1-2 The following steps are used to build a cobweb diagram. Follow them
for the given discrete-time dynamical system based on bacterial
populations. a. Graph the updating function. b. Use your graph of the
updating function to find the point ( b 0 , b 1 ). c. Reflect it off the
diagonal to find the point ( b 1 , b 1 ). d. Use the graph of the
updating function to find ( b 1 , b 2 ). e. Reflect off the diagonal to
find the point ( b 2 , b 2 ). f. Use the graph of the updating function
to find ( b 2 , b 3 ). g. Sketch the solution as a function of time.
1. The discrete-time dynamical system b t +1 = 2.0b t with b 0 = 1.0.
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2.
The discrete-time dynamical system n t +1 = 0.5 n t with n 0 = 1.0.
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3-6 Cobweb the following discrete-time dynamical systems for three steps
starting from the given initial condition. Compare with the solution
found earlier.
3. v t+1 = 1.5 v t , starting from v 0 = 1220 µm 3 (as in Section 1.5, Exercise 5 ).
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4. l t +1 = l t - 1.7, starting from l 0 = 13.1 cm (as in Section 1.5, Exercise 6 ).
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5. n t +1 = 0.5 n t , starting from n 0 = 1200 (as in Section 1.5, Exercise 7 ).
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6.
M t +1 = 0.75 M t + 2.0 starting from the initial condition M 0 = 16.0
(as in Section 1.5, Exercise 8 ).
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7-12 Graph the updating functions associated with the following
discrete-time dynamical systems, and cobweb for five steps starting from
the given initial condition.
7. x t +1 = 2 x t - 1, starting from x 0 = 2.
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8. Z t +1 = 0.9 z t + 1, starting from z 0 = 3.
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9. w t +1 = -0.5 w t + 3, starting from w 0 = 0.
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10. x t +1 = 4 - x t , starting from x 0 = 1 (as in Section 1.5, Exercise 25 ).
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11. starting from x 0 = 1 (as in Section 1.5, Exercise 23 ).
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12.
for x t > 1, starting from x 0 = 3 (as in Section 1.5, Exercise 26
).
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13-16 Find the equilibria of the following discrete-time dynamical
system from the graphs of their updating functions Label the coordinates
of the equilibria.
13.
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14.
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15.
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16.
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17-22 Sketch graphs of the following updating functions over the given
range and mark the equilibria. Find the equilibria algebraically if
possible.
17. f ( x ) = x 2 for 0 = x = 2.
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18.
g ( y )= y 2 - 1 for 0 = y = 2.
19-22 Graph the following discrete-time dynamical systems. Solve for the
equilibria algebraically, and identify equilibria and the regions where
the updating function lies above the diagonal on your graph.
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19. c t +1 = 0.5 c t + 8.0, for 0 = c t = 30.
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20. b t +1 = 3 b t , for 0 = b t = 10.
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21. b t +1 =0.3 b t , for 0 = b t = 10.
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22.
b t +1 = 2.0 b t - 5.0, for 0 = b t = 10.
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23-30 Find the equilibria of the following discrete-time dynamical
systems. Compare with the results of your cobweb diagram from the
earlier problem.
23. v t +1 = 1.5 v t (as in Section 1.5, Exercise 5 ).
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24. l t+1 = l t - 1.7 (as in Section 1.5, Exercise 6 ).
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25. x t+1 = 2 x t - 1 (as in Exercise 7 ).
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26. Z t +1 = 0.9 z t + 1 (as in Exercise 8 ).
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27. w t +1 = -0.5 w t + 3 (as in Exercise 9 ).
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28. x t +1 = 4 - x t (as in Exercise 10 ).
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29. (as in Exercise 11 ).
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30.
(as in Exercise 12 ).
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31-34 Find the equilibria of the following discrete-time dynamical
systems that include parameters. Identify values of the parameter for
which there is no equilibrium, for which the equilibrium is negative,
and for which there is more than one equilibrium.
31. w t +1 = aw t + 3.
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32. x t +1 = b - x t .
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33.
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34.
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35-40 Cobweb the following discrete-time dynamical systems for five steps starting from the given initial condition.
35. An alternative tree growth discrete-time dynamical system with form h t +1 = h t + 5.0 with initial condition h 0 = 10.
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36. The lizard-mite system ( Example 1.5.3 ) x t +1 = 2 x t + 30 with initial condition x 0 = 0.
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37. The model defined in Section 1.5, Exercise 37 starting from an initial volume of 1420.
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38. The model defined in Section 1.5, Exercise 38 starting from an initial mass of 13.1.
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39. The model defined in Section 1.5, Exercise 39 starting from an initial population of 800.
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40.
The model defined in Section 1.5, Exercise 40 starting from an initial
yield of 20.
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41-42 Reconsider the data describing the levels of a medication in the
blood of two patients over the course of several days (measured in mg
per liter), used in Section 1.5, Exercises 53 and 54 .
41.
For the first patient, graph the updating function and cobweb starting
from the initial condition on day 0. Find the equilibrium.
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42.
For the second patient, graph the updating function and cobweb starting
from the initial condition on day 0. Find the equilibrium.
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43-44 Cobweb and find the equilibrium of the following discrete-time
dynamical system.
43.
Consider a bacterial population that doubles every hour, but 1.0 × 10 6
individuals are removed after reproduction ( Section 1.5, Exercise 57
). Cobweb starting from b 0 = 3.0 × 10 6 bacteria.
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44.
Consider a bacterial population that doubles every hour, but 1.0 × 10 6
individuals are removed before reproduction ( Section 1.5, Exercise 58
). Cobweb starting from b 0 = 3.0 × 10 6 bacteria.
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45-46 Consider the following general models for bacterial populations
with harvest.
45.
Consider a bacterial population that doubles every hour, but h
individuals are removed after reproduction. Find the equilibrium. Does
it make sense?
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46.
Consider a bacterial population that increases by a factor of r every
hour, but 1.0 × 10 6 individuals are removed after reproduction. Find
the equilibrium. What values of r produce a positive equilibrium?
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47-48 Consider the general model M t +1 = (1 - a ) M t + S for
medication ( Example 1.6.11 ). Find the loading dose ( Example 1.6.7 )
in the following cases.
47. a = 0.2, S = 2.
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48. a = 0.8, S = 4.
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49.
Use your computer (it may have a special feature for this) to find and
graph the first 10 points on the solutions of the following
discrete-time dynamical systems. The first two describe populations with
reproduction and immigration of 100 individuals per generation, and the
last two describe populations that have 100 individuals harvested or
removed each generation. a. b t +1 = 0.5 b t + 100 starting from b 0 =
100. b. b t +1 = 1.5 b t + 100 starting from b 0 = 100. c. b t +1 = 1.5 b
t - 100 starting from b 0 = 201. d. b t +1 = 1.5 b t - 100 starting
from b 0 = 199. e. What happens if you run the last one for 15 steps?
What is wrong with the model?
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50.
Compose the medication discrete-time dynamical system M t +1 = 0.5 M t +
1.0 with itself 10 times. Plot the resulting function. Use this
composition to find the concentration after 10 days starting from
concentrations of 1.0, 5.0, and 18.0 milligrams per liter. If the goal
is to reach a stable concentration of 2.0 milligrams per liter, do you
think this is a good therapy?
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