The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 120 billion. (Assume that the difference in birth and death rates is 20 million/year = 0.02 billion/year.) (a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate. Let P be the population in billions and t be the time in years, where t = 0 corresponds to 1990.) dP = dt (b) Use the logistic model to estimate the world population in the year 2000. Compare with the actual population of 6.1 billion. (Round the answer to two decimal places.) P = billion (c) Use the logistic model to predict the world population in the years 2100 and 2500. (Round your answer to two decimal places.) year 2100 billion year 2500 P = billion (d) What are your predictions if the carrying capacity is 60 billion? (Round your answers to two decimal places.) year 2000 P = billion year 2100 P = billion year 2500 billion

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 120 billion. (Assume that the difference in birth and death rates is 20 million/year = 0.02 billion/year.) (a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate. Let P be the population in billions and t be the time in years, where t = 0 corresponds to 1990.) dP = dt (b) Use the logistic model to estimate the world population in the year 2000. Compare with the actual population of 6.1 billion. (Round the answer to two decimal places.) P = billion (c) Use the logistic model to predict the world population in the years 2100 and 2500. (Round your answer to two decimal places.) year 2100 billion year 2500 P = billion (d) What are your predictions if the carrying capacity is 60 billion? (Round your answers to two decimal places.) year 2000 P = billion year 2100 P = billion year 2500 billion

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let's assume that the carrying capacity for world population is
120 billion. (Assume that the difference in birth and death rates is 20 million/year = 0.02 billion/year.)
(a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate. Let P be the population in
billions and t be the time in years, where t = 0 corresponds to 1990.)
dP
dt
(b) Use the logistic model to estimate the world population in the year 2000. Compare with the actual population of 6.1 billion. (Round the answer to two decimal places.)
P =
billion
(c) Use the logistic model to predict the world population in the years 2100 and 2500. (Round your answer to two decimal places.)
year 2100
P
billion
year 2500
P =
billion
(d) What are your predictions if the carrying capacity is 60 billion? (Round your answers to two decimal places.)
year 2000
billion
year 2100
P =
billion
year 2500
P =
billion

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Putting the given values in the general form of logistic differential equation

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Solving the obtained differential equation by variable separable and then integrating it

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Using general integration rules

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Putting the values in the obtained expression

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