in (9), (10) we obtain 
es — eaopy +(% + = 48) ny (248) ye. a (13) 
0 
dt B Bo 
dy _ { (2 2 A (2: .) | 
Se cio +(S +E eet Beet xy t.. , (14) 
