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The population of the world was about 6.1 billion in 2000. Birth rates around that time 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 20 billion.(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.)(b) Use the logistic model to estimate the world population in the year 2010 and compare with the actual population of 6.9 billion.(c) Use the logistic model to predict the world population in the years 2100 and 2500.

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 4

Models for Population Growth

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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01:02

The population of the worl…

02:46

The relative growth rate o…

04:16

(a) Assume that the carryi…

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World Population The relat…

heads Clara. So onion, right here. So we know M is equal to 20 and K is equal to 220 over 1000 over a 6.1, which is equal to one over 305. So we know the different Schauble equation. To be deep key over. DT is equal to P over 305 times one minus p over 20. We got we get this equation and we're gonna find for a We get the carrying capacity week to be 20 minus p of zero to be six point on over 6.1. So we got about 2.28 So we got P of tea is equal to 20 over one plus 2.2 a. He to the negative. 0.0 33 p. And we're looking at the population in 2010. So we plug in 10 to get about 6.24 billion, which is less than the real population for part C. We got that. We're using the equation we got from part, eh? So we're looking at the population and 4100 to get about 7.5458 and in 2500 we just plugging 500 when we get 13.91

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