Applications of Systems of Equations 455where r, s, Hand Qare all positive constants. Solving the two equations
rx-Hxy= x(r-Hy)=O
- sy + Qxy = y( -s + Qx) = 0
simultaneously, we see that system (6) h as two critical points- namely, (0, 0)
and (s/Q, r / H).
Let us now consider initial conditions (xo, yo) where xo 2: 0 and Yo ;:::: 0. Ifxo = 0, then the solution of system (6) is (x(t), y(t)) = (0, y 0 e-st). That is, if
xo = 0, then the y-axis is a trajectory in the phase-plane and a solution which
starts on the y-axis remains on the y-axis and moves toward the origin. Notice
that this result is a consequence of assumption 7 which says: "In the absence
of prey, the predator population decreases according to the Malthusian law."
Likewise, ifyo = 0, the solution to system (6) is (x(t),y(t)) = (x 0 ert,o). Thus,if Yo = 0, the x-axis is a trajectory in the phase-plane and a solution which
starts on the x-axis remains on the x-axis and moves away from the origin.
Observe that this result is a consequ ence of assumption 6. Since traj ectories
in the phase-plane cannot intersect one a not her and since the x-axis and the
y-axis are both trajectories, any trajectory which begins in the first quadrant
must remain in the first quadrant for all time.
Calculating first partial derivatives off and g of system (6), we find fx =r - Hy, fy = -Hx, 9x = Qy, and gy = -s + Qx. At the critical point
( s / Q, r / H) the associated linear system has coefficient matrixA_ (fx(s/Q,r/H)
- 9x ( s / Q, r / H)
fy(s/Q, r/ H)) _ ( 0
9y ( s / Q , r / H) - Qr/ H- sH/Q)
0.
The eigenvalues of this matrix are A = ±J=I'S = ±iJr'S, so (s/Q, r / H) is a
neutrally stable center of the associated linear system. Consequently, without
any further analysis, we cannot tell if this critical point remains a stable center
or becomes a stable or unstable spiral point of the nonlinear system (6).
DEFINITION Periodic Trajectory
A trajectory (x(t), y(t)) of an autonomous system is said to be periodic,
if for some positive T, (x(t + T), y(t + T)) = (x(t), y(t)) for all t.Volterra summarized his findings for the prey-predator model (6) in three
basic principles. First, based on the assumption that the coefficients of growth,
r and s, and the coefficients of interaction, H and Q, remain constant, h e