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Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
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 right-hand edge of the figure?Another peculiarity of an updating function that is tangent to the diagonal at an equilibrium is
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.
Move the curve slightly up (again keeping the diagonal in the same place). How many equilibria are there? Describe their stability.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
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 f(x) = x/1 + x
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 f(x) = x/1 - x
Recall the updating function with the fraction p of mutant bacteria given bywhere 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
Recall the updating function with the fraction p of mutant bacteria given bywhere 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
Recall the updating function with the fraction p of mutant bacteria given bywhere 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
In general (without substituting numerical values for s and r), can both equilibria be stable? What happens if r = s?Recall the updating function with the fraction p of mutant bacteria given bywhere s is the per capita production of the mutant and r is the per capita production of the wild type.
A population of bacteria has per capita production r = 0.6 and 1.0 × 106 bacteria are added each generation. 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
A population of bacteria has per capita production r = 0.2 and 5.0 × 106 bacteria are added each generation. 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
Suppose that r = 1.5 and h = 1.0 × 106 bacteria. A lab is growing and harvesting a culture of valuable bacteria described by the discrete-time dynamical system bt + 1 = rbt - h. The bacteria have per capita production r, and h are harvested each generation. Graph the updating function for each,
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.
Without setting r and h to particular values, find the equilibrium algebraically. When is the equilibrium stable? A lab is growing and harvesting a culture of valuable bacteria described by the discrete-time dynamical system bt + 1 = rbt - h. The bacteria have per capita production r, and h are
The nonlinear discrete-time dynamical systemThe model describing the dynamics of the concentration of medication in the bloodstream,Mt + 1 = 0.5Mt + 1.0,becomes nonlinear if the fraction of medication used is a function of the concentration. In each case, use the Slope Criterion for stability to
The nonlinear discrete-time dynamical systemHow does this differ from the model in Exercise 31? Why is the equilibrium smaller?The model describing the dynamics of the concentration of medication in the bloodstream,Mt + 1 = 0.5Mt + 1.0,becomes nonlinear if the fraction of medication used is a
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 discrete-time dynamical systems, and find the derivative at the equilibria. ct + 1 = 0.5ct + 8.0, for 0 ≤ ct ≤ 30.
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 discrete-time dynamical systems, and find the derivative at the equilibria. bt + 1 = 3bt.
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 discrete-time dynamical systems, and find the derivative at the equilibria.
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 discrete-time dynamical systems, and find the derivative at the equilibria.
Consider a population xt with per capita production ofAfter 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
Consider a population xt with per capita production ofAfter 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
Vc = 20.0 millivolts, u = 10.0 millivolts, c = 0.5.Consider the discrete-time dynamical system for a heart studied in Section 1.11.Sketch the updating function. Why is the equilibrium stable when it exists?
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.
Vc = 20.0 millivolts, u = 10.0 millivolts, c = 0.6.Consider the discrete-time dynamical system for a heart studied in Section 1.11.Sketch the updating function. Why is the equilibrium stable when it exists?
Vc = 20.0 mill volts, u = 10.0 mill volts, c = 0.7.Consider the discrete-time dynamical system for a heart studied in Section 1.11.Sketch the updating function. Why is the equilibrium stable when it exists?
Vc = 20.0 millivolts, u = 10.0 millivolts, c = 0.8.Consider the discrete-time dynamical system for a heart studied in Section 1.11.Sketch the updating function. Why is the equilibrium stable when it exists?
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 cob-webbing. ct+1 = 0.5ct + 8.0, for 0 ≤ ct ≤ 30
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 cob-webbing. bt+1 = 3bt, for 0 ≤ bt ≤ 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 cob-webbing. bt+l = 0.3bt, for 0 ≤ bt ≤ 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 cob-webbing. bt+l = 2.0bt - 5.0, for 0 ≤ bt ≤ 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 cob-webbing. f(x) = x2 for 0 ≤ x ≤ 2
The bacterial selection equation pt+1 = 1.5pt / l.5pt + 2.0(1 - pt) at the equilibrium p* = 0. Draw the tangent line approximating the given system at the specified equilibrium and compare the cobweb diagrams. Use the stability theorem to check whether the equilibrium is stable.
Consider the linear discrete-time dynamical system yt+1 = 1. 0 + m(yt - 1.0). For each of the following values of m, a. Find the equilibrium. b. Graph and cobweb. c. Compare your results with the stability condition. m = 1.5.
Consider the linear discrete-time dynamical system yt+1 = 1. 0 + m(yt - 1.0). For each of the following values of m, a. Find the equilibrium. b. Graph and cobweb. c. Compare your results with the stability condition. m = -0.5.
Consider the linear discrete-time dynamical system yt+1 = 1. 0 + m(yt - 1.0). For each of the following values of m, a. Find the equilibrium. b. Graph and cobweb. c. Compare your results with the stability condition. m = -1.5.
The following discrete-time dynamical systems have slope of exactly -1 at the equilibrium. Check this, and then iterate the function for a few steps starting from near the equilibrium to see whether it is stable, unstable, or neither. xt + 1 = 4 - xt.
The following discrete-time dynamical systems have slope of exactly -1 at the equilibrium. Check this, and then iterate the function for a few steps starting from near the equilibrium to see whether it is stable, unstable, or neither. xt + 1 = xt / xt - 1 for xt > 1.
The following discrete-time dynamical systems have slope of exactly -1 at the equilibrium. Check this, and then iterate the function for a few steps starting from near the equilibrium to see whether it is stable, unstable, or neither. xt + 1 = 3xt (1 - xt), the logistic system with r = 3.
The following discrete-time dynamical systems have slope of exactly -1 at the equilibrium. Check this, and then iterate the function for a few steps starting from near the equilibrium to see whether it is stable, unstable, or neither. xt + 1 = 2 / 1 + x2t (the equilibrium is x* = l).
The logistic dynamical system with r = 2.
The dynamical system xt + 1 = xt e1-xt.
We have studied several systems where the fraction of medication absorbed depends on the concentration of medication in the bloodstream. These take the form Mt + 1 = Mt - f(Mt)Mt + 1.0
As in Exercise 1, but at the equilibrium p* = 1. Draw the tangent line approximating the given system at the specified equilibrium and compare the cobweb diagrams. Use the stability theorem to check whether the equilibrium is stable. Exercise 1 The bacterial selection equation pt+1 = 1.5pt / l.5pt
We have studied several systems where the fraction of medication absorbed depends on the concentration of medication in the bloodstream. These take the form Mt + 1 = Mt - f(Mt)Mt + 1.0 Discuss.
We have studied several systems where the fraction of medication absorbed depends on the concentration of medication in the bloodstream. These take the form Mt + 1 = Mt - f(Mt)Mt + 1.0 Discuss in detail.
We have studied several systems where the fraction of medication absorbed depends on the concentration of medication in the bloodstream. These take the form Mt + 1 = Mt - f(Mt)Mt + 1.0 Discuss briefly.
The value of r where the positive equilibrium switches from having a positive to a negative slope. Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb.
One value of r between 1 and the value found in Exercise 47. Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb.
One value of r between the value found in Exercise 47 and r = e2. Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb.
One value of r greater than r = e2. Find values of r satisfying the following conditions for the Ricker model. In each case, graph and cobweb.
Per capita production = 0.5 + 0.5bt.The logistic model quantifies a competitive interaction, wherein per capita production is a decreasing function of population size. In some situations, per capita production is enhanced by population size. For each of the following cases where per capita growth
Per capita production = 0.5 + 0.5b2t.The logistic model quantifies a competitive interaction, wherein per capita production is a decreasing function of population size. In some situations, per capita production is enhanced by population size. For each of the following cases where per capita growth
Consider a modified version of the logistic dynamical system xt + 1 = rxt (1 - xnt). For the following values of n,a. Sketch the updating function with r = 2.b. Find the equilibria.c. Find the derivative of the updating function at the equilibria.d. For what values of r is the x = 0 equilibrium
xt + 1 = 1.5xt(1 - xt) at the equilibrium x* = 0. Draw the tangent line approximating the given system at the specified equilibrium and compare the cobweb diagrams. Use the stability theorem to check whether the equilibrium is stable.
Consider a modified version of the logistic dynamical system xt + 1 = rxt (1 - xnt). For the following values of n, a. Sketch the updating function with r = 2. b. Find the equilibria. c. Find the derivative of the updating function at the equilibria. d. For what values of r is the x = 0 equilibrium
You come in one morning and find that the temperature is 21°C. To correct this, you move the thermostat down by 1°C to 19°C. But the next day the temperature has dropped to 18°C.Expanding oscillations can result from improperly tuned feedback systems. Suppose that a thermostat is supposed to
You come in one morning and find that the temperature is 21°C. To correct this, you move the thermostat down by 1°C to 19°C. But the next day the temperature has dropped to 18.5°C.Expanding oscillations can result from improperly tuned feedback systems. Suppose that a thermostat is supposed to
The model of bacterial selection includes no frequency-dependence, meaning that the per capita production of the different types does not depend on the fraction of types in the population. Each of the following discrete-time dynamical systems for the number of mutants at and the number of wild type
The model of bacterial selection includes no frequency-dependence, meaning that the per capita production of the different types does not depend on the fraction of types in the population. Each of the following discrete-time dynamical systems for the number of mutants at and the number of wild type
If there are n seeds, each sprouts and grows to a size s = 100.0/n. An adult of size s produces s - 1.0 seeds (because it must use 1.0 units of energy to survive).Crowded plants grow to smaller size. Smaller plants make fewer seeds. The following describe the dynamics of a population described by
If there are n seeds, each sprouts and grows to a size s = 100/n. An adult of size 5 produces s - 0.5 seeds. Crowded plants grow to smaller size. Smaller plants make fewer seeds. The following describe the dynamics of a population described by n, the total number of seeds, and s, the size of the
If there are n seeds, each sprouts and grows to a size s = 100/n. Suppose that an adult of size s produces s - 2.0 seeds.Crowded plants grow to smaller size. Smaller plants make fewer seeds. The following describe the dynamics of a population described by n, the total number of seeds, and s, the
If there are n seeds, each sprouts and grows to a size s = 100/n + 5. An adult of size s produces s - 1 seeds.Crowded plants grow to smaller size. Smaller plants make fewer seeds. The following describe the dynamics of a population described by n, the total number of seeds, and s, the size of the
xt + 1 = 1.5xt(1 - xt) at the equilibrium x* = 1/3. Draw the tangent line approximating the given system at the specified equilibrium and compare the cobweb diagrams. Use the stability theorem to check whether the equilibrium is stable.
yt + 1 = 1.2yt with y0 = 2.0. When will the value exceed 100?
yt + 1 = -1.2yt with y0 = 2.0. When will the value exceed 100?
yt + 1 = 0.8yt with y0 = 2.0. When will the value be less than 0.2?
yt + 1 = -0.8yt with y0 = 2.0. When will the value be between 0.0 and 0.2?
Consider the linear discrete-time dynamical system yt+1 = 1. 0 + m(yt - 1.0). For each of the following values of m, a. Find the equilibrium. b. Graph and cobweb. c. Compare your results with the stability condition. m = 0.9.
Find all critical points of the following functions.1. a(x) = x/1 + x.2. f(x) = 1 + 2x - 2x2.3. c(w) = w3 - 3w.4. g(y) = y/1 + y2.5. h(z) = ez2.6. c(θ) = cos(2πθ).
Find the second derivative at the critical points of the following functions. Classify the critical points as minima and maxima. Use the second derivative to draw an accurate graph of the function for the given range.1. a(x) = x/1 + x for 0 ≤ x ≤ 1.2. f(x) = 1 + 2x - 2x2 for 0 ≤ x ≤ 2.3.
Suppose f(x) is a positive function with a maximum at x*. We can often find maxima and minima of other functions composed with f(x). For each of the functions h(x) = g(f(x)),a. Show that h has a critical point at x*.b. Compute the second derivative at this point.c. Check whether your function has a
Organic waste deposited in a lake at t = 0 decreases the oxygen content of the water. Suppose the oxygen content is C(t) = t3 - 30t2 + 6000 for 0 ≤ t ≤ 25. Find the maximum and minimum oxygen content during this time. Solve the above optimization problem.
The size of a population of bacteria introduced to a nutrient grows according toFind the maximum size of this population for t ‰¥ 0.Solve the above optimization problem.
A farmer owns 1000 m of fence and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and a perpendicular and perfectly straight ancient stone wall on another. The area thus needs to be fenced on only two sides. What is the largest area
A farmer owns 1000 m of fence and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. What is the largest area she can enclose?
Consider the bee faced by the problem in Subsection 3.3.2. Find the optimal strategy with the following travel times τ and illustrate the graphical method of solution. For each particular value of τ, find the equation of the tangent line at the optimal t and show that it goes through the point
Suppose that the total food collected by a bee followsF(t) = t / c + twhere c is some parameter. If τ = 1.0, find the optimal departure time in the following circumstances. Sketch the plot from the graphical method.1. c = 2.0.2. c = 1.0.3. c = 0.1.
Mathematical models can help us estimate values that are difficult to measure. Consider again a bee sucking nectar from a flower, withF(t) = t / 0.5 + t.We also measure that the bee remains a length of time t on the flower. Estimate the travel time τ assuming that the bee understands the Marginal
We never showed that the value found in computing the optimal t with the Marginal Value Theorem is in fact a maximum. For each of the following forms for the function F(t), find the second derivative of R(t) and the point where F'(t) = F(t) / t + τ and check whether the solution is a maximum.1.
Animals must survive predation in addition to maximizing their rate of food intake. One theory assumes that they try to maximize the ratio of food collected to predation risk. Suppose that different flowers with nectar of quality n (the rate of food collection) attract P(n) predators. For example,
Find the maximum harvest from a population following the discrete-time dynamical systemNt + 1 = rNt(1 - Nt) - hNtfor the given values of r.a. Find the equilibrium population as a function of h. What is the largest h consistent with a positive equilibrium?b. Find the equilibrium harvest as a
Find the conditions for stability of the equilibrium ofNt + 1 = rNt(1 - Nt) - hNtfor the following values of r. Show that the equilibrium N* is stable when h is set to the value that maximizes the long-term harvest. Graph the updating function and cobweb.1. r = 2.5, as in the text.2. r = 1.5, as in
Calculate the maximum long-term harvest for an alternative model of competition obeying the discrete-time dynamical systemTry the following steps for the given values of the parameters r and k.a. Find the, equilibrium as a function of h.b. What is the largest value of h consistent with a positive
The model of fish harvesting studied in the text includes nothing about harvesting cost. Suppose that the population followsNt + 1 = 2.5Nt(1 - Nt) - hNtas in the text, but that the payoff isP(h) = hN* - chwhere c is the cost per unit effort of harvesting. Find the optimal harvest for the following
Find the global minimum and maximum of the following functions on the interval given. Don't forget to check the endpoints.1. a(x) = x/1 + x for 0 ≤ x ≤ 1.2. f(x) = 1 + 2x - 2x2 for 0 ≤ x ≤ 2.3. c(w) = w3 - 3w for -2 ≤ w ≤ 2.4. g(y) = y/1 + y2 for 0 ≤ y ≤ 2.5. h(z) = ez2 for 0 ≤ z
Use the Intermediate Value Theorem to show that the following equations have solutions for 0 ≤ x ≤ 1.1. ex + x2 - 2 = 0.2. ex - 3x2 = 0.3. ex + x2 - 2 = x.4. ex + x2 - 2 = cos(2πx) - 1.5. xe-3(x - 1) - 2 = 0 (you will need to check an intermediate point).6. x3e-4(x - 1) - 1.1 = 0 (you will
The slope of the function f(x) = x2 must match the slope of the secant connecting x = 0 and x = 1. Find the points guaranteed by the Mean Value Theorem and sketch the associated graph.
The slope of the function f(x) = x2 must match the slope of the secant connecting x = 0 and x = 2. Find the points guaranteed by the Mean Value Theorem and sketch the associated graph.
The slope of the function g(x) = √x must match the slope of the secant connecting x = 0 and x = 1. Find the points guaranteed by the Mean Value Theorem and sketch the associated graph.
The slope of the function g(x) = √x must match the slope of the secant connecting x = 0 and x = 2. Find the points guaranteed by the Mean Value Theorem and sketch the associated graph.
A function with a global minimum and global maximum between the endpoints. Draw functions with the above properties.
A function with a global maximum at the left endpoint and global minimum between the endpoints. Draw functions with the above properties.
A differentiable function with a global maximum at the left endpoint, a global minimum at the right endpoint, and no critical points. Draw functions with the above properties.
A function with a global maximum at the left endpoint, a global minimum at the right endpoint, and at least one critical point. Draw functions with the above properties.
A function with a global minimum and global maximum between the endpoints, but no critical points. Draw functions with the above properties.
A function that never reaches a global maximum. Draw functions with the above properties.
Consider the Heaviside function defined by
Consider the absolute value function g(x) = |x|. Does this satisfy the conditions for the Intermediate Value Theorem? Show that there is no tangent that matches the slope of the secant connecting x = -1 and x = 2. Check whether the Intermediate Value Theorem and Mean Value Theorem fail in the above
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