5.3.1 From
the following graphs of the rate of change as a function of the state
variable, identify stable and unstable equilibria by checking whether
the rate of change is an increasing or decreasing function of the state
variable. ...
Get solution
5.3.2 From
the following graphs of the rate of change as a function of the state
variable, identify stable and unstable equilibria by checking whether
the rate of change is an increasing or decreasing function of the state
variable. ...
Get solution
5.3.3
Use the stability theorem to evaluate the stability of the equilibria
of the following autonomous differential equations. ... (as in
Section 5.2, Exercise 1). Compare your results with the phase line in
Section 5.2, Exercise 15.
Get solution
5.3.4
Use the stability theorem to evaluate the stability of the equilibria
of the following autonomous differential equations. ... (as
in Section 5.2, Exercise 2). Compare your results with the phase line
in Section 5.2, Exercise 16. Reference stability theorem ...
Get solution
5.3.5
Use the stability theorem to evaluate the stability of the equilibria
of the following autonomous differential equations. ... = y cos(y) (as
in Section 5.2, Exercise 3). Compare your results with the phase line
in Section 5.2, Exercise 17. Reference stability theorem ...
Get solution
5.3.6
Use the stability theorem to evaluate the stability of the equilibria
of the following autonomous differential equations. ... (as
in Section 5.2, Exercise 4). Compare your results with the phase line
in Section 5.2, Exercise 18. Reference stability theorem ...
Get solution
5.3.7
Find the stability of the equilibria of the following autonomous
differential equations that include parameters. ` ... (as in
Section 5.2, Exercise 5). Suppose that a > 0.
Get solution
5.3.8
Find the stability of the equilibria of the following autonomous
differential equations that include parameters. ` ... (as in
Section 5.2, Exercise 6). Suppose that c >0.
Get solution
5.3.9
Find the stability of the equilibria of the following autonomous
differential equations that include parameters. ... (as in Section
5.2, Exercise 7). Suppose that α>0 and β <0.
Get solution
5.3.10
Find the stability of the equilibria of the following autonomous
differential equations that include parameters. Find the stability of
the equilibria of the following autonomous differential equations that
include parameters. ... (as in Section 5.2, Exercise 8). Suppose
that β < 0 and a > 1.
Get solution
5.3.11 As
with discrete-time dynamical systems, equilibria can act strange when
the slope of the rate-of-change function is exactly equal to the
critical value of zero. Consider the differential equation .... Find the equilibrium, graph the rate of change as a function of x, and draw the phase-line diagram. Would you consider the equilibrium to be stable or unstable?
Get solution
5.3.12 As
with discrete-time dynamical systems, equilibria can act strange when
the slope of the rate-of-change function is exactly equal to the
critical value of zero. Consider the differential equation .... Find the equilibrium, graph the rate of change as a function of x, and draw the phase-line diagram. Would you consider the equilibrium to be stable or unstable?
Get solution
5.3.13 As
with discrete-time dynamical systems, equilibria can act strange when
the slope of the rate-of-change function is exactly equal to the
critical value of zero. Graph a rate-of-change function that has
a slope of 0 at the equilibrium but the equilibrium is stable. What is
the sign of the second derivative at the equilibrium? What is the sign
of the third derivative at the equilibrium?
Get solution
5.3.14 As
with discrete-time dynamical systems, equilibria can act strange when
the slope of the rate-of-change function is exactly equal to the
critical value of zero. Graph a rate-of-change function that has
a slope of 0 at the equilibrium but the equilibrium is unstable. What
is the sign of the second derivative at the equilibrium? What is the
sign of the third derivative at the equilibrium?
Get solution
5.3.15 The
fact that the rate-of-change function is continuous means that many
behaviors are impossible for an autonomous differential equation. Try
to draw a phase-line diagram with two stable equilibria in a row. Use
the Intermediate Value Theorem to sketch a proof of why this is
impossible.
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5.3.16
The
fact that the rate-of-change function is continuous means that many
behaviors are impossible for an autonomous differential equation. Why
is it impossible for a solution of an autonomous differential equation
to oscillate?
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5.3.17
When
parameter values change, the number and stability of equilibria
sometimes change. Such changes are called bifurcations and play a
central role in the study of differential equations. The following
illustrate several of the more important bifurcations. In each case,
graph the value of equilibria as functions of the parameter value, using
a solid line when an equilibrium is stable and a dashed line when an
equilibrium is unstable. This picture is called a bifurcation diagram.
Consider the equation ... for both positive and negative values
of x. Find the equilibria as functions of a for values of a between −1 and 1. Draw a bifurcation diagram and describe in words what happens at a = 0. The change that occurs at a = 0 is called a transcritical bifurcation.
Get solution
5.3.18
When
parameter values change, the number and stability of equilibria
sometimes change. Such changes are called bifurcations and play a
central role in the study of differential equations. The following
illustrate several of the more important bifurcations. In each case,
graph the value of equilibria as functions of the parameter value, using
a solid line when an equilibrium is stable and a dashed line when an
equilibrium is unstable. This picture is called a bifurcation diagram.
Consider the equation ... for both positive and negative values
of x. Find the equilibria as functions of a for values of a between −1 and 1. Draw a bifurcation diagram and describe in words what happens at a = 0. The change that occurs at a = 0 is called a saddle-node bifurcation.
Get solution
5.3.19
When
parameter values change, the number and stability of equilibria
sometimes change. Such changes are called bifurcations and play a
central role in the study of differential equations. The following
illustrate several of the more important bifurcations. In each case,
graph the value of equilibria as functions of the parameter value, using
a solid line when an equilibrium is stable and a dashed line when an
equilibrium is unstable. This picture is called a bifurcation diagram.
Consider the equation ... for both positive and negative values
of x. Find the equilibria as functions of a for values of a between −1 and 1. Draw a bifurcation diagram and describe in words what happens at a = 0. The change that occurs at a =0 is called a pitchfork bifurcation.
Get solution
5.3.20
When
parameter values change, the number and stability of equilibria
sometimes change. Such changes are called bifurcations and play a
central role in the study of differential equations. The following
illustrate several of the more important bifurcations. In each case,
graph the value of equilibria as functions of the parameter value, using
a solid line when an equilibrium is stable and a dashed line when an
equilibrium is unstable. This picture is called a bifurcation diagram.
Consider the equation ... for both positive and negative values
of x. Find the equilibria as functions of a for values of a between −1 and 1. Draw a bifurcation diagram and describe in words what happens at a = 0. The change that occurs at a =
0 is a slightly different type of pitchfork bifurcation (Exercise 19)
called a subcritical bifurcation (Exercise 19 is supercritical). How
does your picture differ from the simple mirror image of that in
Exercise 19?
Get solution
5.3.21
Use the stability theorem to check the phase-line diagrams for the
following models of bacterial population growth.Reference Section 5.1,
Exercise 27 The model in Section 5.1, Exercise 27 and Section 5.2,
Exercise 21.Reference Section 5.2, Exercise 21 The model in Section
5.1, Exercise 27. Check that your arrows are consistent with the
behavior of b(t) at b = 10 and b = 1000. Reference stability theorem ...
Get solution
5.3.22
Use the stability theorem to check the phase-line diagrams for the
following models of bacterial population growth. The model in Section
5.1, Exercise 28 and Section 5.2, Exercise 22.Reference Section 5.1,
Exercise 28 Suppose that per capita production decreases linearly from a
maximum of λ(0) = 4 with slope −0.001. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 1000? Is b(t) increasing when b =
5000?Reference Section 5.2, Exercise 22 The model in Section 5.1,
Exercise 28. Check that your arrows are consistent with the behavior of b(t) at b = 1000 and b = 5000. Reference stability theorem ...
Get solution
5.3.23
Use the stability theorem to check the phase-line diagrams for the
following models of bacterial population growth. The model in Section
5.1, Exercise 29 and Section 5.2, Exercise 23. In
some circumstances, individuals reproduce better when the population
size is large, and fail to reproduce when the population size is small
(the Allee effect introduced in Exercise 46). Suppose that per capita
production is an increasing linear function with λ(0)=−2 and a slope of 0.01. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 100? Is b(t) increasing when b = 300? The model in Section 5.1, Exercise 29. Check that your arrows are consistent with the behavior of b(t) at b = 100 and b = 300. Reference stability theorem ...
Get solution
5.3.24
Use the stability theorem to check the phase-line diagrams for the
following models of bacterial population growth. The model in Section
5.1, Exercise 30 and Section 5.2, Exercise 24.Reference Section 5.1,
Exercise 30 Suppose that per capita production increases linearly with λ(0)=−5 and a slope of 0.001. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 1000? Is b(t) increasing when b =
3000? Reference Section 5.2, Exercise 24 The model in Section 5.1,
Exercise 30. Check that your arrows are consistent with the behavior of b(t) at b = 1000 and b = 3000. Reference stability theorem ...
Get solution
5.3.25
Use the stability theorem to check the phase-line diagrams for the
following models of bacterial population growth. The model in Section
5.1, Exercise 37 and Section 5.2, Exercise 27. Section 5.1, Exercise 37
The per capita production of each type is reduced by a factor of 1 − p by a factor of 1 − p, so that the per capita production of type a is 2(1 − p). This is a case where a large proportion of type a reduces the production of both types. Will type a take
over? Section 5.2, Exercise 27The model in Section 5.1, Exercise 37.
What happens to a solution starting from a small, but positive, value of
p?
Get solution
5.3.26
Use the stability theorem to check the phase-line diagrams for the
following models of bacterial population growth. The model in Section
5.1, Exercise 38 and Section 5.2, Exercise 28. Reference Section 5.1,
Exercise 38 The per capita production of type a is reduced by a factor of 1 − p and the per capita production of type b is reduced by a factor of p. This is a case where a large proportion of type a reduces the production of type a, and a large proportion of type b reduces the production of type b. Do you think that type a will
still take over? Reference Section 5.2, Exercise 28 The model in
Section 5.1, Exercise 38. What happens to a solution starting from a
small, but positive, value of p?
Get solution
5.3.27 A reaction-diffusion equation describes
how chemical concentration changes due to two factors simultaneously,
reaction and movement. A simple model has the form ... The first term describes diffusion, and the second term R(C) is
the reaction, which could have a positive or negative sign (depending
on whether chemical is being created or destroyed). Suppose that β = 1.0/min, and Γ = 5.0 mol/L. For each of the following forms of R(C),
a. Describe how the reaction rate depends on the concentration.
b. Find the equilibria and their stability.
c. Describe how absorption changes the results. Suppose that R(C)=−C.
Get solution
5.3.28 A reaction-diffusion equation describes
how chemical concentration changes due to two factors simultaneously,
reaction and movement. A simple model has the form ... The first term describes diffusion, and the second term R(C) is
the reaction, which could have a positive or negative sign (depending
on whether chemical is being created or destroyed). Suppose that β = 1.0/min, and Γ = 5.0 mol/L. For each of the following forms of R(C),
a. Describe how the reaction rate depends on the concentration.
b. Find the equilibria and their stability.
c. Describe how absorption changes the results. Suppose that R(C) = 0.5C.
Get solution
5.3.29 A reaction-diffusion equation describes
how chemical concentration changes due to two factors simultaneously,
reaction and movement. A simple model has the form ... The first term describes diffusion, and the second term R(C) is
the reaction, which could have a positive or negative sign (depending
on whether chemical is being created or destroyed). Suppose that β = 1.0/min, and Γ = 5.0 mol/L. For each of the following forms of R(C),
a. Describe how the reaction rate depends on the concentration.
b. Find the equilibria and their stability.
c. Describe how absorption changes the results. Suppose that ...
Get solution
5.3.30 A reaction-diffusion equation describes
how chemical concentration changes due to two factors simultaneously,
reaction and movement. A simple model has the form ... The first term describes diffusion, and the second term R(C) is
the reaction, which could have a positive or negative sign (depending
on whether chemical is being created or destroyed). Suppose that β = 1.0/min, and Γ = 5.0 mol/L. For each of the following forms of R(C),
a. Describe how the reaction rate depends on the concentration.
b. Find the equilibria and their stability.
c. Describe how absorption changes the results. Suppose that ...
Get solution
5.3.31 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.32 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.33 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.34 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.35 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.36
Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. The
equation from the previous problem but with water entering at a rate of
4.0 cm/s (Section 5.2, Exercise 34).
Get solution
5.3.37 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.39 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.39 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.40 Apply
the stability theorem for autonomous differential equations to the
following equations. Show that your results match what you found in your
phase-line diagrams, and give a biological interpretation. ...
Get solution
5.3.41
Exercise
show how the number and stability of equilibria can change when a
parameter changes. Often, bifurcations have important biological
applications, and bifurcation diagrams help in explaining how the
dynamics of a system can suddenly change when a parameter changes only
slightly. In each case, graph the equilibria against the parameter
value, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable to draw the bifurcation diagram.
Consider the logistic differential equation (Section 5.1, Exercise 27)
with harvesting proportional to population size, or ...where h represents the fraction harvested. Graph the equilibria as functions of h for values of h between
0 and 2, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable. Even though they do not make
biological sense, include negative values of the equilibria on your
graph. You should find a transcritical bifurcation (Exercise 17) at h = 1.
Get solution
5.3.42 Exercise
show how the number and stability of equilibria can change when a
parameter changes. Often, bifurcations have important biological
applications, and bifurcation diagrams help in explaining how the
dynamics of a system can suddenly change when a parameter changes only
slightly. In each case, graph the equilibria against the parameter
value, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable to draw the bifurcation diagram. Suppose μ = 1 in the basic disease model ...μI . Graph the two equilibria as functions of α for values of α between
0 and 2, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable. Even though they do not make
biological sense, include negative values of the equilibria on your
graph. You should find a transcritical bifurcation (Exercise 17) at α = 1.
Get solution
5.3.43
Exercise
show how the number and stability of equilibria can change when a
parameter changes. Often, bifurcations have important biological
applications, and bifurcation diagrams help in explaining how the
dynamics of a system can suddenly change when a parameter changes only
slightly. In each case, graph the equilibria against the parameter
value, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable to draw the bifurcation diagram.
Consider a version of the equation in Section 5.2, Exercise 29 that
includes the parameter r , ... Graph the equilibria as functions of r for values of r between
0 and 3, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable. The algebra for checking stability
is messy, so it is only necessary to check stability at r = 3. You should find a saddle-node bifurcation (Exercise 18) at r = 2.
Get solution
5.3.44
Exercise
show how the number and stability of equilibria can change when a
parameter changes. Often, bifurcations have important biological
applications, and bifurcation diagrams help in explaining how the
dynamics of a system can suddenly change when a parameter changes only
slightly. In each case, graph the equilibria against the parameter
value, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable to draw the bifurcation diagram.
Consider a variant of the basic disease model given by ... Graph
the equilibria as functions of α for values of α between
0 and 5, using a solid line when an equilibrium is stable and a dashed
line when an equilibrium is unstable. The algebra for checking stability
is messy, so it is only necessary to check stability at α = 5. You should find a saddle-node bifurcation (Exercise 18) at α = 4.
Get solution
5.3.45
Right
at a bifurcation point, the stability theorem fails because the slope
of the rate-of-change function at the equilibrium is exactly zero. In
each of the following cases, check that the stability theorem fails, and
then draw a phase-line diagram to find the stability. Analyze the
stability of the positive equilibrium in the model from Exercise 44 when
α = 4, the point where the bifurcation occurs.
Get solution
5.3.46 Right
at a bifurcation point, the stability theorem fails because the slope
of the rate-of-change function at the equilibrium is exactly zero. In
each of the following cases, check that the stability theorem fails, and
then draw a phase-line diagram to find the stability. Analyze the stability of the disease model when α = μ= 1, the point where the bifurcation occurs in Exercise 42.
Get solution
