Each of Problems 1 through 6 can be interpreted as describeing the interaction of two species with populations
a) Draw a direction field and describe how solutions seem to behave.
b) Find the critical points.
c) For each critical points, find the corresponding linear sytem. Find the eigenvectors of the linear system, classify each critical points as to type, and determine whether it is asymototically stable, stable, or unstable.
d) Sketch thetrajectories in the neighbourhood of each critical points.
e) Compute and plot enough trajectories of the given system to show clearly the behaviour of the solutions.
f) Determine the limiting behaviour of
Want to see the full answer?
Check out a sample textbook solutionChapter 7 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Additional Math Textbook Solutions
Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)
Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Calculus Volume 3
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
- 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)arrow_forwardConsider 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.arrow_forwarda.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.Draw a phase portrait for the system. f.Determine the limiting behavior of x and y as t → ∞. g.Interpret the results in terms of the populations of the two species.3.dx/dt=x(1−0.5x−0.5y)dy/dt=y(−0.25+0.5x)arrow_forward
- 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.Draw a phase portrait for the system. f.Determine the limiting behavior of x and y as t → ∞. g.Interpret the results in terms of the populations of the two species. 1.dx/dt=x(1.5−0.5y)dy/dt=y(−0.5+x)arrow_forwardIn each of Problems 1 through 20: (a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. (e) Draw a sketch of, or describe in words, the basin of attraction of each asymptotically stable critical point. 1. dx/dt = -2x+y, dy/dt = x² - yarrow_forwardAn 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 willarrow_forward
- For the following systems, the origin is the equilibrium point. 3. a) Write each system in matrix form b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. dx dt e) State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. dx dt dy dt = Ax. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) = 4x - 13y = 2x - 6yarrow_forwardFor the following systems, the origin is the equilibrium point. dx a) Write each system in matrix form = Ax. dt 5. b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. e) State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) dx dt dy dt = -3x + 4y = 2x - 5yarrow_forwardConstruct 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.arrow_forward
- For the following systems, the origin is the equilibrium point. dx a) Write each system in matrix form = Ax. dt b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. e) State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) 6. dx dt dy dt = 2x - 8y = x - 2yarrow_forwardFor the following systems, the origin is the equilibrium point. a) Write each system in matrix form b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. e) State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. 4. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) dx dt dy dt = x + 4y dt = 4x + y = Ax.arrow_forward(3.3) Find the fixed points of the following dynamical system: -+v +v, v= 0+v? +1, and examine their stability.arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage