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 9.5.28b

Chapter 9, Problem 9.5.28b

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.

x′(t) = 2x − 4xy, y′(t) = −y + 2xy

Verified step by step guidance

Identify the given system of differential equations: \(x'(t) = 2x - 4xy\) and \(y'(t) = -y + 2xy\).

To find the lines where \(x'(t) = 0\), set the right-hand side of the first equation equal to zero: \(2x - 4xy = 0\).

Factor the equation \(2x - 4xy = 0\) to get \(2x(1 - 2y) = 0\). This implies either \(x = 0\) or \(1 - 2y = 0\).

Solve \(1 - 2y = 0\) to find \(y = \frac{1}{2}\). So, the lines where \(x'(t) = 0\) are \(x = 0\) and \(y = \frac{1}{2}\).

Next, find the lines where \(y'(t) = 0\) by setting \(-y + 2xy = 0\). Factor to get \(y(-1 + 2x) = 0\), which implies \(y = 0\) or \(x = \frac{1}{2}\).

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Key Concepts

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

Predator-Prey Model

A predator-prey model describes the interaction between two species: one as prey (x) and the other as predator (y). The populations change over time based on birth, death, and interaction rates, often modeled by coupled differential equations. Understanding these dynamics helps analyze population stability and oscillations.

Nullclines (Lines where derivatives are zero)

Nullclines are curves in the phase plane where the rate of change of one variable is zero (x' = 0 or y' = 0). They help identify equilibrium points and the system's behavior by showing where populations neither increase nor decrease. Finding nullclines is essential for analyzing system stability.

Solving for Equilibrium Conditions

Equilibrium points occur where both derivatives are zero simultaneously (x' = 0 and y' = 0). Solving these conditions involves setting the differential equations to zero and finding the corresponding population values. These points indicate steady states of the system and are key to understanding long-term behavior.

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Related Practice

Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The general solution of the separable equation y'(t) = t/(y' + 10y⁴) can be expressed explicitly with y in terms of t.

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Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Tumor growth The growth of cancer tumors may be modeled by the Gompertz growth equation. Let M(t) be the mass of a tumor, for t ≥ 0. The relevant initial value problem is:

dM/dt = -rM(t)ln(M(t)/K), M(0) = M₀,

where r and K are positive constants and 0 < M₀ < K.

b. Graph the solution for M₀ = 100 and r = 0.05.

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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

b. Solve the initial value problem.

A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

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Textbook Question

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

b. In what regions are solutions increasing? Decreasing?

y'(t) = (y−1)(1+y)

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Textbook Question

Properties of stirred tank solutions

b. Verify that M(0) = M₀

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