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Questions and Answers of Statistics Econometrics

Cell cycle dynamics The process of cell division is periodic, with repeated growth and division phases as the cell population multiplies. It has been suggested that the division phase is triggered by
H abitat destruction The model of Exercise 23 can be extended to include the effects of habitat destruction.Suppose that only a fraction h of the patches are habitable (0 , h , 1). The equations
Competition-colonization models The metapopulation model from Exercise 15 can be extended to include two species, where one is a superior competitor. The equations are dp1 dt− c1p1s1 2 p1d 2 m1p1
Cancer progression The development of many cancers, such as colorectal cancer, proceeds through a series of precancerous stages. Suppose there are n 2 1 precancerous stages before cancer develops at
Suppose the model of Exercise 20 is replaced by the equations dx dt− 0.4xs1 2 0.000005xd 2 0.002xy dy dt− 20.2y 1 0.000008xy(a) According to these equations, what happens to the insect population
P redator-prey dynamics Populations of birds and insects are modeled by the equations dx dt− 0.4x 2 0.002xy dy dt− 20.2y 1 0.000008xy(a) Which of the variables, x or y, represents the bird
H ormone transport In lung physiology, the transport of a substance across a capillary wall has been modeled by the differential equation dh dt− 2 RV S h k 1 hD where h is the hormone concentration
A tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter enters the tank at a rate of 10 Lymin.The solution is kept thoroughly mixed and drains from the tank at the same
Lung preoxygenation Some medical procedures require a patient’s airway to be temporarily blocked, preventing the inspiration of oxygen. The duration of time over which such procedures can be
T he Brentano-Stevens Law in psychology models the way that a subject reacts to a stimulus. It states that if R represents the reaction to an amount S of stimulus, then the relative rates of increase
Levins’ metapopulation model from Exercise 7.2.15 describes a population consisting of patches that can be either occupied or vacant. Occupied patches create more occupied patches by sending
Seasonality and population dynamics The per capita growth rate of a population varies seasonally and habitat destuction is also occurring. This is modeled aswhere nstd is the population size at time
Seasonality and population dynamics The per capita growth rate of a population varies seasonally. The population dynamics are modeled aswhere nstd is the population size at time t (measured in days).
s1 1 cos xdy9 − s1 1 e2yd sin x, ys0d − 0 Solve the initial-value problem.
dr dt 1 2tr − r, rs0d − 5 Solve the initial-value problem.
dx dt− 1 2 t 1 x 2 tx Solve the differential equation.
2yey2y9 − 2x 1 3sx Solve the differential equation.
(a) Use Euler’s method with step size 0.2 to estimate ys0.4d, where ysxd is the solution of the initial-value problem y9 − 2xy2 ys0d − 1(b) Repeat part (a) with step size 0.1.(c) Find the exact
(a) A direction field for the differential equation y9 − x2 2 y2 is shown. Sketch the solution of the initial-value problem y9 − x2 2 y2 ys0d − 1 Use your graph to estimate the value of
(a) Sketch a direction field for the differential equation y9 − xyy. Then use it to sketch the four solutions that satisfy the initial conditions ys0d − 1, ys0d − 21, ys2d − 1, and ys22d −
(a) A direction field for the differential equation y9 − ys y 2 2ds y 2 4d is shown. Sketch the graphs of the solutions that satisfy the given initial conditions.(i) ys0d − 20.3 (ii) ys0d −
x9 − ax 1 x3 A differential equation is given.(a) Determine all equilibria as a function of the constant a.(b) Construct a phase plot and use it to determine the stability of the equilibria found
x9 − ax 2 x3 A differential equation is given.(a) Determine all equilibria as a function of the constant a.(b) Construct a phase plot and use it to determine the stability of the equilibria found
x9 − a 2 x2 A differential equation is given.(a) Determine all equilibria as a function of the constant a.(b) Construct a phase plot and use it to determine the stability of the equilibria found in
x9 − ax 2 x2 A differential equation is given.(a) Determine all equilibria as a function of the constant a.(b) Construct a phase plot and use it to determine the stability of the equilibria found
If y is the solution of the initial-value problem dy dt− 2yS1 2 y5D ys0d − 1 then lim tl`y − 5.Determine whether the statement is true or false. If it is true, explain why. If it is false,
The equation y9 − 3y 2 2x 1 6xy 2 1 is separable.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the
The equation y9 − x 1 y is separable.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
Consider the differential equation y9 − ts yd where ts yd is a differentiable function of y. It is not possible for y to exhibit oscillatory behavior.Determine whether the statement is true or
The function f sxd − sln xdyx is a solution of the differential equation x2y9 1 xy − 1.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or
All solutions of the differential equation y9 − 21 2 y4 are decreasing functions.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an
(a) Write Lotka-Volterra competition equations for two competing fish species, x and y.(b) What would the nullclines have to look like for species x to always outcompete species y?
What is a nullcline?
(a) Write Lotka-Volterra equations to model populations of sharks S and their food F.(b) What do these equations say about each population in the absence of the other?
(a) Write the logistic equation.(b) Under what circumstances is this an appropriate model for population growth?
What is a separable differential equation? How do you solve it?
Explain how Euler’s method works.
What is a direction field for the differential equation y9 − Fsx, yd?
What is a phase plot for the differential equation y9 − ts yd?
What can you say about the solutions of the equation y9 − x2 1 y2 just by looking at the differential equation?
(a) What is a differential equation?(b) What is the order of a differential equation?(c) What is an initial condition?(d) What are the differences between pure-time, autonomous, and nonautonomous
The Rosenzweig-MacArthur model is a consumerresource model similar to that from Exercise 30, but with a different consumption function. A simplified version isSuppose that all constants are positive
Fitzhugh-Nagumo equations Consider the following alternative form of the Fitzhugh-Nagumo equations from Example 3:where « . 0 and 0 , a , 1. Construct the phase plane, including all nullclines,
H emodialysis is a process by which a machine is used to filter urea and other waste products from a patient’s blood if their kidneys fail. The concentration of a patient’s urea during dialysis
A model for self-reproducing resources with limited growth is obtained by choosing f sRd − rRs1 2 RyKd, tsR, Cd − bRC, and hsCd − C. Assume all constants are positive and K .
A model for self-reproducing resources is obtained by choosing f sRd − rR, tsR, Cd − bRC, and hsCd − C, where all constants are positive.Consumer-resource models often have the following
A chemostat is an experimental consumer-resource system.If the resource is not self-reproducing, then it can be modeled by choosing f sRd − , tsR, Cd − bRC, and hsCd − C, where all constants
Lotka-Volterra competition equations For each case, derive the equations for all nullclines of the Lotka-Volterra model in Example 1 and use them to construct the phase plane, including all
Metastasis of malignant tumors Metastasis is the process by which cancer cells spread throughout the body and initiate tumors in various organs. This sometimes happens via the bloodstream, where
The Michaelis-Menten equations describe a biochemical reaction in which an enzyme E and substrate S bind to form a complex C. This complex can then either dissociate back into its original components
The Kermack-McKendrick equations from Exercise 23 can be extended to model persistent diseases rather than single outbreaks by including an inflow of susceptible individuals and their natural death.
The Kermack-McKendrick equations are first-order differential equations describing an infectious disease outbreak. Using S and I to denote the number of susceptible and infected people in a
The van der Pol equation is a second-order differential equation describing oscillatory dynamics in a variable x:dx2 dt 2 2 s1 2 x2d dx dt 1 x − 0 where is a positive constant. This equation was
Hooke’s Law states that the force F exerted by a spring on a mass is proportional to the displacement from its resting position.From the figure we have F − 2kp for some positive constant k, where
x9 − 2s y 2 1d 2 asx 2 1d, y9 − 2s y 2 1d 2 1a sx 2 1d, a ± 21, 0, 1, x, y . 0 A system of differential equations is given.(a) Use a phase plane analysis to determine the values of the constant
x9 − ay2 2 x 1 1, y9 − 2s1 2 yd A system of differential equations is given.(a) Use a phase plane analysis to determine the values of the constant a for which the sole equilibrium of the
x9 − asx 2 ad, y9 − 4 2 y 2 x, a ± 0 A system of differential equations is given.(a) Use a phase plane analysis to determine the values of the constant a for which the sole equilibrium of the
x9 − y 2 ax, y9 − x 2 y, a . 0, a ± 1 A system of differential equations is given.(a) Use a phase plane analysis to determine the values of the constant a for which the sole equilibrium of the
x9 − asx 2 3d, y9 − 5 2 y, a ± 0 A system of differential equations is given.(a) Use a phase plane analysis to determine the values of the constant a for which the sole equilibrium of the
x9 − 2sx 2 2d lnsxyd, y9 − exsx 2 yd, x, y . 0 A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the
z9 − z 3 2 4z 2 1 3z 2 2w, w9 − z 2 w 2 1 A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction
x9 − 5 2 2x 2 xy, y9 − xy 2 y, x, y > 0 A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction
p9 − 2q 2 1, q9 − q2 2 q 2 p A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction of
p9 − 2p2 1 q 2 1, q9 − qs2 2 p 2 qd A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction of
x9 − xs2 2 xd, y9 − ys3 2 yd A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction of
n9 − ns1 2 2md, m9 − ms2 2 2n 2 md, n, m > 0 A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the
p9 − ps1 2 p 2 qd, q9 − qs2 2 3p 2 qd, p, q > 0 A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the
x9 − xs3 2 x 2 yd, y9 − ys2 2 x 2 yd, x, y > 0 A system of differential equations is given.(a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the
The variable x is increasing in quadrants I and III and decreasing in quadrants II and IV. The variable y is increasing everywhere. 0 IV x
The variable x is increasing to the left of its nullcline and decreasing to the right of it. The variable y is decreasing above its nullcline and increasing below it. x 0
The variable x is increasing below its nullcline and decreasing above it. The variable y is decreasing above its nullcline and increasing below it. y 0 x
The variable x is increasing above its nullcline and decreasing below it. The variable y is decreasing above its nullcline and increasing below it. 0 x
The variable x is increasing between zero and its nullcline and decreasing elsewhere. The variable y is increasing below its nullcline and decreasing above it. y. 0 x
Modified aphid-ladybug dynamics In Exercise 22 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows:dA dt− 2As1 2 0.0001Ad 2
Modified predator-prey dynamics In Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let’s modify those equations as follows:(a) According to these equations,
A phid-ladybug dynamics Populations of aphids and ladybugs are modeled by the equations(a) Find an expression for dLydA.(b) The direction field for the differential equation in part (b) is shown. Use
Lotka-Volterra equations In Example 1(a) we showed that parametric curves describing the rabbit and wolf populations in the phase plane satisfy the differential equation dW dR−20.02W 1 0.00002RW
Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory. 20. 1200 1000 800 600- 400- 200 0 5 species 1 species 2 10 15 t
Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory. 19. y species 1 200 150+ 100- 50- 0 species 2
Rabbits and foxes A phase trajectory is shown for populations of rabbits sRd and foxes sFd.(a) Describe how each population changes as time goes by.(b) Use your description to make a rough sketch of
Rabbits and foxes A phase trajectory is shown for populations of rabbits sRd and foxes sFd.(a) Describe how each population changes as time goes by.(b) Use your description to make a rough sketch of
A food web Lynx eat snowshoe hares, and snowshoe hares eat woody plants, such as willows. Suppose that, in the absence of hares, the willow population will grow exponentially and the lynx population
Cooperation, competition, or predation? The system of differential equationsis a model for the populations of two species. Does the model describe cooperation, or competition, or a predatorprey
Competition and cooperation Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit(flowering plants and insect
P redator-prey equations For each predator-prey system, determine which of the variables, x or y, represents the prey population and which represents the predator population. Is the growth of the
x − sin t, y − cos2t, 22 < t < 2Describe the motion of a particle with position sx, yd as t varies in the given interval.
x − 5 sin t, y − 2 cos t, 2 < t < 5Describe the motion of a particle with position sx, yd as t varies in the given interval.
x − 2 sin t, y − 4 1 cos t, 0 < t < 3y2 Describe the motion of a particle with position sx, yd as t varies in the given interval.
x − 3 1 2 cos t, y − 1 1 2 sin t, y2 < t < 3y2 Describe the motion of a particle with position sx, yd as t varies in the given interval.
x − t 2, y − t 3(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.(b) Eliminate the parameter
x − st , y − 1 2 t(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.(b) Eliminate the
x − 1 1 3t, y − 2 2 t 2(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.(b) Eliminate the
x − 3t 2 5, y − 2t 1 1(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.(b) Eliminate the
x − e2t 1 t, y − et 2 t, 22 < t < 2 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.
x − cos2t, y − 1 2 sin t, 0 < t < y2 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.
x − t 2, y − t 3 2 4t, 23 < t < 3 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.
x − t 2 1 t, y − t 2 2 t, 22 < t < 2 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.
Species–area relationship The number of species found on an island typically increases with the area of the island.■Suppose that this relationship is such that the rate of increase with island
According to Newton’s Law of Universal Gravitation, the gravitational force on an object of mass m that has been projected vertically upward from the earth’s surface is F −mtR2 sx 1 Rd2 where x
T issue culture Let Astd be the area of a tissue culture at time t and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the

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