5.6.1 Finding equilibria of coupled differential equations requires solving simultaneous equations.
The following are linear equations, where the only possibilities are no
solutions, one solution, or a whole line of solutions. For each of the
following pairs, solve and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. −3 − y + 3x = 0 −2 + 2y − 4x = 0
Get solution
5.6.2 Finding equilibria of coupled differential equations requires solving simultaneous equations.
The following are linear equations, where the only possibilities are no
solutions, one solution, or a whole line of solutions. For each of the
following pairs, solve and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. −6 + 3y + 3x = 0 −2 − 2y − 6x = 0
Get solution
5.6.3 Finding equilibria of coupled differential equations requires solving simultaneous equations.
The following are linear equations, where the only possibilities are no
solutions, one solution, or a whole line of solutions. For each of the
following pairs, solve and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. 3 − 3y + 3x = 0 −2 + 2y − 2x = 0 What goes wrong? Use your graph with y on the vertical axis to explain the problem.
Get solution
5.6.4 Finding equilibria of coupled differential equations requires solving simultaneous equations.
The following are linear equations, where the only possibilities are no
solutions, one solution, or a whole line of solutions. For each of the
following pairs, solve and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. 8 − 2y + 4x = 0 −2 + y − 2x = 0 What goes wrong? Use your graph with y on the vertical axis to explain the problem.
Get solution
5.6.5 Finding
equilibria of nonlinear coupled differential equations requires solving
nonlinear simultaneous equations that can have any number of solutions.
For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. ...
Get solution
5.6.6 Finding
equilibria of nonlinear coupled differential equations requires solving
nonlinear simultaneous equations that can have any number of solutions.
For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. ...
Get solution
5.6.7 Finding
equilibria of nonlinear coupled differential equations requires solving
nonlinear simultaneous equations that can have any number of solutions.
For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. ... Solving the first equation for y in terms of x does not give a function. Graph the relation and find the solutions.
Get solution
5.6.8 Finding
equilibria of nonlinear coupled differential equations requires solving
nonlinear simultaneous equations that can have any number of solutions.
For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. ... Solving the first equation for y in terms of x does not give a function. Graph the relation and find the solutions.
Get solution
5.6.9 Finding
equilibria of nonlinear coupled differential equations requires solving
nonlinear simultaneous equations that can have any number of solutions.
For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. x ( y − x) = 0 −6 + 2y − 4x = 0 Solving the first equation for y in terms of x does not give a function (and includes a vertical section). Graph the relation and find the solutions.
Get solution
5.6.10 Finding
equilibria of nonlinear coupled differential equations requires solving
nonlinear simultaneous equations that can have any number of solutions.
For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both equations give the same value for y. Sketch a graph with y on the vertical axis. ... Solving the first equation for y in terms of x does not give a function and includes a vertical section. Graph the relation and find the solutions.
Get solution
5.6.11
Graph the nullclines in the phase plane and find the equilibria of
the following. Predator-prey model ... Equation 5.5.1) with λ=1.0, δ =3.0, ε =0.002, η = 0.005.
Get solution
5.6.12
Graph the nullclines in the phase plane and find the equilibria of
the following. Predator-prey model ... Predator-prey model with
λ=1.0, δ =3.0, ε =0.005, η = 0.002.
Get solution
5.6.13
Graph the nullclines in the phase plane and find the equilibria of
the following. Predator-prey model ... Newton’s law of cooling
...
Get solution
5.6.14
Graph the nullclines in the phase plane and find the equilibria of
the following. Predator-prey model ... Newton’s law of cooling
with α = 0.1 and ...= 0.5.
Get solution
5.6.15
Graph the nullclines in the phase plane and find the equilibria of
the following. Predator-prey model ... Competition model
...
Get solution
5.6.16 Graph the nullclines in the phase plane and find the equilibria of the following. Predator-prey model ... ...
Get solution
5.6.17
Redraw
the phase planes for the following problems, but make the other choice
for the vertical variable. Check that you get the same equilibrium. The
equations in Exercise 11. Reference Exercise 11 Predator-prey model
... Equation 5.5.1) with λ=1.0, δ =3.0, ε =0.002, η = 0.005.
Get solution
5.6.18
Redraw
the phase planes for the following problems, but make the other choice
for the vertical variable. Check that you get the same equilibrium. The
equations in Exercise 12. Reference Exercise 12 Predator-prey model
... Predator-prey model with λ=1.0, δ =3.0, ε =0.005, η = 0.002.
Get solution
5.6.19
Redraw
the phase planes for the following problems, but make the other choice
for the vertical variable. Check that you get the same equilibrium. The
equations in Exercise 15. Reference Exercise 15 Predator-prey model
... Competition model ...
Get solution
5.6.20
Redraw
the phase planes for the following problems, but make the other choice
for the vertical variable. Check that you get the same equilibrium. he
equations in Exercise 16. Reference Exercise 16 Predator-prey model
... ...
Get solution
5.6.21
If
each state variable in a system of autonomous differential equations
does not respond to changes in the value of the other, but depends only
on a constant value, the two equations can be considered separately. In
this case, the phase plane is particularly simple. Find the null clines
and equilibria in the following cases. The situation in Section 5.5,
Exercise 5. Reference Section 5.5, Exercise 5. Suppose that types a and b do not interact (equivalent to setting b = 0 in the differential equation for a).
Get solution
5.6.22
If
each state variable in a system of autonomous differential equations
does not respond to changes in the value of the other, but depends only
on a constant value, the two equations can be considered separately. In
this case, the phase plane is particularly simple. Find the null clines
and equilibria in the following cases. The situation in Section 5.5,
Exercise 6. Reference Section 5.5, Exercise 6. Suppose that types a and b interact with a fixed population of 500 of the other (set b = 500 in the differential equation for a).
Get solution
5.6.23
Find the nullclines and equilibria for the following predator-prey
models. The model in Section 5.5, Exercise 23. Reference Section 5.5,
Exercise 23per capita growth of prey = 1.0 − 0.05p per capita growth of predators=−1.0 + 0.02b.
Get solution
5.6.24
Find the nullclines and equilibria for the following predator-prey
models. The model in Section 5.5, Exercise 24. Reference Section 5.5,
Exercise 24. 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
5.6.25
Find the nullclines and equilibria for the following predator-prey
models. The model in Section 5.5, Exercise 25. Reference Section 5.5,
Exercise 25. per capita growth of prey = 2.0 − 0.0001 ... per capita growth of predators=−1.0 + 0.01b
Get solution
5.6.26
Find the nullclines and equilibria for the following predator-prey
models. The model in Section 5.5, Exercise 26. Reference Section 5.5,
Exercise 26 per capita growth of prey = 2.0 − 0.01p per capita growth of predators=−1.0 + 0.0001 ...
Get solution
5.6.27
Find and graph the nullclines, and find the equilibria for the
following models. The model found in Section 5.5, Exercise 27.
Reference Section 5.5, Exercise 27 Two predators that must eat each
other to survive.
Get solution
5.6.28
Find and graph the nullclines, and find the equilibria for the
following models. The model found in Section 5.5, Exercise 28.
Reference Section 5.5, Exercise 28 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
5.6.29
The model found in Section 5.5, Exercise 29. The model found in
Section 5.5, Exercise 30. Reference Section 5.5, Exercise 29 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
5.6.30
The model found in Section 5.5, Exercise 30. Reference Section 5.5,
Exercise 30 Tw o competitors where the per capita production of each
type is affected only by the population size of the other type.
Get solution
5.6.31 The
models of diffusion derived in Section 5.5, Exercises 31 and 32 assume
that the membrane between the vessels is equally permeable in both
directions. Suppose instead that the constant of proportionality
governing the rate at which chemical moves differs in the two
directions. In each of the following cases,
a. Find the rate at which chemical moves from the smaller to the larger vessel.
b. Find the rate at which chemical moves from the larger to the smaller vessel.
c. Find the rate of change of the amount of chemical in each vessel.
d. Divide by the volumes ...to find the rate of change of concentration.
e. Find and graph the nullclines.
f. What are the equilibria? Do they make sense? The
constant of proportionality governing the rate at which chemical enters
the cell is three times as large as the constant governing the rate at
which it leaves (as in Section 5.1, Exercise 33).
Get solution
5.6.32 The
models of diffusion derived in Section 5.5, Exercises 31 and 32 assume
that the membrane between the vessels is equally permeable in both
directions. Suppose instead that the constant of proportionality
governing the rate at which chemical moves differs in the two
directions. In each of the following cases,
a. Find the rate at which chemical moves from the smaller to the larger vessel.
b. Find the rate at which chemical moves from the larger to the smaller vessel.
c. Find the rate of change of the amount of chemical in each vessel.
d. Divide by the volumes ...to find the rate of change of concentration.
e. Find and graph the nullclines.
f. What are the equilibria? Do they make sense? The
constant of proportionality governing the rate at which chemical enters
the cell is half as large as the constant governing the rate at which
it leaves (as in Section 5.1, Exercise 34).
Get solution
5.6.33 In our model of competition, the per capita growth rate of types a and b are functions only of the total population size. This means that reproduction is reduced just as much by an individual of type a as by an individual of type b. In many systems, each type interferes differently with type a than with type b.
Check that the given set of equations matches the assumptions in each
of the following cases, and find and graph the equilibria and
nullclines. Suppose that individuals of type b reduce the per capita growth rate of type a by half as much as individuals of type a, and that individuals of type a reduce the per capita type b growth rate by twice as much as individuals of type b. The equations are ...
Get solution
5.6.34 In our model of competition, the per capita growth rate of types a and b are functions only of the total population size. This means that reproduction is reduced just as much by an individual of type a as by an individual of type b. In many systems, each type interferes differently with type a than with type b.
Check that the given set of equations matches the assumptions in each
of the following cases, and find and graph the equilibria and
nullclines. Suppose that individuals of type b reduce the per capita type a growth rate by half as much as individuals of type a, and that individuals of type a reduce the per capita type b growth rate by half as much as individuals of type b. The equations are ... (There should be four equilibria.)
Get solution
5.6.35
Draw the nullclines and find equilibria of the following extensions
of the basic disease model. The model in Section 5.5, Exercise 35. Find
the nullclines and equilibria of this model when α = 2.0, μ = 1.0, and k = 1.0.
Get solution
5.6.36
Draw the nullclines and find equilibria of the following extensions
of the basic disease model. The model in Section 5.5, Exercise 36. Find
the nullclines and equilibria of this model when α = 2.0 and μ = 1.0.
Get solution
5.6.37
Draw the nullclines and find equilibria of the following extensions
of the basic disease model. The model in Section 5.5, Exercise 37. Find
the nullclines and equilibria of this model when α = 2.0, μ = 1.0, and k = 0.5.
Get solution
5.6.38
Draw the nullclines and find equilibria of the following extensions
of the basic disease model. The model in Section 5.5, Exercise 38. Find
the nullclines and equilibria of this model when α = 2.0, μ = 1.0, and k = 4.0.
Get solution
5.6.39
Draw the nullclines and find equilibria of the following extensions
of the basic disease model. The model in Section 5.5, Exercise 39. Find
the nullclines and equilibria of this model when α = 2.0, μ = 1.0, and b = 2.0.
Get solution
5.6.40
Draw the nullclines and find equilibria of the following extensions
of the basic disease model. The model in Section 5.5, Exercise 40. Find
the nullclines and equilibria of this model when α = 2.0, μ = 1.0, and b = 1.0.
Get solution
5.6.41 One complicated equation for chemical kinetics is the Schnakenberg reaction. Let A and B denote the concentrations of two chemicals A and B. A is added at constant rate ..., B is added at constant rate ..., A breaks down at rate ..., and B is converted into A with an autocatalytic reaction. The equations are ... The final term is somewhat like the term αI S in
the epidemic equation studied in the previous three problems, but
differs in that the rate of the reaction becomes faster the larger the
concentration of A. Suppose that ...
a. Have your computer draw the nullclines and find the equilibria in the case ...
b. Do the same with ...
c. Try to explain your results.
Get solution
