5.7.1 Suppose
the following functions are solutions of some differential equation.
Graph these as functions of time and as phase plane trajectories for 0≤ t ≤ 2. Mark the position at t = 0, t = 1, and t =2. x ( t )=t, y(t)=3t.
Get solution
5.7.2 Suppose
the following functions are solutions of some differential equation.
Graph these as functions of time and as phase plane trajectories for 0≤ t ≤ 2. Mark the position at t = 0, t = 1, and t =2. ...
Get solution
5.7.3 Suppose
the following functions are solutions of some differential equation.
Graph these as functions of time and as phase plane trajectories for 0≤ t ≤ 2. Mark the position at t = 0, t = 1, and t =2. ...
Get solution
5.7.4 Suppose
the following functions are solutions of some differential equation.
Graph these as functions of time and as phase plane trajectories for 0≤ t ≤ 2. Mark the position at t = 0, t = 1, and t =2. x ( t )=1 + t (t − 2), y(t)=t (3 − t).
Get solution
5.7.5 From
the following graphs of solutions of differential equations as
functions of time, graph the matching phase-plane trajectory. ...
Get solution
5.7.6 From
the following graphs of solutions of differential equations as
functions of time, graph the matching phase-plane trajectory. ...
Get solution
5.7.7 From
the following graphs of phase-plane trajectories, graph the matching
solutions of differential equations as functions of time. ...
Get solution
5.7.8 From
the following graphs of phase-plane trajectories, graph the matching
solutions of differential equations as functions of time. ...
Get solution
5.7.9 On
the following phase-plane diagrams, use the direction arrows to sketch
phase-plane trajectories starting from two different initial conditions.
...
Get solution
5.7.10 On
the following phase-plane diagrams, use the direction arrows to sketch
phase-plane trajectories starting from two different initial conditions.
...
Get solution
5.7.11
Use the information in the phase-plane diagram to draw direction
arrows on the nullclines. The diagram in Exercise 9. Reference
Exercise 9 ...
Get solution
5.7.12
Use the information in the phase-plane diagram to draw direction
arrows on the nullclines. The diagram in Exercise 10. Reference
Exercise 10 ...
Get solution
5.7.13 Compare
solutions estimated with Euler’s method with the phase-plane diagram
and direction arrows found in the text for the competition equations
(Figure 5.7.13) ... starting from the given initial conditions. Assume that μ=2.0, λ = 2.0, .... Start from a = 750 and b = 500. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 7.
Get solution
5.7.14 Compare
solutions estimated with Euler’s method with the phase-plane diagram
and direction arrows found in the text for the competition equations
(Figure 5.7.13) ... starting from the given initial conditions. Assume that μ=2.0, λ = 2.0, .... Start from a = 250 and b = 500. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 8.
Get solution
5.7.15 Compare
solutions estimated with Euler’s method with the phase-plane diagram
and direction arrows found in the text for the competition equations
(Figure 5.7.13) ... with the given parameter values and starting from the given initial conditions. Suppose α = 0.3 and ... = 0.1. Start from H = 60 and A = 20. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 11.
Get solution
5.7.16 Compare
solutions estimated with Euler’s method with the phase-plane diagram
and direction arrows found in the text for the competition equations
(Figure 5.7.13) ... with the given parameter values and starting from the given initial conditions. Suppose α = 0.3 and ...= 0.1. Start from H = 0 and A = 20. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 12.
Get solution
5.7.17 Compare
solutions estimated with Euler’s method with the phase-plane diagram
and direction arrows found in the text for the competition equations
(Figure 5.7.13) ... with the given parameter values and starting from the given initial conditions. Suppose α = 3.0 and ... = 1.0. Start from H = 60 and A = 20. Take two steps, with a step length of Δt = 0.25, as in Section 5.5, Exercise 13. Does this diagram help explain what went wrong?
Get solution
5.7.18 Compare
solutions estimated with Euler’s method with the phase-plane diagram
and direction arrows found in the text for the competition equations
(Figure 5.7.13) ... with the given parameter values and starting from the given initial conditions. Suppose α = 3.0 and ... = 1.0. Start from H = 0 and A = 20. Take two steps, with a step length of Δt = 0.5, as in Section 5.5, Exercise 14. Does this diagram help explain what went wrong?
Get solution
5.7.19 Draw
the nullclines and direction arrows for the following models of
springs. Make sure to include positive and negative values for the
position x and the velocity v. The model in Section 5.5,
Exercise 15. Reference Section 5.5, Exercise 15 studied in Subsection
2.10.3 describes how acceleration (the second derivative of the position
x) is equal to the negative of the position. The spring constant k has
been set to 1 for simplicity. This one equation for the second
derivative can be written as a system of two autonomous differential
equations. Write the velocity v in terms of the derivative of the position x, and the acceleration in terms of the derivative of the velocity v. Use the spring equation to write the derivative of the velocity in terms of the position x. Write the spring equation as a pair of equations for position and velocity.
Get solution
5.7.20 Draw
the nullclines and direction arrows for the following models of
springs. Make sure to include positive and negative values for the
position x and the velocity v. The model in Section 5.5,
Exercise 16. Reference Section 5.5, Exercise 16 Friction
also creates acceleration proportional to the negative of the velocity
(see Section 2.10, Exercise 39). A simple case obeys the equation
... Write this as a pair of coupled differential equations for x and v.
Get solution
5.7.21 Sketch
the given solution of the following models of springs first as a pair
of functions of time, and then in the phase plane. Check that the
solution follows the arrows. The solution x(t)= cos(t) (Section 5.5, Exercise 17) of the spring equation in Section 5.5, Exercise 15.
Get solution
5.7.22 Sketch
the given solution of the following models of springs first as a pair
of functions of time, and then in the phase plane. Check that the
solution follows the arrows. he solution ... (Section 5.5, Exercise 18) of the spring equation with friction in Section 5.5, Exercise 16.
Get solution
5.7.23
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 23. Reference Exercise 23 Consider
the following types of predator-prey interactions. Graph the per capita
rates of change and write the associated system of autonomous
differential equations. per capita growth of prey = 1.0 − 0.05p per capita growth of predators=−1.0 + 0.02b.
Get solution
5.7.24
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 24. Reference Exercise 24 Consider
the following types of predator-prey interactions. Graph the per capita
rates of change and write the associated system of autonomous
differential equations. per capita growth of prey = 2.0 − 0.01p per capita growth of predators = 1.0 + 0.01b. How does this differ from the basic predator-prey system (Equation 5.5.1)? (Equation 5.5.1) ...
Get solution
5.7.25
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 27. Reference Exercise 27 Write
systems of differential equations describing the following situations.
Feel free to make up parameter values as needed. Two predators that
must eat each other to survive.
Get solution
5.7.26
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 28. Reference Exercise 28 Write
systems of differential equations describing the following situations.
Feel free to make up parameter values as needed. Two
predators that must eat each other to survive, but with per capita
production of each reduced by competition with its own species.
Get solution
5.7.27
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 29. Reference Exercise 29 Write
systems of differential equations describing the following situations.
Feel free to make up parameter values as needed. Two competitors where
the per capita production of a is decreased by the total population, and the total population of b is decreased by the population of b.
Get solution
5.7.28
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 30. Reference Exercise 30 Write
systems of differential equations describing the following situations.
Feel free to make up parameter values as needed. Tw o competitors where
the per capita production of each type is affected only by the
population size of the other type.
Get solution
5.7.29
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 33. Reference Exercise 33 Write
systems of autonomous differential equations describing the temperature
of an object and the temperature of the room in the following cases.
The
size of the room is 10.0 times that of the object, but the specific
heat of the room is 0.2 times that of the object (meaning that a small
amount of heat can produce a large change in the temperature of the
room).
Get solution
5.7.31
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 35. Reference Exercise 35 There
are many important extensions of the two dimensional disease model
(Equation 5.5.2) that include more categories of people, and that model
processes of birth, death, and immunity. Suppose that individuals who
leave the infected class through recovery (at per capita rate μ) become susceptible again, but that individuals in the infected class also die at additional per capita rate k.
Get solution
5.7.32
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 36. Reference Exercise 36 There
are many important extensions of the two dimensional disease model
(Equation 5.5.2) that include more categories of people, and that model
processes of birth, death, and immunity. Suppose that one-third
of the individuals who leave the infected class through recovery become
permanently immune, and the other two-thirds become susceptible again.
Get solution
5.7.32
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 36. Reference Exercise 36 There
are many important extensions of the two dimensional disease model
(Equation 5.5.2) that include more categories of people, and that model
processes of birth, death, and immunity. Suppose that one-third
of the individuals who leave the infected class through recovery become
permanently immune, and the other two-thirds become susceptible again.
Get solution
5.7.33
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 37. Reference Exercise 37 There
are many important extensions of the two dimensional disease model
(Equation 5.5.2) that include more categories of people, and that model
processes of birth, death, and immunity. Suppose that all
individuals become susceptible upon recovery (as in the basic model) but
that there is a source of mortality, so that both infected and
susceptible individuals die at per capita rate k.
Get solution
5.7.34
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 38. Reference Exercise 38
p>There are many important extensions of the two dimensional
disease model (Equation 5.5.2) that include more categories of people,
and that model processes of birth, death, and immunity. Suppose
that all individuals become susceptible upon recovery (as in the basic
model) but that there is a source of mortality, where susceptible
individuals die at per capita rate k, but infected individuals die at a per capita rate that is twice as large.
Get solution
5.7.35
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 39. Reference Exercise 39 There
are many important extensions of the two dimensional disease model
(Equation 5.5.2) that include more categories of people, and that model
processes of birth, death, and immunity. Suppose that all
individuals become susceptible upon recovery (as in the basic model) but
that all individuals give birth at rate b. The offspring of susceptible individuals are susceptible and the offspring of infected individuals are infected.
Get solution
5.7.36
For the following problems, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 40. Reference Exercise 40 There
are many important extensions of the two dimensional disease model
(Equation 5.5.2) that include more categories of people, and that model
processes of birth, death, and immunity. Suppose that all
individuals become susceptible upon recovery (as in the basic model) but
that all individuals give birth at rate b, and that all offspring are susceptible.
Get solution
5.7.37 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The
model in Exercise 25 starting from (1500, 200). Is there another path
for the solution that is consistent with the direction arrows?
Get solution
5.7.38 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The model in Exercise 26 starting from (1500, 200).
Get solution
5.7.39 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The model in Exercise 27 starting from (200, 300).
Get solution
5.7.40 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The model in Exercise 28 starting from (200, 300).
Get solution
5.7.41 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The model in Exercise 31 starting from (0.5, 1).
Get solution
5.7.42 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The model in Exercise 32 starting from (0.5, 1).
Get solution
5.7.43 For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. he model in Exercise 35 starting from (0.5, 1).
Get solution
5.7.44
For
the following problems, use the direction arrows on your phase-plane to
sketch a solution starting from the given initial condition. The model
in Exercise 36 starting from (0.5, 0,5). Can you be sure that the
solution behaves exactly like your picture?
Get solution
5.7.45
Consider the following differential equations describing diffusion
and utilization of a chemical.... The parameters have the following
meanings. ...
a. Set δ = 4 and the rest of the parameters to their designated values. Plot the nullclines and find the equilibrium.
b. Follow the same steps with δ =1. Is there an equilibrium? Can you say why not? (No math jargon allowed.) Sketch C and Γ as functions of time.
c. Try to figure out the critical value of δ where the behaviour changes.
Get solution
5.7.46
Many biological systems need to be able to respond to changes in
the level of some signal (like a hormone) without responding to the
actual level. For example, a cell might have no response to a low level
of hormone. If the hormone level rapidly increases, the cell responds.
But if the hormone level then remains constant at the higher level, the
cell again stops responding. This process is sometimes called
adaptation. One mechanism for this process is summarized in the
following model. Internal response is a function of the fraction p of cell surface receptors that are bound by the hormone. This fraction increases when the hormone level H is high. However, hormone also dissociates from bound receptors. Assume this happens at a rate A, but that this rate is controlled by the cell. One possible set of equations is ... Suppose that ... = 0.5 and that ε is a small value (like 0.1 or 0.01). The value of H is determined by conditions external to the cell and does not have its own differential equation.
a. Find the nullclines and equilibria of this model assuming that H is a constant. Does H appear in your final results? Explain why the cell might want to respond in the same way to any constant level of H.
b. Use your computer to simulate the response when the level of H jumps quickly from H = 1 to H = 10. One way to do this is to solve the equations with H = 10 using as initial conditions the equilibrium values of p and A when H = 1. Draw graphs of p and A in the phase-plane and as functions of time. Explain what is happening.
c. Do the same when the level of H drops rapidly from H =10 to H =1.
Get solution
