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 9.5.29a

Chapter 9, Problem 9.5.29a

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.
a. Identify which equation corresponds to the predator and which corresponds to the prey.

x′(t) = −3x + xy, y′(t) = 2y − xy

Verified step by step guidance

Examine the given system of differential equations: $x'(t) = -3x + xy$ and $y'(t) = 2y - xy$.

Recall that in predator-prey models, the prey population typically grows exponentially in the absence of predators, and the predator population declines without prey.

Look at the terms without interaction: For $x'(t)$, the term $-3x$ suggests that $x$ decreases exponentially when $y=0$, indicating $x$ is the predator population.

For $y'(t)$, the term \$2y$ suggests that $y$ grows exponentially when $x=0$, indicating $y$ is the prey population.

Confirm that the interaction terms $xy$ have opposite signs in each equation, representing the predator-prey interaction where predators benefit from prey and prey are harmed by predators.

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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: one as prey and the other as predator. The prey population typically grows in absence of predators, while the predator population depends on consuming prey. These models use differential equations to capture how populations change over time based on these interactions.

Interpreting Differential Equations in Population Models

Each differential equation represents the rate of change of a population. Positive terms indicate growth factors, while negative terms represent decline. Interaction terms, often products of both populations, model how one species affects the other's growth or decline, helping identify which equation corresponds to predator or prey.

Equilibrium and Stability in Predator-Prey Systems

Equilibrium points occur when population rates of change are zero, indicating steady states. Analyzing these points helps understand long-term behavior of the system. Stability analysis determines whether populations return to equilibrium after disturbances, crucial for predicting population survival or extinction.

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Textbook Question

Growth rate functions

a. Show that the logistic growth rate function f(P)=rP(1−P/K) has a maximum value of rK/4 at the point P=K/2.

Textbook Question

43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.

a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).

v₀ = 49 m/s, s₀ = 60 m

Textbook Question

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

a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).

y'(t) = (y−2)(y+1)

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Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

a. Find the general solution of the equation.

y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

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Textbook Question

{Use of Tech} Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t)=rB−m, for t≥0, with B(0)=B0. The constant r>0 reflects the annual interest rate, m>0 is the annual rate of withdrawal, B0 is the initial balance in the account, and t is measured in years.

a. Solve the initial value problem with r=0.05, m=$1000/year, and B0=$15,000 Does the balance in the account increase or decrease?

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Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions.

y′(t) = 6 - 2y

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