1550078481-Ordinary_Differential_Equations__Roberts_

468 Ordinary Differential Equations

If the growth assumption is changed from the Malthusian law to the logistic
law, then the competitive hunters model becomes

(2)

dx 2

- = ax - bxy - rx

dt

dy - = cy - dxy - sy 2

dt

where a, b, c, d, r and s are positive constants.

EXERCISES

1. a. Show that the competitive hunters model, system (1), has critical points
   at (0,0) and (C/D,A/B).

b. Show that both critical points are unstable.

c. Show if x > C / D and y < A/ B, then the x population will increase

indefinitely and the y population will b ecome extinct.

d. Show if x < C / D and y > A/ B, then the x population will become

extinct and the y population will increase indefinitely.

2. Use computer software to so lve the competitive hunters model, system (1),
   with A= 2, B = 1, C = 3, and D = 1 on the interval [O, 5] for the following
   initial conditions:

a. x(O) = 1, y(O) = .5 b. x(O) = 1.5, y(O) = .5 c. x(O) = 10 , y(O) = 9

d. x(O) = 10 , y(O) = 8

In each case, determine which species b ecomes extinct.

3. Show that the logistic law, competitive hunters model, system (2), has
   critical points at (0, 0), (0, c/ s), (a/r, 0) , and (p/ D , q/ D) where p =as - be,
   q = er - ad, and D =rs - bd.


4. Consider the competitive hunters model

(3)

dx

• = 2x - 4xy - 4x^2
  dt

dy = 2y - 2xy - 2y^2.

dt

a. Find the critical points of system (3) in the first quadrant.

b. Use computer software to solve system (3) on the interval [O, 5] for the
following initial conditions:

(i) x(O) = .3, y(O) = .l (ii) x(O) = 2, y(O) = .l

In each case, estimate limt-><xi x(t) and limt_, 00 y(t) and determine which
species becomes extinct.