456 Ordinary Differential Equations
(s/Q,r/H)." That is, the critical point (s/Q,r/H) of the nonlinear sys-
tem (6) remains a stable center and the prey and predator populations vary
periodically. The p eriod T depends only on the coefficients r, s, H , and Q
and on the initial conditions Xo > 0 and Yo > 0.
Let (x(t), y(t)) be a p eriodic solution of (6) with period T > 0. The average
number of prey x and the average number of predators fj over the period T is
1 {T
x = T J
0
x ( t) dt
1 {T
fi = T lo y(t) dt.
Volterra's second principle, also based on the assumption that r, s, H, and Q
remain constant, is that x = s/Q and fj = r/H. Thus, his second principle,
the law of conservation of averages, says that "the average values of
x(t) and y(t) over the period T is equal to their critical point values."
This law is fairly easy to verify as the following computations show. Dividing
the first equation of system (6) by x and integrating from 0 to T, we obtain
{ :«~! dt ~ lnx({ ~ lnx(T) -lnx(O) ~ { (r -Hy) dt
Since xis periodic with period T, we have x(T) = x(O) and lnx(T) = lnx(O);
therefore,
Consequently,
1 T(r-Hy)dt=0.
rT = H 1T ydt or
1 t r
fj = T J 0 y dt = H.
Dividing the second equation of system (6) by y and proceeding as above, we
also find x = s/Q.
We are now ready to determine the effect of fishing on the prey-predator
model (6). The simplest model which includes the effects of fishing (harvest-
ing) assumes indiscriminate, constant-effort harvesting in which fisher-
men keep whatever fish they catch. Therefore, in indiscriminate, constant-
effort harvesting it is assumed that the number of fish harvested (caught) of
each species is proportional to the population of that species. The prey-
predator model with indiscriminate, constant-effort harvesting is
(7)
dx
dt
rx - Hxy-hx = (r - h)x - Hxy
dy
- = -sy + Qxy - hy = (-s - h)y + Qxy
dt