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 9.5.28d

Chapter 9, Problem 9.5.28d

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.
d. Identify the four regions in the first quadrant of the xy-plane in which x' and y' are positive or negative.

x′(t) = 2x − 4xy, y′(t) = −y + 2xy

Verified step by step guidance

First, write down the given system of differential equations clearly: $x\prime(t) = 2x - 4xy$ $y\prime(t) = -y + 2xy$

To find where $x\prime$ and $y\prime$ are positive or negative, analyze the sign of each derivative separately by factoring and considering the variables $x$ and $y$ in the first quadrant (where $x > 0$ and $y > 0$).

For $x\prime = 2x - 4xy$, factor out \$2x$: \(x\prime = 2x(1 - 2y)$. Since \)x > 0$, the sign of \(x\prime$ depends on \)(1 - 2y)$. So, - $x\prime > 0$ when $1 - 2y > 0 \Rightarrow y < \frac{1}{2}$ - $x\prime < 0$ when $y > \frac{1}{2}$

For $y\prime = -y + 2xy$, factor out $y$: $y\prime = y(-1 + 2x)$. Since $y > 0$, the sign of $y\prime$ depends on $(-1 + 2x)$. So, - $y\prime > 0$ when $-1 + 2x > 0 \Rightarrow x > \frac{1}{2}$ - $y\prime < 0$ when $x < \frac{1}{2}$

Using these inequalities, divide the first quadrant into four regions based on the lines $x = \frac{1}{2}$ and $y = \frac{1}{2}$: 1. Region where $x < \frac{1}{2}$ and $y < \frac{1}{2}$: determine signs of $x\prime$ and $y\prime$ here. 2. Region where $x < \frac{1}{2}$ and $y > \frac{1}{2}$: determine signs. 3. Region where $x > \frac{1}{2}$ and $y < \frac{1}{2}$: determine signs. 4. Region where $x > \frac{1}{2}$ and $y > \frac{1}{2}$: determine signs. This classification identifies where each population is increasing or decreasing.

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Key Concepts

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

Predator-Prey Model Dynamics

Predator-prey models describe interactions between two species: prey (x) and predator (y). The prey population grows naturally but decreases due to predation, while the predator population declines without prey but increases when consuming prey. Understanding these dynamics helps analyze how populations change over time.

Sign Analysis of Derivatives (x' and y')

The signs of x' and y' indicate whether the populations are increasing or decreasing. By determining where x' and y' are positive or negative in the xy-plane, we can partition the plane into regions that describe growth or decline of each species, which is essential for understanding system behavior.

Phase Plane and Quadrant Analysis

The phase plane plots prey population (x) against predator population (y). Dividing the first quadrant (where populations are positive) into regions based on the signs of x' and y' reveals the system's qualitative behavior. This helps visualize trajectories and predict long-term outcomes.

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Related Practice

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.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

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Textbook Question

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

d. According to Newton’s Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time.

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Textbook Question

U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:

d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 410 million rather than 398 million. What is the value of the carrying capacity in this case?

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Textbook Question

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.

d. After how many minutes does the drug mass reach 90% of its steady-state level?

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Textbook Question

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.

d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁cosh(kt) + C₂sinh(kt), where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section 7.3).

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Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

d. In general, how does halving the time step affect the error at t=0.2 and t=0.4?

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

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