1.10.1 A population consists of 200 red birds and 800 blue
birds. Find the fraction of red birds and blue birds after the
following. Check that the fractions add up to 1. The population of red
birds doubles, and the population of blue birds remains the same.
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1.10.2
A population consists of 200 red birds and 800 blue birds. Find the
fraction of red birds and blue birds after the following. Check that
the fractions add up to 1. The population of blue birds doubles, and
the population of red birds remains the same.
Get solution
1.10.3
A population consists of 200 red birds and 800 blue birds. Find the
fraction of red birds and blue birds after the following. Check that
the fractions add up to 1. The population of red birds is multiplied by
a factor of r , and the population of blue birds remains the same.
Get solution
1.10.4
A population consists of 200 red birds and 800 blue birds. Find the
fraction of red birds and blue birds after the following. Check that
the fractions add up to 1. The population of blue birds is multiplied
by a factor of s, and the population of red birds remains the same.
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1.10.5 Sketch graphs of the following functions. ... (the updating function in Section
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1.10.6 Sketch graphs of the following functions. ...
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1.10.7 Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find ...in the following situations. Equation 1.10.7 ... ...
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1.10.8 Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find ...in the following situations. Equation 1.10.7 ... ...
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1.10.9 Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find ...in the following situations. Equation 1.10.7 ... ...
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1.10.10 Using the discrete-time dynamical system and the derivation of Equation 1.10.7, find ...in the following situations. Equation 1.10.7 ... ...
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1.10.11 Solve for the equilibria of the following discrete-time dynamical systems. ...
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1.10.12 Solve for the equilibria of the following discrete-time dynamical systems. ...
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1.10.13 Find all non-negative equilibria of the following mathematically elegant discrete-time dynamical systems. ... where a is a positive parameter. What happens to this system if a = 0?
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1.10.14 Find all non-negative equilibria of the following mathematically elegant discrete-time dynamical systems. ... where a is a positive parameter. What happens to this system if a = 0?
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1.10.15 Identify stable and unstable equilibria on the following graphs of updating functions. ...
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1.10.16 Identify stable and unstable equilibria on the following graphs of updating functions. ...
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1.10.17 Identify stable and unstable equilibria on the following graphs of updating functions. ...
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1.10.18 Identify stable and unstable equilibria on the following graphs of updating functions. ...
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1.10.19 Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from ...Which equilibria are stable? s = 1.2, r = 2.0. Equation 1.10.7 ...
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1.10.20 Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from ...Which equilibria are stable? s = 1.8, r = 0.8. Equation 1.10.7 ...
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1.10.21 Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from ...Which equilibria are stable? s = 0.3, r = 0.5. Compare with Exercise 19. Equation 1.10.7 ... Reference Exercise 19. s = 1.2, r = 2.0.
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1.10.22 Find and graph the updating functions for the following cases of the selection model (Equation 1.10.7). Cobweb starting from ...Which equilibria are stable? s = 1.8, r = 1.8. Equation 1.10.7 ...
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1.10.23
For each of the following discrete-time dynamical systems, indicate
which of the equilibria are stable and which are unstable. ...
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1.10.24
For each of the following discrete-time dynamical systems, indicate
which of the equilibria are stable and which are unstable. ...
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1.10.25 This
section has ignored the important evolutionary force of mutation. This
series of problems builds models that consider mutation without
reproduction. Suppose that 20% of wild-type bacteria transform into
mutants and that 10% of mutants transform back into wild type
(“revert”). In each case, find the following.
a. The number of wild type bacteria that mutate and the number of mutants that revert.
b. The number of wild-type bacteria and the number of mutants after mutation and reversion.
c. The total number of bacteria before and after mutation. Why is it the same?
d. The fraction of mutants before and after mutation. Begin with 1.0 ×... wild type and 1.0 ×... mutants.
Get solution
1.10.26 This
section has ignored the important evolutionary force of mutation. This
series of problems builds models that consider mutation without
reproduction. Suppose that 20% of wild-type bacteria transform into
mutants and that 10% of mutants transform back into wild type
(“revert”). In each case, find the following.
a. The number of wild type bacteria that mutate and the number of mutants that revert.
b. The number of wild-type bacteria and the number of mutants after mutation and reversion.
c. The total number of bacteria before and after mutation. Why is it the same?
d. The fraction of mutants before and after mutation. Begin with 1.0 ×... wild type and 1.0 ×... mutants.
Get solution
1.10.27 This
section has ignored the important evolutionary force of mutation. This
series of problems builds models that consider mutation without
reproduction. Suppose that 20% of wild-type bacteria transform into
mutants and that 10% of mutants transform back into wild type
(“revert”). In each case, find the following.
a. The number of wild type bacteria that mutate and the number of mutants that revert.
b. The number of wild-type bacteria and the number of mutants after mutation and reversion.
c. The total number of bacteria before and after mutation. Why is it the same?
d. The fraction of mutants before and after mutation. Begin with ... of
mutants. Find the equilibrium fraction of mutants. Cobweb starting from
the initial condition in Exercise 25. Is the equilibrium stable?
Get solution
1.10.28 This
section has ignored the important evolutionary force of mutation. This
series of problems builds models that consider mutation without
reproduction. Suppose that 20% of wild-type bacteria transform into
mutants and that 10% of mutants transform back into wild type
(“revert”). In each case, find the following.
a. The number of wild type bacteria that mutate and the number of mutants that revert.
b. The number of wild-type bacteria and the number of mutants after mutation and reversion.
c. The total number of bacteria before and after mutation. Why is it the same?
d. The fraction of mutants before and after mutation. Begin with ... mutants,
but suppose that a fraction 0.1 mutate and a fraction 0.2 revert. Find
the discrete time dynamical system and the equilibrium fraction of
mutants.
Get solution
1.10.29 This
series of problems combines mutation with selection. In one simple
scenario, mutations occur in only one direction (wild type turn into
mutants but not vice versa), but wild type and mutants have different
levels of per capita production. Suppose that a fraction 0.1 of wild
type mutate each generation, but that each wildtype individual produces
2.0 offspring while each mutant produces only 1.5 offspring. In each
case, find the following.
a. The number of wild-type bacteria that mutate.
b. The number of wild-type bacteria and the number of mutants after mutation.
c. The number of wild-type bacteria and the number of mutants after reproduction.
d. The total number of bacteria after mutation and reproduction.
e. The fraction of mutants after mutation and reproduction. Begin with 1.0 ×... wild type and 1.0 ×... mutants.
Get solution
1.10.30 This
series of problems combines mutation with selection. In one simple
scenario, mutations occur in only one direction (wild type turn into
mutants but not vice versa), but wild type and mutants have different
levels of per capita production. Suppose that a fraction 0.1 of wild
type mutate each generation, but that each wildtype individual produces
2.0 offspring while each mutant produces only 1.5 offspring. In each
case, find the following.
a. The number of wild-type bacteria that mutate.
b. The number of wild-type bacteria and the number of mutants after mutation.
c. The number of wild-type bacteria and the number of mutants after reproduction.
d. The total number of bacteria after mutation and reproduction.
e. The fraction of mutants after mutation and reproduction. Begin with 1.0 ×...wild type and 1.0 ×... mutants.
Get solution
1.10.31 This
series of problems combines mutation with selection. In one simple
scenario, mutations occur in only one direction (wild type turn into
mutants but not vice versa), but wild type and mutants have different
levels of per capita production. Suppose that a fraction 0.1 of wild
type mutate each generation, but that each wildtype individual produces
2.0 offspring while each mutant produces only 1.5 offspring. In each
case, find the following.
a. The number of wild-type bacteria that mutate.
b. The number of wild-type bacteria and the number of mutants after mutation.
c. The number of wild-type bacteria and the number of mutants after reproduction.
d. The total number of bacteria after mutation and reproduction.
e. The fraction of mutants after mutation and reproduction. Begin with ...wild type and ... mutants. Find the discretetime dynamical system for the fraction ... of
mutants. Find the equilibrium fraction of mutants. Cobweb starting from
the initial condition in Exercise 29. Is the equilibrium stable?
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1.10.32 This
series of problems combines mutation with selection. In one simple
scenario, mutations occur in only one direction (wild type turn into
mutants but not vice versa), but wild type and mutants have different
levels of per capita production. Suppose that a fraction 0.1 of wild
type mutate each generation, but that each wildtype individual produces
2.0 offspring while each mutant produces only 1.5 offspring. In each
case, find the following.
a. The number of wild-type bacteria that mutate.
b. The number of wild-type bacteria and the number of mutants after mutation.
c. The number of wild-type bacteria and the number of mutants after reproduction.
d. The total number of bacteria after mutation and reproduction.
e. The fraction of mutants after mutation and reproduction. Begin with ...wild type and ... mutants,
but suppose that a fraction 0.2 mutate and that the per capita
production of mutants is 1.0. Find the discrete-time dynamical system
and the equilibrium fraction of mutants.
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1.10.33 The
model of selection studied in this section is similar to a model of
migration. Suppose two nearby islands have populations of butterflies,
with ...on the first island and ... on
the second. Each year, 20% of the butterflies from the first island fly
to the second and 30% of the butterflies from the second fly to the
first. Suppose there are no births and deaths. Suppose there are 100 butterflies on each island at time t = 0. How many are on each island at t = 1? At t = 2?
Get solution
1.10.34 The
model of selection studied in this section is similar to a model of
migration. Suppose two nearby islands have populations of butterflies,
with ...on the first island and ... on
the second. Each year, 20% of the butterflies from the first island fly
to the second and 30% of the butterflies from the second fly to the
first. Suppose there are no births and deaths. Suppose there are 200
butterflies on the first island and none on the second at time t = 0. How many are on each island at t =1? At t =2?
Get solution
1.10.35 The
model of selection studied in this section is similar to a model of
migration. Suppose two nearby islands have populations of butterflies,
with ... on the second. Each
year, 20% of the butterflies from the first island fly to the second and
30% of the butterflies from the second fly to the first. Suppose there
are no births and deaths. Find equations for ...
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1.10.36 The
model of selection studied in this section is similar to a model of
migration. Suppose two nearby islands have populations of butterflies,
with ...on the first island and ... on
the second. Each year, 20% of the butterflies from the first island fly
to the second and 30% of the butterflies from the second fly to the
first. Suppose there are no births and deaths. Divide both sides of the discrete-time dynamical system for ... by ...to find a discrete-time dynamical system for the fraction ... on the first island. What is the equilibrium fraction?
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1.10.37 The
following two problems extend the migration models to include some
reproduction. Each year, 20% of the butterflies from the first island
fly to the second and 30% of the butterflies from the second fly to the
first. Again, ...represents the number of butterflies on the first island, ...represents the number of butterflies on the second island, and ... represents the fraction of butterflies on the first island. In each case, find the following:
a. Start with 100 butterflies on each island and find the number after migration and after reproduction.
b. Find equations for ...
c. Find the discrete-time dynamical system for pt+1 in terms of ... .
d. Find the equilibrium p∗.
e. Sketch a graph and cobweb from a reasonable initial condition. Each
butterfly that begins the year on the first island produces one
offspring after migration (whether they find themselves on the first or
the second island). Those that begin the year on the second island do
not reproduce. Assume that no butterflies die.
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1.10.38 Now
suppose that the butterflies that do not migrate reproduce (making one
additional butterfly each) and those that do migrate fail to reproduce
from exhaustion. No butterflies die.
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1.10.39
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. In the basic model, half is used no matter how much there
is. More generally, new concentration = old concentration − fraction
used × old concentration + supplement Suppose that the fraction used
is a decreasing function of the concentration. The following problems
look at two cases. In each, find the equilibrium concentration. Suppose
that ... Write the discrete-time dynamical system and solve for
the equilibrium and compare with M∗ = 2.0 for the basic model.
Get solution
1.10.40
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. In the basic model, half is used no matter how much there
is. More generally, new concentration = old concentration − fraction
used × old concentration + supplement Suppose that ... Write
the discrete-time dynamical system, solve for the equilibrium, and
compare with M∗ = 2.0 for the basic model.
Get solution
1.10.41
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. In the basic model, half is used no matter how much there
is. More generally, new concentration = old concentration − fraction
used × old concentration + supplement Suppose that ... for
some parameter β ≤ 1. Write the discrete-time dynamical system and solve for the equilibrium. Sketch a graph of the equilibrium as a function of β. Cobweb starting from ... = 1.0 in the cases β = 0.05 and β = 0.5.
Get solution
1.10.42
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. In the basic model, half is used no matter how much there
is. More generally, new concentration = old concentration − fraction
used × old concentration + supplement Suppose that ... for
some parameter α. Write the discrete-time dynamical system and solve for the equilibrium. Sketch a graph of the equilibrium as a function of α. What happens whenα>0.5? Can you explain this in biological terms? Cobweb starting from ... = 1.0 in the cases α = 0.1 and α = 1.0.
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1.10.43 Our
models of bacterial population growth neglect the fact that bacteria
produce fewer offspring in large populations. The following problems
introduce two important models of this process, having the form ... where the per capita production r is a function of the population size bt . In each case,
a. Graph the per capita production as a function of population size.
b. Write the discrete-time dynamical system and graph the updating function.
c. Find the equilibria.
d. Cobweb and say whether the equilibrium seems to be stable. One
widely used nonlinear model of competition is the “logistic” model,
where per capita production is a linearly decreasing function of
population size. Suppose that the per capita production is ...
Get solution
1.10.44 Our
models of bacterial population growth neglect the fact that bacteria
produce fewer offspring in large populations. The following problems
introduce two important models of this process, having the form ... where the per capita production r is a function of the population size bt . In each case,
a. Graph the per capita production as a function of population size.
b. Write the discrete-time dynamical system and graph the updating function.
c. Find the equilibria.
d. Cobweb and say whether the equilibrium seems to be stable. In
an alternative model, the per capita production decreases as the
reciprocal of a linear function. Suppose that the per capita production
is ...
Get solution
1.10.45 Our
models of bacterial population growth neglect the fact that bacteria
produce fewer offspring in large populations. The following problems
introduce two important models of this process, having the form ... where the per capita production r is a function of the population size bt . In each case,
a. Graph the per capita production as a function of population size.
b. Write the discrete-time dynamical system and graph the updating function.
c. Find the equilibria.
d. Cobweb and say whether the equilibrium seems to be stable. In
another alternative model, called the Ricker model, the per capita
production decreases exponentially. Suppose that per capita production
is ...
Get solution
1.10.46 Our
models of bacterial population growth neglect the fact that bacteria
produce fewer offspring in large populations. The following problems
introduce two important models of this process, having the form ... where the per capita production r is a function of the population size bt . In each case,
a. Graph the per capita production as a function of population size.
b. Write the discrete-time dynamical system and graph the updating function.
c. Find the equilibria.
d.
Cobweb and say whether the equilibrium seems to be stable. In a model
with an Allee effect, organisms reproduce poorly when the population is
small. In one case, per capita production follows ...
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1.10.47
Consider the discrete-time dynamical system ... similar to
the form in Exercise 43. Plot the updating function and have your
computer find solutions for 50 steps starting from ... = 0.3 for the following values of r :
a. Some value of r between 0 and 1. What is the only equilibrium?
b. Some value of r between 1 and 2. Where are the equilibria? Which one seems to be stable?
c. Some value of r between 2 and 3. Where are the equilibria? Which one seems to be stable?
d. Try several values of r between 3 and 4. What is happening to the solution? Is there any stable equilibrium?
e. The solution is chaotic when r =4. One property of chaos is sensitive dependence on initial conditions. Compare a solution starting from ... = 0.3 with one starting at ... =0.30001.
Even though they start off very close, they soon separate and become
completely different. Why might this be a problem for a scientific
experiment?
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1.10.48 Consider the discrete-time dynamical system ... for the following values of the parameter a. Use your computer to graph the function and the diagonal to look for equilibria. Cobweb starting from ...= 1 in each case.
a. a = 0.3.
b. a = 0.4.
c. a = 1/e.
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1.10.49 Consider the equation describing the dynamics of selection ... but
with two cultures 1 and 2. In 1, the mutant does better than the wild
type, and in 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 discrete-time dynamical systems ...to describe the dynamics in the two cultures.
a. Graph the functions ...along with the identity function. Find the first five values of solutions starting from ...in each culture. Explain in words what each solution is doing and why.
b. Suppose you change the experiment. Begin by taking a population with a fraction ...
of mutants. Split this population in half, and place one half in
culture 1 and the other half in culture 2. Let the bacteria reproduce
once in each culture, and then mix them together. Split the mix in half
and repeat the process. The updating function is ... Can
you derive this? Plot this updating function along with the identity
function. Have your computer find the equilibria and label them on your
graph. Do they make sense?
c. Use cobwebbing to figure out which equilibria are stable.
d. Find one solution starting from ... = 0.001 and another starting from ... =0.999. Are these results consistent with the stability of the equilibria? Explain in words why the solutions do what they do.
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