3.1.1 Find
the equilibria of the following discrete-time dynamical systems from
their graphs and apply the Graphical Criterion for stability to find
which are stable. Check by cobwebbing. ...
Get solution
3.1.2 Find
the equilibria of the following discrete-time dynamical systems from
their graphs and apply the Graphical Criterion for stability to find
which are stable. Check by cobwebbing. ...
Get solution
3.1.3 Find
the equilibria of the following discrete-time dynamical systems from
their graphs and apply the Graphical Criterion for stability to find
which are stable. Check by cobwebbing. ...
Get solution
3.1.4 Find
the equilibria of the following discrete-time dynamical systems from
their graphs and apply the Graphical Criterion for stability to find
which are stable. Check by cobwebbing. ...
Get solution
3.1.5 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...
Get solution
3.1.6 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...
Get solution
3.1.7 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...
Get solution
3.1.8 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...
Get solution
3.1.9 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...
Get solution
3.1.10 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...
Get solution
3.1.11 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ... (from Section 1.6, Exercise 29).
Get solution
3.1.12 Graph
the following discrete-time dynamical systems, find the equilibria
algebraically, and check whether the stability derived from the Slope
Criterion for stability matches that found with cobwebbing. ...= 1).
Get solution
3.1.13 The
unusual equilibrium in the text has an updating function that lies
below the diagonal both to the left and to the right of the equilibrium.
There are several other ways that an updating function can be tangent
to the diagonal at an equilibrium. In each case, cobweb starting from
points to the left and to the right of the equilibrium, and describe the
stability. Graph an updating function that lies above the diagonal both to the left and to the right of an equilibrium.
Get solution
3.1.14 The
unusual equilibrium in the text has an updating function that lies
below the diagonal both to the left and to the right of the equilibrium.
There are several other ways that an updating function can be tangent
to the diagonal at an equilibrium. In each case, cobweb starting from
points to the left and to the right of the equilibrium, and describe the
stability. Graph an updating function that is tangent to the
diagonal at an equilibrium but crosses from below to above. Show by
cobwebbing that the equilibrium is unstable. What is the second
derivative at the equilibrium?
Get solution
3.1.15 The
unusual equilibrium in the text has an updating function that lies
below the diagonal both to the left and to the right of the equilibrium.
There are several other ways that an updating function can be tangent
to the diagonal at an equilibrium. In each case, cobweb starting from
points to the left and to the right of the equilibrium, and describe the
stability. Graph an updating function that is tangent to the
diagonal at an equilibrium but crosses from above to below. Show by
cobwebbing that the equilibrium is stable. What is the second derivative
at the equilibrium?
Get solution
3.1.16 The
unusual equilibrium in the text has an updating function that lies
below the diagonal both to the left and to the right of the equilibrium.
There are several other ways that an updating function can be tangent
to the diagonal at an equilibrium. In each case, cobweb starting from
points to the left and to the right of the equilibrium, and describe the
stability. Sketch the graph of an updating function that has a corner at an equilibrium and is stable.
Get solution
3.1.17 The
unusual equilibrium in the text has an updating function that lies
below the diagonal both to the left and to the right of the equilibrium.
There are several other ways that an updating function can be tangent
to the diagonal at an equilibrium. In each case, cobweb starting from
points to the left and to the right of the equilibrium, and describe the
stability. Sketch the graph of an updating function that has a corner at an equilibrium and is unstable.
Get solution
3.1.18 The
unusual equilibrium in the text has an updating function that lies
below the diagonal both to the left and to the right of the equilibrium.
There are several other ways that an updating function can be tangent
to the diagonal at an equilibrium. In each case, cobweb starting from
points to the left and to the right of the equilibrium, and describe the
stability. Sketch the graph of an updating function that has a corner at an equilibrium and is neither stable nor unstable.
Get solution
3.1.19 Another
peculiarity of an updating function that is tangent to the diagonal at
an equilibrium is that slight changes in the graph can produce big
changes in the number of equilibria. The following are based on Figure
3.1.10. ... Move
the curve slightly down (while keeping the diagonal in the same place).
How many equilibria are there now? What happens when you cobweb starting
from a point at the righthand edge of the figure?
Get solution
3.1.20 Another
peculiarity of an updating function that is tangent to the diagonal at
an equilibrium is that slight changes in the graph can produce big
changes in the number of equilibria. The following are based on Figure
3.1.10. ... Move
the curve slightly up (again keeping the diagonal in the same place).
How many equilibria are there? Describe their stability.
Get solution
3.1.21 Find
the inverse of each of the following updating functions, and compute
the slope of both the original updating function and the derivative at
the equilibrium. The updating function ... (as in Section 3.1, Exercise 21).
Get solution
3.1.22 Find
the inverse of each of the following updating functions, and compute
the slope of both the original updating function and the derivative at
the equilibrium. The updating function ... (as in Section 3.1, Exercise 22).
Get solution
3.1.23 Recall the updating function with the fraction p of mutant bacteria given by ... where s is the per capita production of the mutant and r is the per capita production of the wild type. Find the derivative in the following cases, and evaluate at the equilibria p = 0 and p = 1. Are the equilibria stable? s = 1.2, r = 2.0
Get solution
3.1.24 Recall the updating function with the fraction p of mutant bacteria given by where s is the per capita ...production of the mutant and r is the per capita production of the wild type. Find the derivative in the following cases, and evaluate at the equilibria p = 0 and p = 1. Are the equilibria stable? s = 3.0, r = 1.2
Get solution
3.1.25 Recall the updating function with the fraction p of mutant bacteria given by where s is the per capita ...production of the mutant and r is the per capita production of the wild type. Find the derivative in the following cases, and evaluate at the equilibria p = 0 and p = 1. Are the equilibria stable? s = 1.5, r = 1.5
Get solution
3.1.26 Recall the updating function with the fraction p of mutant bacteria given by where s is the per capita ...production of the mutant and r is the per capita production of the wild type. Find the derivative in the following cases, and evaluate at the equilibria p = 0 and p = 1. Are the equilibria stable? In general (without substituting numerical values for s and r ), can both equilibria be stable? What happens if r = s?
Get solution
3.1.27 Find
the equilibrium population of bacteria in the following cases with
supplementation. Graph the updating function for each, and use the Slope
Criterion for stability to check the stability. A population of bacteria has per capita production r = 0.6 and ...bacteria are added each generation (as in Section 1.9, Exercise 35).
Get solution
3.1.29
A lab is growing and harvesting a culture of valuable bacteria
described by the discrete-time dynamical system ... The bacteria
have per capita production r , and h are
harvested each generation (as in Section 1.9, Exercises 49 and 50).
Graph the updating function for each, and use the Slope Criterion for
stability to check the stability. Suppose that r = 1.5 and h = ...bacteria.
Get solution
3.1.29
A lab is growing and harvesting a culture of valuable bacteria
described by the discrete-time dynamical system ... The bacteria
have per capita production r , and h are
harvested each generation (as in Section 1.9, Exercises 49 and 50).
Graph the updating function for each, and use the Slope Criterion for
stability to check the stability. Suppose that r = 1.5 and h = ...bacteria.
Get solution
3.1.30
A lab is growing and harvesting a culture of valuable bacteria
described by the discrete-time dynamical system ... The bacteria
have per capita production r , and h are
harvested each generation (as in Section 1.9, Exercises 49 and 50).
Graph the updating function for each, and use the Slope Criterion for
stability to check the stability. Without setting r and h to particular values, find the equilibrium algebraically. When is the equilibrium stable?
Get solution
3.1.31
The model describing the dynamics of the concentration of medication
in the bloodstream,... becomes
nonlinear if the fraction of medication used is a function of the
concentration (as in Section 1.10, Exercise 39). In each case, use the
Slope Criterion for stability to check the stability of the equilibrium.
The nonlinear discrete-time dynamical system ...
Get solution
3.1.32
The model describing the dynamics of the concentration of medication
in the bloodstream,... becomes
nonlinear if the fraction of medication used is a function of the
concentration (as in Section 1.10, Exercise 39). In each case, use the
Slope Criterion for stability to check the stability of the equilibrium.
The nonlinear discrete-time dynamical system ... How does this
differ from the model in Exercise 31? Why is the equilibrium smaller?
Get solution
3.1.33 An
equilibrium that is stable when time goes forward should be unstable
when time goes backward. Find the inverses of the updating functions
associated with the following discretetime dynamical systems, and find
the derivative at the equilibria. ...
Get solution
3.1.34 An
equilibrium that is stable when time goes forward should be unstable
when time goes backward. Find the inverses of the updating functions
associated with the following discretetime dynamical systems, and find
the derivative at the equilibria. ...
Get solution
3.1.35 An
equilibrium that is stable when time goes forward should be unstable
when time goes backward. Find the inverses of the updating functions
associated with the following discretetime dynamical systems, and find
the derivative at the equilibria. ...
Get solution
3.1.36 An
equilibrium that is stable when time goes forward should be unstable
when time goes backward. Find the inverses of the updating functions
associated with the following discretetime dynamical systems, and find
the derivative at the equilibria. ...
Get solution
3.1.37 Consider a population ... with
per capita production of.... After writing the discrete-time dynamical
system, compute the following for the given values of the parameter r .
a. Find the equilibria.
b. Graph the updating function.
c. Indicate which equilibria are stable and which are unstable, and check with the Slope Criterion for stability.
d. Describe in words how the population would behave. r = 1.0
Get solution
3.1.38 Consider a population ... with
per capita production of.... After writing the discrete-time dynamical
system, compute the following for the given values of the parameter r .
a. Find the equilibria.
b. Graph the updating function.
c. Indicate which equilibria are stable and which are unstable, and check with the Slope Criterion for stability.
d. Describe in words how the population would behave. r = 2.5
Get solution
3.1.39
Consider the discrete-time dynamical system for a heart studied in
Section 1.11. ... In each case, sketch the updating function.
Why is the equilibrium stable when it exists? ... = 20.0 millivolts, u = 10.0 millivolts, c = 0.5.
Get solution
3.1.40
Consider the discrete-time dynamical system for a heart studied in
Section 1.11. ... In each case, sketch the updating function.
Why is the equilibrium stable when it exists? ... = 20.0 millivolts, u = 10.0 millivolts, c = 0.6.
Get solution
3.1.41
Consider the discrete-time dynamical system for a heart studied in
Section 1.11. ... In each case, sketch the updating function.
Why is the equilibrium stable when it exists? ... = 20.0 millivolts, u = 10.0 millivolts, c = 0.7.
Get solution
3.1.42
Consider the discrete-time dynamical system for a heart studied in
Section 1.11. ... In each case, sketch the updating function.
Why is the equilibrium stable when it exists? ... = 20.0 millivolts, u = 10.0 millivolts, c = 0.8.
Get solution
3.1.43 Consider
the discrete-time dynamical system found in Section 1.10, Exercise 49.
In that exercise, there were two cultures, 1 and 2. In culture 1, the
mutant does better than the wild type, and in culture 2 the wild type
does better. In particular, suppose that s = 2.0 and r = 0.3 in culture 1, and that s = 0.6 and r = 2.0 in culture 2. Define updating functions ...to
describe the dynamics in the two cultures. The overall updating
function after mixing equal amounts from the two is ...
a. Write the updating function explicitly.
b. Find the equilibria.
c. Find the derivative of the updating function.
d. Evaluate the stability of the equilibrium.
Get solution
3.1.44 Consider
the discrete-time dynamical systems with the following discrete-time
dynamical systems. Check the stability of the equilibria. ...
Get solution
3.1.45 Consider the discrete-time dynamical system ... (thanks to Larry Okun). We will study this for different values of a.
a. Follow the dynamics starting from initial condition ... = 1.0 for a = 1.0, 1.1, 1.2, 1.3, 1.4. Keep running the system until it seems to reach an equilibrium.
b. Do the same, but increase a slowly
past a critical value of approximately 1.4446679. Solutions should
creep up for a while, and then increase very quickly. Graph the updating
function for values above and below the critical value and try to
explain why.
c. At the critical value, the slope of the updating function is 1 at the equilibrium. Show that this occurs with ...What is the equilibrium?
Get solution
3.1.46 Consider a population with per capita production = ... Start with r = 3.0 and find the equilibrium. Then try smaller and smaller values of r and track what happens to the equilibrium, using the previous equilibrium as a starting point. What happens when r crosses 2? Follow the equilibrium population down to r = 1. If this were a real population and r was a measure of the quality of habitat, how would you interpret this behavior? Then do the same but start with r = 1 and increase r up to 3. Can you explain what is going on?
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