1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations

5. a. Find the critical points of the competitive hunters model

(4)

dx 2

- = 4x - .5xy - .2x

dt

dy y2

dt = 4y -. 25xy -

3

.

469

b. Use SOLVESYS or your computer software to solve system (4) on the

interva l [O, 5] for the initial conditions:

(i) x(O) = .1, y(O) = 1 (ii) x(O) = 1, y(O) = .1 (iii) x(O) = 20 , y(O) = 5

(iv) x(O) = 20, y(O) = 10

In each case, estimate limt-->exi x(t) and limt-+oo y(t) and determine which
species b ecomes extinct.

6. a. Find the critical points of the competitive hunters model

(5)

dx 2

dt = 2x - 2xy - 2x

dy 2

dt = 4y - 2xy - 6y.

b. Use computer software to solve system (5) on the interval [O, 5] for the
initia l conditions:

(i) x(O) = .1, y(O) = .1 (ii) x(O) = .1, y(O) = 1 (iii) x(O) = 1, y(O) = .1

(iv) x(O) = 1, y(O) = 1

In each case, estimate limt-+oo x(t) and limt-+oo y(t). Does either species be-
come extinct?

7. This exercise is designed to illustrate the radica l changes which constant-
   effort harvesting of one of the competing species can bring about. One of
   the species, for example, might have a fur which humans find very desirable.
   Consider the following competitive hunters model in which the x sp ecies is
   harvested with positive harvesting constant H.

(6)

dx 2

• = lOx - xy - x - H x
  dt

dy 2

dt = 8y - xy - 2y.