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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.5.27e
Chapter 9, Problem 9.5.27e

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.


e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves.


x′(t) = −3x + 6xy, y′(t) = y − 4xy

Verified step by step guidance
1
Identify the system of differential equations given: \(x'(t) = -3x + 6xy\) and \(y'(t) = y - 4xy\). Here, \(x(t)\) and \(y(t)\) represent the populations of prey and predator, respectively.
Find the equilibrium points by setting \(x'(t) = 0\) and \(y'(t) = 0\). Solve the system of algebraic equations: \(-3x + 6xy = 0\) and \(y - 4xy = 0\) to find points where the populations do not change.
Analyze the direction of the vector field around the equilibrium points by considering the signs of \(x'(t)\) and \(y'(t)\) in different regions of the \(xy\)-plane. This helps determine how the populations evolve over time near these points.
Sketch the nullclines: curves where \(x'(t) = 0\) and \(y'(t) = 0\). These nullclines divide the plane into regions where the direction of change of \(x\) and \(y\) can be determined, guiding the shape of the solution trajectories.
Draw representative solution curves in the \(xy\)-plane by following the direction of the vector field, starting from various initial points. Indicate the direction of time evolution with arrows along the curves to show how the populations change.

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Key Concepts

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

Predator-Prey Differential Equations

Predator-prey models use coupled differential equations to describe interactions between two species: prey (x) and predator (y). The equations capture growth and decline rates influenced by each population, reflecting biological interactions such as predation and reproduction.
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Phase Plane and Solution Curves

The phase plane is a graphical representation where each point corresponds to a state (x, y) of the system. Solution curves trace the trajectory of populations over time, showing how the system evolves. Sketching these curves helps visualize dynamics and stability.
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Differentiation of Parametric Curves

Direction Fields and Vector Fields

Direction fields assign a vector to each point in the phase plane, indicating the instantaneous direction of the system's evolution. They guide the drawing of solution curves and show whether populations increase or decrease, helping to understand the flow of the system.
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Related Practice
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.

Textbook Question

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

Textbook Question

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


d. Sketch the direction field and verify that it is consistent with parts (a)–(c).


y'(t) = (y−2)(y+1)