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 9.5.6

Chapter 9, Problem 9.5.6

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

Verified step by step guidance

Understand that the predator-prey model, often represented by the Lotka-Volterra equations, is a system of differential equations modeling the interaction between two species: predators and their prey.

Identify the key assumptions about the prey population: it grows exponentially in the absence of predators, meaning unlimited resources and no other limiting factors affect its growth.

Recognize the assumptions about the predator population: it declines exponentially in the absence of prey, implying that predators rely solely on the prey for food and survival.

Note the interaction assumptions: the rate at which predators consume prey is proportional to the product of their populations, indicating encounters between predators and prey happen randomly and frequently.

Acknowledge that the model assumes no environmental complexity such as migration, age structure, or other species interactions, and that the environment is constant over time.

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Key Concepts

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

Lotka-Volterra Predator-Prey Model

This mathematical model describes the interaction between predator and prey populations using differential equations. It assumes that prey population grows exponentially in the absence of predators, while predators depend solely on prey for food, affecting their growth rate.

Assumptions of Population Dynamics

The model assumes constant environmental conditions, no immigration or emigration, and that the populations are homogeneous and well-mixed. It also presumes that the predator's consumption rate is proportional to the encounter rate with prey.

Closed System and Interaction Rates

The predator-prey model assumes a closed ecosystem with no external influences. Interaction rates, such as predation and reproduction, are constant and directly proportional to the sizes of the respective populations, simplifying real-world complexities.

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Related Practice

Textbook Question

25–28. Two steps of Euler’s method For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.

y′(t) = 2−y, y(0) = 1; Δt = 0.1

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Textbook Question

12–16. Sketching direction fields Use the window [-2, 2] x [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.

y(x) = sin y, y(−2) = 1/2

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Textbook Question

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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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

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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Textbook Question

39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form

a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t).

Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems.

t³y′(t) + 3t²y = (1 + t)/t, y(1) = 6

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