1550078481-Ordinary_Differential_Equations__Roberts_

466 Ordinary Differential Equations

b. Determine the stability characteristics of the critical point.
c. Use computer software to solve system (2) on the interval [O, 5] for the
initial condit ions x(O) = 1 , y(O) = l. Display a single graph with x(t) and y(t)
on the interval [O, 5] and display a phase-plane graph of y versus x. Estimat e
limt, 00 x(t) and limt, 00 y(t).

2. In the absence of predators, the prey population of system (2) grows
   according to the Malthusian law. Suppose t his growth assumption is changed
   to lo gistic law growth and the prey-predator system becomes

dx = 4x - 4x2 - 8xy

dt 1+2x

(3)

dy _

2

8xy

dt - - y + 1 + 2x ·

a. Find the critical point of system (3) in the first quadrant. How does it
compare with the critical point of system (2)?

b. Determine the stability characteristics of the critical point of system (3).

c. Use computer software to solve system (3) on the interval [O, 5] for the

initial conditions x(O) = 1 , y(O) = l. Display a graph of x(t) and y(t) and a

phase-plane graph of y versus x. Estimate limt_, 00 x(t) and limt_, 00 y(t).

3. Use SOLVESYS or your computer software to solve the prey-predator
   system

dx = 4x - 4x2 - 16 xy

dt 1+4x

(4)

dy _

2

16xy

dt - - y + 1 + 4x

on the interval [O, 10] for the following initial conditions:

a. x(O) = .25, y(O) = .4 b. x(O) = .2 5, y(O) = .675

c. x(O) =. 25 , y(O) = 1

In each case, produce a phase-plane graph of y versus x. What do you notice
about the phase-plane graphs?

May's Prey-Predator Model

The following prey-predator model was proposed by R. M. May

(1)

dx 2 cxy

• =ax-bx - --
  dt x+k

dy = dy-ey2.

dt x