Explanation Given: The given values are a = 5 , b = 1 and h = 7 . Calculation: Consider the differential equation for the model of the population P ( t ) . d P d t = P ( a − b P ) − h (1) Substitute 5 for a , 1 for b and 7 for h in Equation (1). d P d t = P ( 5 − ( 1 ) P ) − 7 = P ( 5 − P ) − 7 (2) Use quadratic formula to solve for P . P = 5 ± 5 2 − 4 ( 1 ) ( 7 ) 2 ( 1 ) = 5 ± − 3 2 = 5 ± i 3 2 The solutions for P are complex and therefore there are no critical points. Substitute 1 for P in Equation (2). d P d t = 1 ( 5 − 1 ) − 7 = 4 − 7 = − 3 The following figure shows the phase diagram and the solution curve. Figure (1) Consider the differential equation. d P d t = P ( 5 − P ) − 7 Separate the variables and integrate. d P d t = − P 2 + 5 P − 7 d P P 2 − 5 P + 7 = − d t ∫ d P ( P − 5 2 ) 2 + 3 4 = − ∫ d t Let u = P − 5 2 ∫ d P ( u ) 2 + 3 4 = − ∫ d t 2 3 3 tan − 1 ( 2 u 3 3 ) = − t + c Substitute P − 5 2 for u in the above equation. 2 3 3 tan − 1 ( ( 2 P − 5 ) 3 3 ) = − t + c tan − 1 ( ( 2 P − 5 ) 3 3 ) = − 3 2 t + c 1 (3) Simplify the above equation

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.8, Problem 7E

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To investigate: The harvesting model both qualitatively and analytically in the given conditions and determine whether the population becomes extinct in finite time.

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Chapter 2.8, Problem 6E

Chapter 2.8, Problem 8E

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Chapter 2 Solutions

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To investigate: The harvesting model both qualitatively and analytically in the given conditions and determine whether the population becomes extinct in finite time.

Solution Summary: The author analyzes the harvesting model qualitatively and analytically, and determines whether the population becomes extinct in finite time.

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To investigate: The harvesting model both qualitatively and analytically in the given conditions and determine whether the population becomes extinct in finite time.

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Chapter 2 Solutions