412 
BIOLOGY: A. J. LOTKA 
Proc. N. a. S. 
in- 
of 
Mass of newly 
formed 52 per 
unit of time 
(derived from 
Si as food in- 
gested) 
Or, in analytical symbols, 
dXi 
dt 
Rate of 
crease of 
per unit 
time 
Mass of S2 
destroyed or 
eliminated per 
unit of time 
= A\Xr-B,X,X2-A'\Xi 
(4) 
{A\-A\)X,-B,X,X^, 
A \X\ — B1X1X2 
Xl(A^-B^X,) 
(5) 
(6) 
(7) 
(8) 
dX2 
It 
A2X1X2-B2X2 (9) 
= X2{A2X,-B2) (10) 
The coefficients Ai, A2, Bi, B2 are in general functions of Xi and X2' 
The reasons for selecting the form (5), (9) for the analytical formula- 
tion of (3), (4) require perhaps a little explanation. For small changes 
the rate of formation of new material of a given species of organism under 
given conditions is proportional to the existing mass of that species. In 
other words, the growth of living matter is a typically autocatakinetic^ 
process. This term has, therefore, been put in the form A' Xi for the 
species 5i. Proportionality does not hold for large changes of Xi, X2, 
and this is duly provided for in that A'l is a function of Xi, X2. 
Similarly the mass of matter rejected per unit of time from the species 
Si is proportional to Xi, and has been put in the form A'\Xi, where A" 
is in general a function of Xi, X2. 
Again, the mass of 5i destroyed by S2 feeding upon it will, for small 
changes, be proportional to X2 and also to Xi. This term has, therefore, 
been set down in the form B1X1X2. Here again the departures from pro- 
portionality are taken care of by the variations of Bi with Xi and X2, of 
which variables Bi is a function. 
Similar remarks apply to the formulation (9) of (4). If there were no 
waste in the feeding process, and assuming that 52 consumes no other 
substance than 5i, we would have Bi = A2; but in the more general case 
Bi =»= A2. Approaching now the analytical treatment of equations (5), 
(9), or their equivalents (8), (10), we note first of all that there are two 
ways of satisfying the condition for equilibrium, namely: 
Xi = X2 = 0 (11) 
and 
B2 V _ ^1 
A^'^'-B, 
Xi = 
(12) 
