Page 362 - Advanced engineering mathematics

342 CHAPTER 10 Systems of Linear Differential Equations

12. Use a software package to generate a phase portrait 14. A more sophisticated approach to a competing species
for each of the following predator/prey models. model is to incorporate a logistic term, leading to the
(a) x = x − 0.5xy, y = 2xy − 1.2y model

(b) x = 3x − 1.5xy, y = xy − 1.6y 2 2

x = ax − bx − kxy, y = cy − dy −rxy,

(c) x = 1.6x − 2.1xy, y = 1.9xy − 0.4y

with the coefficients positive constants. Generate

(d) x = 1.8 − 0.2xy, y = 3.1xy − 0.4y
phase portraits for the following systems.
13. Generate a phase portrait for each of the following (a) x = x(1 − x − 0.5y),

competing species models. y = y(1 − 0.5y − 0.25x)

(a) x = 2x − xy, y = y − 2xy

(b) x = x(1 − x − 0.2y),

(b) x = 1.6y − 1.2xy, y = 2y − 0.4xy y = y(1 − 0.4y − 0.25x)

(c) x = 1.4x − 0.6xy, y = 2y − 0.7xy (c) x = x(2 − x − 0.2y), y = y(1 − 0.4y − x)

(d) x = 3.2x − 1.4xy, y = 4.4y − 0.8xy (d) x = x(1 − 0.5x − y), y = y(2 − 0.5y − 0.4x)

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October 14, 2010 20:32 THM/NEIL Page-342 27410_10_ch10_p295-342