1550078481-Ordinary_Differential_Equations__Roberts_

Answers to Selected Exercises

Exercises 10.4 Lanchester's Combat Models

1. b.
   Winner Time Over Number of Remaining
   (Days) Winning Combatants
   (i) y 1.975 2.35
   (ii) x 2.76 3.78
   (iii) x .995 2.66

3.

Winner Time Over Number of Remaining

(Days) Winning Combatants

(i) y 1.885 1.47

(ii) y 1.85 1.70

5. No

Exercises 10.5 Models for Interacting Species

Volterra-Lotka Prey-Predator Model

1. a. T = 3.148 years

minimum maximum average

x 1.7500 2.2727 2

y 3.5000 4.5455 4

b. T = 3.18 years- slightly longer than T for part a.

x

y

minimum

1.9952

2.8800

maximum

2.6343

3.9791

average

2.3

3.4

567

Minimum, maximum, and average prey population increase; while mini-
mum, maximum, and average predator population decrease.

Modified Prey-Predator Models

1. The critical point (0, 0) is a saddle point.
   r
   The critical point ( C, 0) is in the first quadrant. If Qr - Cs < 0, then
   ( ; , 0) is an asymptotically stable node. If Qr - Cs > 0, then ( ; , 0) is a

saddle point.