1550078481-Ordinary_Differential_Equations__Roberts_

PORTRAIT User's Guide

EXAMPLE Solving Two Initial Value Problems for an

Autonomous, Two Component System

527

Use PORTRAIT to solve the Volterra-Lotka prey-predator system initial
value problems

(2)

dx

- = x - .5xy

dt

dy

dt = -2y + .25xy

(i) on the interval [O, 4.5] for initial conditions x(O) = 5, y(O) = 5

and

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

For (i) display a graph of x(t) and y(t). For (i) and (ii) produce a phase-plane
graph of y(t) versus x(t) on the rectangle

R = { ( x, y) I 0 :::; x :::; 20 and 0 :::; y :::; 6}.

SOLUTION

Notice when using PORTRAIT to solve the system IVP (1), the indepen-
dent variable must always be the variable x and the dependent variables must
always be the variables y 1 and y 2. However, in the given system (2) the in-
dependent variable is t and the dependent variables are x and y. When we
specify the system to be integrated to PORTRAIT, x will be associated with
the dependent variable y 1 , and y will be associated with the dependent vari-
able y 2. Making this association, we see that the system to be solved using
the notation of PORTRAIT is

(3)

dy1

- =Yi - .5Y1Y2

dx

dy2

• = -2y2 + .25y1y2
  dx

In order to use PORTRAIT to solve the system IVP (3), we double click
the PORTRAIT icon and when the screen shown in Figure B.l appears on
the monitor, we single click the RUN PORTRAIT button, which causes the
screen shown in Figure B.2 to appear on the monitor.