Solutions Modeling Dynamics of Life 3ed Adler - Chapter 3.1

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. ...
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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. ...
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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. ...
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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. ...
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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. ...
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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. ...
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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. ...
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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. ...
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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. ...
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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. ...
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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).
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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).
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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.
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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?
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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?
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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.
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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.
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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.
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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?
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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.
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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).
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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).
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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
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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
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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
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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?
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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).
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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.
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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.
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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?
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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 ...
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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?
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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. ...
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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. ...
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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. ...
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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. ...
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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
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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
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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.
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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.
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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.
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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.
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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.
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3.1.44 Consider the discrete-time dynamical systems with the following discrete-time dynamical systems. Check the stability of the equilibria. ...
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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?
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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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