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Chapter 3: Q3E (page 102)

A model for the populationin a suburb of a large city is given by the initial-value problem

Whereis measured in months. What is the limiting value of the population? At what time will the population be equal to one-half of this limiting value?

Short Answer

Expert verified

Answer

Limiting value is

It takes aroundmonths for the population to be equal to one-half of this limiting value.

Step by step solution

01

Rewrite differential equation as

This is a classic logistic differential equation with form

In this caseand

represents the limiting value of the population.

02

We are also given 

The solution is given by the formula

Therefore

03

Now, check that the limiting value is 


04

When the population is equal to one-half of this limiting value, i.e. equal to, we get


Therefore, it takes around 53 months for the population to be equal to one-half of the limiting value.

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Most popular questions from this chapter

When a vertical beam of light passes through a transparent medium, the rate at which its intensity Idecreases is proportional toI(t), where trepresents the thickness of the medium (in feet).In clear seawater, the intensity 3feet below the surface is 25% of the initial intensityI0of the incident beam. What is the intensity of the beam 15feet below the surface?

Rocket Motion Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward, the air resistance is linear, and the rocket consumes its fuel at a constant rate. In Problem 22 of Exercises 1.3 you were asked to use Newton’s second law of motion in the form given in (17) of that exercise set to show that a mathematical model for the velocity v(t) of the rocket is given by

dvdt+k-vm0-λtv=-g+Rm0-λt,

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Question: With the identifications\(a = r,b = r/K\), and\(a/b = K\), Figures 2.1.7 and 3.2.2 show that the logistic population model, (3) of Section\(3.2\), predicts that for an initial population\({P_0},0 < {P_0} < K\), regardless of how small \({P_0}\)is, the population increases over time but does not surpass the carrying capacity\(K\).Also, for \({P_0} > K\)the same model predicts that a population cannot sustain itself over time, so it decreases but yet never falls below the carrying capacity \(K\)of the ecosystem. The American ecologist Warder Clyde Alle (1885-1955) showed that by depleting certain fisheries beyond a certain level, the fish population never recovers. How would you modify the differential equation (3) to describe a population \(P\)that has these same two characteristics of (3) but additionally has a threshold level\(A,0 < A < K\), below which the population cannot sustain itself and approaches extinction over time. (Hint: Construct a phase portrait of what you want and then form a differential equation.)

Suppose a cell is suspended in a solution containing a solute of constant concentrationcS. Suppose further that the cell has constant volume Vand that the area of its permeable membrane is the constant A. By Fick’s law the rate of change of its mass mis directly proportional to the area Aand the difference Cs-C(t), where C(t)is the concentration of the solute inside the cell at time t. Find C(t)if m=V.C(t)andC(0)=C0.See Figure 3.R.2.

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