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

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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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For the Big-M tableau (of a maximization LP and row0 at bottom and M=1000), Z Ꮖ 1 x2 x3 81 82 83 e4 a4 RHS 0 7 0 0 1 0 4 3 -3 20 0 -4.5 0 0 0 1 -8 -2.5 2.5 6 0 7 0 1 0 0 8 3 -3 4 0 -1 50 1 0 0 0-2 -1 1 4 0000 0 30 970 200 If the original value of c₁ is increased by 60, what is the updated value of c₁ (meaning keeping the same set for BV. -10? Having made that change, what is the new optimal value for ž?

Here is the optimal tableau for a standard Max problem. zx1 x2 x3 24 81 82 83 rhs 1 0 5 3 0 6 0 1 .3 7.5 0 - .1 .2 0 0 28 360 0 -8 522 0 2700 0 6 12 1 60 0 0 -1/15-3 1 1/15 -1/10 0 2 Using that the dual solution y = CBy B-1 and finding B = (B-¹)-¹ we find the original CBV and rhs b. The allowable increase for b₂ is If b₂ is increased by 3 then, using Dual Theorem, the new value for * is If c₂ is increased by 10, then the new value for optimal > is i.e. if no change to BV, then just a change to profit on selling product 2. The original coefficients c₁ = =☐ a and c4 = 5 If c4 is changed to 512, then (first adjusting other columns of row0 by adding Delta times row belonging to x4 or using B-matrix method to update row0) the new optimal value, after doing more simplex algorithm, for > is

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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