Each of Problems 1 through 5 can be interpreted as describing the interaction of two specieswith population densities x and y. In each of these problems, carry out the following steps:(a) Draw a direction field and describe how solutions seem to behave.(b) Find the critical points.(c) For each critical point, find the corresponding linear system. Find the eigenvalues andeigenvectors of the linear system. Classify each critical point as to type, and determine whether it isasymptotically stable, stable, or unstable.(d) Sketch the trajectories in the neighborhood of each critical point.(e) Draw a phase portrait for the system.(f) Determine the limiting behavior of x and y as t → ∞ and interpret the results in terms of thepopulations of the two species.1. dx/dt = x(1.5 -0.5y), dy/dt = y(-0.5 + x)

Since you have posted a question with multiple subparts, we will solve the first three subparts for…

Answered: (a) Draw a direction field and describe…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 16EQ

See similar textbooks

Similar questions

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations à and y. In each of these problems, carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point, find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system, classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t → ∞, and interpret the results in terms of the populations of the two species. 1. dx/dt = x(1.5 — x 0.5y), dy/dt = y(2-y-0.75x)
Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations & and y. In each of these problems, carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point, find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system, classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable.
Consider the example of injection moulding of a rubber component as shown in Figure Q3(b). The process engineer would like to optimise the strength of the component by optimising the following factors: temperature = 190°C and 210°C, pressure = 50 MPa and 100 MPa, and speed of injection = 10 mm/s and 50 mm/s. What type of mathematical model that the engineer can develop if the relationship is linear and no interactions are significant? Write down the general equation that relates the strength of the component with the process factors.
describing the interaction of two species with populations x and y. In each of these problems, carry out the following steps. a.Draw a direction field and describe how solutions behave. b.Find the critical points. c.For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. d.Sketch trajectories in the neighborhood of each critical point. e.Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. f.Determine the limiting behavior of x and y as t → ∞, and interpret the results in terms of the populations of the two species. 2.dx/dt=x(1.5−x−0.5y)dy/dt=y(2−0.5y−1.5x)
PLEASE HELP WITH QUESTION 4 Find the critical point of each of the non homogeneous linear system given. Then determine the type of critical point and stability.
(3.3) Find the fixed points of the following dynamical system: -+v +v, v= 0+v? +1, and examine their stability.
Find all the critical point(s) of each system given. Then determine the type and stability -3 2 4
An ecologist models the interaction between the tree frog (P) and insect (N) populations of a small region of a rainforest using the Lotka-Volterra predator prey model. The insects are food for the tree frogs. The model has nullclines at N=0, N=500, P=0, and P=75. Suppose the small region of the rainforest currently has 800 insects and 50 tree frogs. In the short term, the model predicts the insect population will • and the tree frog population will At another point time, a researcher finds the region has 300 insects and 70 tree frogs. In the short term, the model predicts the insect population will * and the tree frog population will
Construct a model for the number of cats, y, after x months that make use of the following assumptions: 1. It begins with two cats – one female and one male, both unneutered. 2. Each litter is composed of 4 kittens – 3 males and 1 female. 3. It takes four months before a new generation of cats is born. 4. No cat dies (all are healthy) and no new cats are introduced.
(B. Janssen, KTH, 2014) Consider the linear system 0.550x+0.423y = 0.127 0.484x + 0.372y = 0.112 Suppose we are given two possible solutions, u = [_11] and v- -1.91. 1.01 0.9 a. Decide based on the residuals b - Au and b - Av which of the two possible solutions is the 'better' solution. b. Calculate the exact solution x. c. Compute the errors to the exact solution. That is, compute the infinity norms of u-x and v-x. Do the results change your answer to 7a?
Find the production schedule for the technology matrix and demand vector given below: 0.4 0.1 5 A = 0.4 0.6 0.8 D =| 8 0.1 0.3 0.2 X = 3.
For the following linear system: a. Find all the critical points for the system b. find the corresponding linear system near each critical point (the linearized system and the Jacobian matrix. c. classify the ponts as stable or unstable.

SEE MORE QUESTIONS

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS