Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.R.28e

Chapter 9, Problem 9.R.28e

Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.
e. Discuss some possible shortcomings of this model. Why might the carrying capacity be either greater than or less than the value predicted by the model?

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Understand that the logistic growth model assumes a fixed carrying capacity, which is the maximum population the environment can sustain indefinitely.

Recognize that the carrying capacity predicted by the model is based on current assumptions about resources, technology, and environmental conditions, which can change over time.

Consider that the carrying capacity might be greater than predicted if technological advances improve resource availability or if policies enhance sustainability.

Consider that the carrying capacity might be less than predicted if environmental degradation, resource depletion, or unforeseen social factors reduce the ability to support the population.

Acknowledge that the logistic model simplifies complex population dynamics and may not account for sudden changes, migration, or other demographic factors, which are potential shortcomings of the model.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logistic Growth Model

The logistic growth model describes population growth that starts exponentially but slows as it approaches a maximum limit called the carrying capacity. It is represented by an S-shaped curve, reflecting limited resources and environmental constraints that prevent indefinite growth.

Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustain indefinitely given available resources, space, and other ecological factors. It is not fixed and can change due to technological, environmental, or social factors affecting resource availability.

Model Limitations and Assumptions

Mathematical models like logistic growth rely on assumptions such as constant carrying capacity and uniform resource distribution. Real-world factors like migration, policy changes, technological advances, and environmental shifts can cause deviations, making predictions uncertain or inaccurate.

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Related Practice

Textbook Question

Euler’s metho d Consider the initial value problem y′(t)=1/2y,y(0)=1.

a. Use Euler’s method with Δt=0.1 to compute approximations to y(0.1) and y(0.2).

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Textbook Question

Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.

d. For what values of A are the corresponding solutions decreasing, for t≥0

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Textbook Question

Direction fields The direction field for the equation y′(t)=t−y, for |t|≤4 and |y|≤4, is shown in the figure.

b. Use the direction field to sketch the solution curve that passes through the point (0,−1/2).

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Textbook Question

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(t) = -3y + 9, y(0) = 4

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Textbook Question

What are the assumptions underlying the predator-prey model discussed in this section?

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Textbook Question

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³

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