1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 457

r - h and -s replaced by -s - h. So, t he critical point in the first quad-

rant of system (7) is at ((s + h)/Q, (r - h)/ H). Since h > 0, we see that

(s + h)/Q > s/Q and (r - h)/H < r/H. Thus, we arrive at Volterra's

third principle which states: "indiscriminate, constant-effort harvesting

increases the average prey population and decreases the average

predator population." This third principle substant iates the conclusion

reached by D'Ancona- namely, that the predator population, on the aver-
age, increases when fishing decreases and decreases when fishing increases
and, conversely, the prey population, on the average, decreases when fishing
decreases and increases when fishing increases.
Since its initial formulation, the Volterra-Lotka prey-predator model has
been supported and chall enged by many ecologists and biologists. Critics
of the model cite the fact that most prey-predator systems found in nature
tend to an equilibrium state. However, this model i s a good model for the
prey-predator fish system of the Adriatic Sea, since these fish do not comp ete
within their own species for available resources. This model also adequately
represents population dynamics for other prey-predator systems in which the
individual populations do not complete within their own species for resources.
Shortly, we will examine systems which include terms to reflect internal com-
petition.

EXAMPLE 11 Computer Solution of a Volterra-Lotka

Prey-Predator System

Solve the Volterra-Lotka prey-predator system

dx

dt = x - .5xy

dy

dt = -2y + .25xy

on the interval [O, 5] for the initial conditions x(O) = 10 , y(O) = 5. Display

x (t) and y(t) on the same graph over the interval [O, 5]. Produce a phase-plane
graph of y versus x.

SOLUTION

We input the two functions defining the system, f(x, y) = x - .5xy and

g(x, y) = -2y + .25xy , into our computer software. Then, we input the

interval of integration [O, 5] and the initial conditions x(O) = 10 and y(O) = 5.

After the integration was completed, we indicated we wanted to graph x(t)
and y(t) on the interval [O, 5] with 0 ::::; y ::::; 20. The resulting graph is displayed
in Figure 10 .15. In this graph, t he solution x(t) li es above t he so lution y(t).

Since we wanted to produce a phase-plane graph of y versus x, we indicated
to our software that we wanted x(t) assigned to the horizontal axis and y(t)