Each of Problems 1 through 5 can be interpreted as describing the interaction of two species with population densities
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 linearsystem. Find the eigenvalues and eigenvectors of the linear system. Classify each critical point as to type, anddetermine whether it is asymptotically stable, stable, orunstable.
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
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
Fundamentals of Differential Equations and Boundary Value Problems
Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)
Calculus Volume 3
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
Excursions in Modern Mathematics (9th Edition)
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
- (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.arrow_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_forwardPlease help solvearrow_forward
- Three engineers are independently estimating the spring constant of a spring, using the linear model specified by Hooke’s law. Engineer A measures the length of the spring under loads of 0, 1, 2, 4, and 6 lb, for a total of five measurements. Engineer B uses the same loads, but repeats the experiment twice, for a total of 10 independent measurements. Engineer C uses loads of 0, 2, 4, 8, and 12 lb, measuring once for each load. The engineers all use the same measurement apparatus and procedure. Each engineer computes a 95% confidence interval for the spring constant. If the width of the interval of engineer A is divided by the width of the interval of engineer B, the quotient will be approximatelyarrow_forwardThree engineers are independently estimating the spring constant of a spring, using the linear model specified by Hooke’s law. Engineer A measures the length of the spring under loads of 0, 1, 3, 4, and 6 lb, for a total of five measurements. Engineer B uses the same loads, but repeats the experiment twice, for a total of 10 independent measurements.Engineer C uses loads of 0, 2, 6, 8, and 12 lb, measuring once for each load. The engineers all use the same measurement apparatus and procedure. Each engineer computes a 95% confidence interval for the spring constant. a) If the width of the interval of engineer A is divided by the width of the interval of engineer B, the quotient will be approximately _____. b) If the width of the interval of engineer A is divided by the width of the interval of engineer C, the quotient will be approximately __________. c) Each engineer computes a 95% confidence interval for the length of the spring under a load of 2.5 lb. Which interval is most likely to…arrow_forwarddo d, e onlyarrow_forward
- mm.2arrow_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_forward.The system x′=3(x+y−13x3−k),y′=−13(x+0.8y−0.7)x′=3(x+y−13x3−k),y′=−13(x+0.8y−0.7) is a special case of the Fitzhugh–Nagumo16 equations, which model the transmission of neural impulses along an axon. The parameter k is the external stimulus. a.Show that the system has one critical point regardless of the value of k.arrow_forward
- Please help with my homeworkarrow_forwardConsider the discrete-time dynamical system modeling the concentration of a chemical in a lung. (Note: round all values at the end of the calculations and use 4 decimal places.)ct+1 = (1-p)ct + pβLet V = 2 L, W = 1 L, and β = 6 mmol/LIf c0 = 7 mmol/L, iterate to find the following values:c1 = ____mmol/Lc2 = ____mmol/Lc3 = ____mmol/Lc4 = ____mmol/LFind the equilibrium of this system:c* = ____mmol/Larrow_forwardIn each of Problems 5 and 6 the coefficient matrix has a zero eigenvalue. As a result, the pattern of trajectories is different from those in the examples in the text. For each system: Ga. Draw a direction field. b. Find the general solution of the given system of equations. G c. Draw a few of the trajectories. 4 -3 8 -6 5. x' = Xarrow_forward
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education