Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

Ch. 9 - Differential Equations

문제 9.5.29a

9장, 문제 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

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

a. The differential equation y′+2y=t is first-order, linear, and separable.

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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.

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

a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

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{Use of Tech} Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation

dP/dt=kP(1−P/A),P0=P_0,

where K is a positive infection rate, A is the number of people in the community, and P0 is the number of infected people at t=0. The model also assumes no recovery.

a. Find the solution of the initial value problem, for t≥0, in terms of K, A, and P0.

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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) = y(2 - y)

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

d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.

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