1550078481-Ordinary_Differential_Equations__Roberts_

568 Ordinary Differential Equations

s Qr - Cs

If Qr - Cs < 0, then the critical point ( Q, QH ) is in the fourth

quadrant and is a saddle point. If Qr - Cs 2: 0, then the critical point
s Qr - Cs....
( Q , Q H ) 1s m t h e first quadrant and 1s an asym ptot1cally stable node

provided C^2 s - 4Q( Qr - Cs) 2: O; otherwise, the critical point is an asymp-
totically stable spiral point.

3. a. The average predator population decreases.

b. The average prey population increases.

c. The average prey population increases and the average predator popu-

lation decreases.

5. (iii)

As t increases a b c d e

x(t) -> 1. 34 1 1.34 1.15 1.36

y(t) -> 0. 34 0 0.00 0.00 0. 00

7. a. (0, 0 , 0),
   a
   (b, 0, 0),

g -d

(o, h ' T),

(~, ae - bd O)

e ce

(

ah-cg g aeh - bdh - ceg)

bh ' h ' bfh

b.

As t increases ( i) (ii)

Y1 ( t) -> 1.15 l. 9

Y2(t)-> .93 1.1

y3(t)->. 11 .9

Leslie's Prey-Predator Model

ae ad
1 a ( ) · stable node

· · be + cd' be + cd '

b. limt, 00 x(t) = .8 limt, 00 y(t) = 1.6

Leslie-Gower Prey-Predator Model

l. a. limt-. 00 x ( t) ~ (^26) limt-. 00 y(t) ~ 10
A Different Uptake Function

l. a. b. unstable spiral point

c. limt-.+oo x(t) = 0 limt-.+oo y(t) = 0