3.2.1 Draw
the tangent line approximating the given system at the specified
equilibrium and compare the cobweb diagrams. Use the stability theorem
to check whether the equilibrium is stable. The bacterial selection equation ... at the equilibrium ...
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3.2.2 Draw
the tangent line approximating the given system at the specified
equilibrium and compare the cobweb diagrams. Use the stability theorem
to check whether the equilibrium is stable. As in Exercise 1, but at the equilibrium ... Reference Exercise 1
Draw the tangent line approximating the given system at the specified
equilibrium and compare the cobweb diagrams. Use the stability theorem
to check whether the equilibrium is stable.As in Exercise 1, but at the
equilibrium Reference Exercise 1
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3.2.3 Draw
the tangent line approximating the given system at the specified
equilibrium and compare the cobweb diagrams. Use the stability theorem
to check whether the equilibrium is stable. ...
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3.2.4 Draw
the tangent line approximating the given system at the specified
equilibrium and compare the cobweb diagrams. Use the stability theorem
to check whether the equilibrium is stable. ...
Get solution
3.2.5
Starting from the given initial condition, find the solution for five
steps of each of the following. ... When will the value exceed
100?
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3.2.6
Starting from the given initial condition, find the solution for five
steps of each of the following. ... When will the value exceed
100?
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3.2.7
Starting from the given initial condition, find the solution for five
steps of each of the following. ... When will the value be less
than 0.2?
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3.2.8
Starting from the given initial condition, find the solution for five
steps of each of the following. ... When will the value be less
than 0.2?
Get solution
3.2.9 Consider the linear discrete-time dynamical system ... .... For each of the following values of m,
a. Find the equilibrium.
b. Graph and cobweb.
c. Compare your results with the stability condition. m = 0.9.
Get solution
3.2.11 Consider the linear discrete-time dynamical system ... .... For each of the following values of m,
a. Find the equilibrium.
b. Graph and cobweb.
c. Compare your results with the stability condition. m=−0.5.
Get solution
3.2.11 Consider the linear discrete-time dynamical system ... .... For each of the following values of m,
a. Find the equilibrium.
b. Graph and cobweb.
c. Compare your results with the stability condition. m=−0.5.
Get solution
3.2.12 Consider the linear discrete-time dynamical system ... .... For each of the following values of m,
a. Find the equilibrium.
b. Graph and cobweb.
c. Compare your results with the stability condition. m=−1.5.
Get solution
3.2.13 The
following discrete-time dynamical systems have slope of exactly −1 at
the equilibrium. Check this, and then iterate the function for a few
steps starting from near the equilibrium to see whether it is stable,
unstable, or neither. ... (as in Section 1.6, Exercise 10).
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3.2.14 The
following discrete-time dynamical systems have slope of exactly −1 at
the equilibrium. Check this, and then iterate the function for a few
steps starting from near the equilibrium to see whether it is stable,
unstable, or neither. ... (as in Section 1.6, Exercise 30).
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3.2.15 The
following discrete-time dynamical systems have slope of exactly −1 at
the equilibrium. Check this, and then iterate the function for a few
steps starting from near the equilibrium to see whether it is stable,
unstable, or neither. ... the logistic system with r = 3.
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3.2.16 The
following discrete-time dynamical systems have slope of exactly −1 at
the equilibrium. Check this, and then iterate the function for a few
steps starting from near the equilibrium to see whether it is stable,
unstable, or neither. ... (the equilibrium is ...
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3.2.17 Equilibria
where the slope of the tangent line is exactly 0 are also special. Show
that the following systems satisfy this special relationship. What does
this say about the stability of the equilibrium? (Think about a linear
dynamical system with a slope of 0. How quickly do solutions approach
the equilibrium?) Draw a cobweb diagram to illustrate these results. The logistic dynamical system with r = 2.
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3.2.18 Equilibria
where the slope of the tangent line is exactly 0 are also special. Show
that the following systems satisfy this special relationship. What does
this say about the stability of the equilibrium? (Think about a linear
dynamical system with a slope of 0. How quickly do solutions approach
the equilibrium?) Draw a cobweb diagram to illustrate these results. The dynamical system ...
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3.2.19 We
have studied several systems where the fraction of medication absorbed
depends on the concentration of medication in the bloodstream. These
take the form ... where ... is
the fraction absorbed and 1.0 is the supplement. If the fraction
absorbed increases, it seems possible that the equilibrium level will
become unstable (high levels are rapidly reduced). For each of the
following forms for ...the equilibrium level is ....
Find the slope of the updating function at the equilibrium and check if
it is stable. In which cases does the solution oscillate? ...
Get solution
3.2.21 We
have studied several systems where the fraction of medication absorbed
depends on the concentration of medication in the bloodstream. These
take the form ... where ... is
the fraction absorbed and 1.0 is the supplement. If the fraction
absorbed increases, it seems possible that the equilibrium level will
become unstable (high levels are rapidly reduced). For each of the
following forms for ...the equilibrium level is ....
Find the slope of the updating function at the equilibrium and check if
it is stable. In which cases does the solution oscillate? ...
Get solution
3.2.21 We
have studied several systems where the fraction of medication absorbed
depends on the concentration of medication in the bloodstream. These
take the form ... where ... is
the fraction absorbed and 1.0 is the supplement. If the fraction
absorbed increases, it seems possible that the equilibrium level will
become unstable (high levels are rapidly reduced). For each of the
following forms for ...the equilibrium level is ....
Find the slope of the updating function at the equilibrium and check if
it is stable. In which cases does the solution oscillate? ...
Get solution
3.2.22 We
have studied several systems where the fraction of medication absorbed
depends on the concentration of medication in the bloodstream. These
take the form ... where ... is
the fraction absorbed and 1.0 is the supplement. If the fraction
absorbed increases, it seems possible that the equilibrium level will
become unstable (high levels are rapidly reduced). For each of the
following forms for ...the equilibrium level is ....
Find the slope of the updating function at the equilibrium and check if
it is stable. In which cases does the solution oscillate? ...
Get solution
3.2.23 Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb. The value of r where the positive equilibrium switches from having a positive to a negative slope.
Get solution
3.2.24 Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb. One value of r between 1 and the value found in Exercise 47.
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3.2.25 Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb. One value of r between the value found in Exercise 47 and r = ....
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3.2.26 Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb. One value of r greater than r = ....
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3.2.27 The
logistic model quantifies a competitive interaction, wherein per capita
production is a decreasing function of population size. In some
situations, per capita production is enhanced by population size. For
each of the following cases where per capita growth increases as a
function of population size,
a. Write the updating function.
b. Find the equilibria and their stability.
c. Graph the updating function and cobweb. Which equilibrium is stable?
d. Explain what this population is doing. Per capita production = ...
Get solution
3.2.28 The
logistic model quantifies a competitive interaction, wherein per capita
production is a decreasing function of population size. In some
situations, per capita production is enhanced by population size. For
each of the following cases where per capita growth increases as a
function of population size,
a. Write the updating function.
b. Find the equilibria and their stability.
c. Graph the updating function and cobweb. Which equilibrium is stable?
d. Explain what this population is doing. Per capita production = ...
Get solution
3.2.29 Consider a modified version of the logistic dynamical system ... For the following values of n,
a. Sketch the updating function with r = 2.
b. Find the equilibria.
c. Find the derivative of the updating function at the equilibria.
d. For what values of r is the x = 0 equilibrium stable? For what values of r is the positive equilibrium stable? n =2.
Get solution
3.2.30 Consider a modified version of the logistic dynamical system ... For the following values of n,
a. Sketch the updating function with r = 2.
b. Find the equilibria.
c. Find the derivative of the updating function at the equilibria.
d. For what values of r is the x = 0 equilibrium stable? For what values of r is the positive equilibrium stable? n =3.
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3.2.31 You
come in one morning and find that the temperature is 21?C. To correct
this, you move the thermostat down by 1?C to 19?C. But the next day the
temperature has dropped to 18?C.
Get solution
3.2.32 You
come in one morning and find that the temperature is 21?C. To correct
this, you move the thermostat down by 1?C to 19?C. But the next day the
temperature has dropped to 18.5?C.
Get solution
3.2.33 The model of bacterial selection includes no frequencydependence,
meaning that the per capita production of the different types does not
depend on the fraction of types in the population. Each of the following
discrete-time dynamical systems for the number of mutants ... and the number of wild type ... depends on the fraction of mutants ....
For each, explain in words how each type is affected by the frequency
of the mutants, find the discrete-time dynamical system for ... , find the equilibria, evaluate their stability, and plot a cobweb diagram. Do any of them oscillate? Suppose ...
Get solution
3.2.34 The model of bacterial selection includes no frequencydependence,
meaning that the per capita production of the different types does not
depend on the fraction of types in the population. Each of the following
discrete-time dynamical systems for the number of mutants ... and the number of wild type ... depends on the fraction of mutants ....
For each, explain in words how each type is affected by the frequency
of the mutants, find the discrete-time dynamical system for ... , find the equilibria, evaluate their stability, and plot a cobweb diagram. Do any of them oscillate? Suppose ...
Get solution
3.2.35 Crowded
plants grow to smaller size. Smaller plants make fewer seeds. The
following describe the dynamics of a population described by n, the total number of seeds, and s, the size of the adult produced. In each case,
a. Start from n = 20.0
and find the total number of seeds for the next 2 yr (if the number of
seeds per plant is a fraction, don’t worry. Just think of it as an
average).
b. Write the discrete-time dynamical system for the number of seeds.
c. Find the equilibrium number of seeds.
d. Graph the updating function and cobweb.
e. How does this result relate to the stability condition? If there are n seeds, each sprouts and grows to a size ... An adult of size s produces s − 1.0 seeds (because it must use 1.0 units of energy to survive).
Get solution
3.2.37 Crowded
plants grow to smaller size. Smaller plants make fewer seeds. The
following describe the dynamics of a population described by n, the total number of seeds, and s, the size of the adult produced. In each case,
a. Start from n = 20.0
and find the total number of seeds for the next 2 yr (if the number of
seeds per plant is a fraction, don’t worry. Just think of it as an
average).
b. Write the discrete-time dynamical system for the number of seeds.
c. Find the equilibrium number of seeds.
d. Graph the updating function and cobweb.
e. How does this result relate to the stability condition? If there are n seeds, each sprouts and grows to a size .... Suppose that an adult of size s produces s − 2.0 seeds.
Get solution
3.2.37 Crowded
plants grow to smaller size. Smaller plants make fewer seeds. The
following describe the dynamics of a population described by n, the total number of seeds, and s, the size of the adult produced. In each case,
a. Start from n = 20.0
and find the total number of seeds for the next 2 yr (if the number of
seeds per plant is a fraction, don’t worry. Just think of it as an
average).
b. Write the discrete-time dynamical system for the number of seeds.
c. Find the equilibrium number of seeds.
d. Graph the updating function and cobweb.
e. How does this result relate to the stability condition? If there are n seeds, each sprouts and grows to a size .... Suppose that an adult of size s produces s − 2.0 seeds.
Get solution
3.2.38 Crowded
plants grow to smaller size. Smaller plants make fewer seeds. The
following describe the dynamics of a population described by n, the total number of seeds, and s, the size of the adult produced. In each case,
a. Start from n = 20.0
and find the total number of seeds for the next 2 yr (if the number of
seeds per plant is a fraction, don’t worry. Just think of it as an
average).
b. Write the discrete-time dynamical system for the number of seeds.
c. Find the equilibrium number of seeds.
d. Graph the updating function and cobweb.
e. How does this result relate to the stability condition? If there are n seeds, each sprouts and grows to a size .... An adult of size s produces s − 1 seeds.
Get solution
3.2.39 Study the behavior of the logistic dynamical system first for values of r near 3.0, and then for values between 3.5 and 4.0. Try the following with five values of r near 3.0 (such as 2.9, 2.99, 3.0, 3.01, and 3.1) and ten values of r between 3.5 and 4.0.
a. Use your computer to find solutions for 100 steps.
b. Look at the last 50 or so points on the solution and try to describe what is going on.
c. The case with r = 4.0
is famously chaotic. One of the properties of chaotic systems is
“sensitivity to initial conditions.” Run the system for 100 steps from
one initial condition, and then run it again from an initial condition
that is very close. If you compare your two solutions, they should be
similar for a while, but eventually become completely different. What if
a real system had this property?
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3.2.41 Consider a population following ...
b. Find the equilibria for these values of r (the
equation cannot be solved in general, but your computer should have a
routine for solving, or just guess). Make sure you find them all.
c. Find the derivative of the updating function at each equilibrium.
d. Find which of the equilibria are stable.
e. When all the equilibria are unstable, how might this model behave differently from the Ricker model with r > ...? Can you explain why?
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3.2.41 Consider a population following ...
b. Find the equilibria for these values of r (the
equation cannot be solved in general, but your computer should have a
routine for solving, or just guess). Make sure you find them all.
c. Find the derivative of the updating function at each equilibrium.
d. Find which of the equilibria are stable.
e. When all the equilibria are unstable, how might this model behave differently from the Ricker model with r > ...? Can you explain why?
Get solution
