1550078481-Ordinary_Differential_Equations__Roberts_

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

where h, the harvesting coefficient, is a positive constant. For r - h > 0

(i.e., for h < r) system (7) is the same as system (6) with r replaced by