Program Explanation Given information: The logistic equation is d P d t = a P − b P 2 . Here, P ( t ) is a rabbit population, B = a P is the time rate at which births occur and D = b P 2 is the rate at which deaths occur. Initial population is 120 with rate of 8 births per month and 6 deaths per month. P 0 = 240 B 0 = 9 D 0 = 12 Take initial time as 0. Explanation: The solution of the logistic differential equation is shown below. P ( t ) = M P 0 P 0 + ( M − P 0 ) e − k M t Here, k = D 0 P 0 2 and M = B 0 P 0 D 0 . Obtain the maximum capacity of the system M as follows: M = 9 ⋅ 240 12 = 180 Obtain the value of k as follows: k = D 0 P 0 2 = 12 240 2 = 1 4800 Now, substitute the known values k and M into the equation P ( t ) = M P 0 P 0 + ( M − P 0 ) e − k M t as, P ( t ) = 180 ⋅ 240 240 + ( 180 − 240 ) e − 180 t 4800 = 43200 240 − 60 e − 3 t 80 To obtain the time at which population reaches to 105% of maximum population, substitute P ( t ) = 1

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Chapter 2.1, Problem 17P

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Program Description: Purpose of problem is to calculate the number of months takes for P(t) to reach 105% of the limiting population ( M ).

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Chapter 2.1, Problem 16P

Chapter 2.1, Problem 18P

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Chapter 2 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Program Description: Purpose of problem is to calculate the number of months takes for P ( t ) to reach 105% of the limiting population ( M ).

Solution Summary: The author explains how to calculate the number of months takes for P(t) to reach 105% of the limiting population.

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Program Description: Purpose of problem is to calculate the number of months takes for P(t) to reach 105% of the limiting population ( M ).

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Chapter 2 Solutions