Each of Problems 1 through 6 can be interpreted as describing the interaction of two specieswith 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 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) Compute and plot enough trajectories of the given system to show clearly the behavior of thesolutions.(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 — x 0.5y), dy/dt = y(2-y-0.75x)

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 32E

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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.
Each of Problems 1 through 5 can be interpreted as describing the interaction of two species with 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 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 → ∞ and 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)
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.
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
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)
(3.3) Find the fixed points of the following dynamical system: -+v +v, v= 0+v? +1, and examine their stability.
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.
a. Given the internal consumption matrix A, and the external demand matrix D as follows. 0.1 0.2 0.2 200 D = A = |0.1 0.3 0.1 100 [0.1 0.2 0.3 150 Solve the system using the open model: X = AX + D or X = (I – A)-'D. b. An economy has two industries, namely, farming and building. For every Php50 of food pro- duced, the farmer uses Php10 and the builder uses Php7.50. For every Php50 worth of build- ing, the builder uses Php12.50 and the farmer uses Php10. If the external demand for food is Php5M, and for building Php10M, what should be the total production for each industry in pesos?
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.
The Lotka-Volterra model is often used to characterize predator-prey interactions. For example, if R is the population of rabbits (which reproduce autocatlytically), G is the amount of grass available for rabbit food (assumed to be constant), L is the population of Lynxes that feeds on the rabbits, and D represents dead lynxes, the following equations represent the dynamic behavior of the populations of rabbits and lynxes: R+G→ 2R (1) L+R→ 2L (2) (3) Each step is irreversible since, for example, rabbits cannot turn back into grass. a) Write down the differential equations that describe how the populations of rabbits (R) and lynxes (L) change with time. b) Assuming G and all of the rate constants are unity, solve the equations for the evolution of the animal populations with time. Let the initial values of R and L be 20 and 1, respectively. Plot your results and discuss how the two populations are related.
For 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.
A manufacturing company has three processing lines. Each processing line combines various raw materials to produce the same output, Y. One of the processing lines can combine units of input (X) to produce units of output (Y) such that there is a linear relationship between X and Y. The Chief Executive Officer wants the output model to be estimated for this processing line. Given the values of X and Y (in thousands of units) in the table below, and as the Director of Data Management, you are to: x 5 7 2 8 4 8 12 6 14 8 Determine the extent to which output changes as the input change and interpret your result. Find the intercept of the output model and interpret it. Write down the equation of the line of best fit for output. Determine the value of the extent of relationship between input (X) and output (Y), and interpret your result.

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