Solutions Modeling Dynamics of Life 3ed Adler - Chapter 5.1

5.1.1 Identify the following as pure-time, autonomous, or nonautonomous differential equations. In each case, identify the state variable. ...
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5.1.3 Identify the following as pure-time, autonomous, or nonautonomous differential equations. In each case, identify the state variable. ...
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5.1.3 Identify the following as pure-time, autonomous, or nonautonomous differential equations. In each case, identify the state variable. ...
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5.1.4 Identify the following as pure-time, autonomous, or nonautonomous differential equations. In each case, identify the state variable. ...
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5.1.5 For the given time, value of the state variable, and values of the parameters, say whether the state variable is increasing, decreasing, or remaining unchanged. t =0, F =1, and k =1 in the differential equation in Exercise 1. Reference Exercise 1. ...
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5.1.6 For the given time, value of the state variable, and values of the parameters, say whether the state variable is increasing, decreasing, or remaining unchanged. t =0, x =1, and λ=2 in the differential equation in Exercise 2. Reference Exercise 2. ...
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5.1.8 For the given time, value of the state variable, and values of the parameters, say whether the state variable is increasing, decreasing, or remaining unchanged. t = 2, m = 0, α = 2, and λ = 1 in the differential equation in Exercise 4. Reference Exercise 4. ...
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5.1.8 For the given time, value of the state variable, and values of the parameters, say whether the state variable is increasing, decreasing, or remaining unchanged. t = 2, m = 0, α = 2, and λ = 1 in the differential equation in Exercise 4. Reference Exercise 4. ...
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5.1.10 Solve the pure-time differential equation starting from the initial condition p(1)=1, find p(2), and add the curve to your graph.
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5.1.10 Solve the pure-time differential equation starting from the initial condition p(1)=1, find p(2), and add the curve to your graph.
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5.1.11 Check that the solution of the autonomous differential equation starting from the initial condition b(0)=1 is b(t)= .... Find b(1) and sketch the solution.
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5.1.12 Check that the solution of the autonomous differential equation starting from the initial condition b(1) = 1 is b(t) = .... Find b(2) and add to the sketch of the solution.
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5.1.13 Exercises 9 and 10 give the value of p one time unit after it took on the value 1. Why don’t the two answers match? (This behavior is typical of pure-time differential equations.)Exercises 9 Use integration to solve the pure-time differential equation starting from the initial condition p(0)= 1, find p(1),and sketch the solution. Exercises 10Solve the pure-time differential equation starting from the initial condition p(1)=1, find p(2), and add the curve to your graph.
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5.1.14 Exercises 11 and 12 give the value of b one time unit after it took on the value 1. Why do the two answers match? (This behavior is typical of autonomous differential equations.)Exercises 11 Check that the solution of the autonomous differential equation starting from the initial condition b(0)=1 is b(t)= .... Find b(1) and sketch the solution.Exercises 12 Check that the solution of the autonomous differential equation starting from the initial condition b(1) = 1 is b(t) = .... Find b(2) and add to the sketch of the solution.
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5.1.15 Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. Check that ...is a solution of the differential equation ...= 1 + 2x with initial condition x(0) = 1.
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5.1.16 Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. Check that ...`is a solution of the differential equation ...with initial condition b(0) = 10.
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5.1.17 Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. Check that G(t) = 1 + ... is a solution of the differential equation ...= G − 1 with initial condition G(0) = 2.
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5.1.18 Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. check that ...is a solution of the differential equation ...with initial condition z(0) = 2.
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5.1.19 Use Euler’s method to estimate the solution of the differential equation at the given time, and compare with the value given by the exact solution. Sketch a graph of the solution along with the lines predicted by Euler’s method. Estimate x(2) if x obeys the differential equation ...= 1 + 2x with initial condition x(0)= 1. Use Euler’s method with Δt = 1 for two steps. Compare with the exact answer in Exercise 15. Reference Euler’s method ... ...
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5.1.20 Use Euler’s method to estimate the solution of the differential equation at the given time, and compare with the value given by the exact solution. Sketch a graph of the solution along with the lines predicted by Euler’s method. Estimate b(1.0) if b obeys the differential equation ...= 3b with initial condition b(0)=10. Use Euler’s method with Δt = 0.5 for two steps. Compare with the exact answer in Exercise 16. Reference Euler’s method ... ...
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5.1.21 Use Euler’s method to estimate the solution of the differential equation at the given time, and compare with the value given by the exact solution. Sketch a graph of the solution along with the lines predicted by Euler’s method. Estimate G(1.0) if G obeys the differential equation ...= G − 1 with initial condition G(0)= 2. Use Euler’s method with Δt = 0.2 for five steps. Compare with the exact answer in Exercise 17. Reference Euler’s method ... ...
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5.1.22 Use Euler’s method to estimate the solution of the differential equation at the given time, and compare with the value given by the exact solution. Sketch a graph of the solution along with the lines predicted by Euler’s method. Estimate z(4.0) if z obeys the differential equation ...with initial condition z(0)=2. Use Euler’s method with Δt = 1.0 for four steps. Compare with the exact answer in Exercise 18. Reference Euler’s method ... ...
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5.1.23 The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the following cases, use the chain rule to derive a new differential equation. Suppose ...= 2x − 1. Set y = 2x − 1 and find a differential equation for y. The end result should be simpler than the original equation.
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5.1.24 The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the following cases, use the chain rule to derive a new differential equation. Suppose ...= 4b + 2. Set z = 4b + 2 and find a differential equation for z. The end result should be simpler than the original equation.
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5.1.25 The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the following cases, use the chain rule to derive a new differential equation. Suppose ...and find a differential equation for y. This transformation changes a nonlinear differential equation for x into a linear differential equation for y (this is called a Bernoulli differential equation).
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5.1.26 The derivation of the differential equation for p in the text requires combining two differential equations for a and b. Often, one can find a differential equation for a new variable derived from a single equation. In the following cases, use the chain rule to derive a new differential equation. Suppose ...and find a differential equation for y. This is another example of a Bernoulli differential equation.
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5.1.27 The simple model of bacterial growth assumes that per capita production does not depend on population size. The following problems help you derive models of the form ... where the per capita production λ is a function of the population size b. 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 per capita production has a maximum at λ(0) = 1 and that it decreases with a slope of −0.002. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 10? Is b(t) increasing when b = 1000?
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5.1.28 The simple model of bacterial growth assumes that per capita production does not depend on population size. The following problems help you derive models of the form ... where the per capita production λ is a function of the population size b. Suppose that per capita production decreases linearly from a maximum of λ(0) = 4 with slope −0.001. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 1000? Is b(t) increasing when b = 5000?
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5.1.29 The simple model of bacterial growth assumes that per capita production does not depend on population size. The following problems help you derive models of the form ... where the per capita production λ is a function of the population size b. In some circumstances, individuals reproduce better when the population size is large, and fail to reproduce when the population size is small (the Allee effect introduced in Exercise 46). Suppose that per capita production is an increasing linear function with λ(0)=−2 and a slope of 0.01. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 100? Is b(t) increasing when b = 300?
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5.1.30 The simple model of bacterial growth assumes that per capita production does not depend on population size. The following problems help you derive models of the form ... where the per capita production λ is a function of the population size b. Suppose that per capita production increases linearly with λ(0)=−5 and a slope of 0.001. Find λ(b) and the differential equation for b. Is b(t) increasing when b = 1000? Is b(t) increasing when b = 3000?
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5.1.31 The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might depend on the concentration itsel
f. In each of the following cases, draw a diagram illustrating the situation and write the associated differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose that no chemical re-enters the cell. This should look like the differential equation for a population. What would be happening to the concentration?
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5.1.32 The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might depend on the concentration itsel
f. In each of the following cases, draw a diagram illustrating the situation and write the associated differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose that no chemical leaves the cell. What would happen to the concentration?
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5.1.33 The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might depend on the concentration itsel
f. In each of the following cases, draw a diagram illustrating the situation and write the associated differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose that the constant of proportionality governing the rate at which chemical enters the cell is three times as large as the constant governing the rate at which it leaves. Would the concentration inside the cell be increasing or decreasing if C = Γ? What would this mean for the cell?
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5.1.34 The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might depend on the concentration itsel
f. In each of the following cases, draw a diagram illustrating the situation and write the associated differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose that the constant of proportionality governing the rate at which chemical enters the cell is half as large as the constant governing the rate at which it leaves. Would the concentration inside the cell be increasing or decreasing if C = Γ? What would this mean for the cell?
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5.1.35 The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might depend on the concentration itsel
f. In each of the following cases, draw a diagram illustrating the situation and write the associated differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose that the constant of proportionality governing the rate at which chemical enters the cell is proportional to 1 + C (because the chemical helps to open special channels). Would the concentration inside the cell be increasing or decreasing if C = Γ? What would this mean for the cell?
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5.1.36 The derivation of the movement of chemical assumed that chemical moved as easily into the cell as out of it. If the membrane can act as a filter, the rates at which chemical enters and leaves might differ, or might depend on the concentration itsel
f. In each of the following cases, draw a diagram illustrating the situation and write the associated differential equation. Let C be the concentration inside the cell, Γ the concentration outside, and β the constant of proportionality relating the concentration and the rate. Suppose that the constant of proportionality governing the rate at which chemical enters the cell is proportional to 1 – C (because the chemical helps to close special channels). Would the concentration inside the cell be increasing or decreasing if C = Γ? What would this mean for the cell?
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5.1.37 The model of selection includes no interaction between bacterial types a and b (the per capita production of each type is a constant). Write a pair of differential equations for a and b with the following forms for the per capita production, and derive an equation for the fraction p of type a. Assume that the basic per capita production for type a is μ= 2, and that for type b is λ = 1.5. The per capita production of each type is reduced by a factor of 1 − p by a factor of 1 − p, so that the per capita production of type a is 2(1 − p). This is a case where a large proportion of type a reduces the production of both types. Will type a take over?
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5.1.38 The model of selection includes no interaction between bacterial types a and b (the per capita production of each type is a constant). Write a pair of differential equations for a and b with the following forms for the per capita production, and derive an equation for the fraction p of type a. Assume that the basic per capita production for type a is μ= 2, and that for type b is λ = 1.5. The per capita production of type a is reduced by a factor of 1 − p and the per capita production of type b is reduced by a factor of p. This is a case where a large proportion of type a reduces the production of type a, and a large proportion of type b reduces the production of type b. Do you think that type a will still take over?
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5.1.39 We will find later (with separation of variables) that the solution for Newton’s law of cooling with initial condition H(0) is ... For each set of given parameter values,
a. Write and check the solution.
b. Find the temperature at t = 1 and t = 2.
c. Sketch of graph of your solution. What happens as t approaches infinity? Set α = 0.2/min, A = 10°C, and H(0) = 40.
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5.1.40 We will find later (with separation of variables) that the solution for Newton’s law of cooling with initial condition H(0) is ... For each set of given parameter values,
a. Write and check the solution.
b. Find the temperature at t = 1 and t = 2.
c. Sketch of graph of your solution. What happens as t approaches infinity? Set α = 0.02/min, A = 30°C, and H(0) = 40.
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5.1.41 Use Euler’s method to estimate the temperature for the following cases of Newton’s law of cooling. Compare with the exact answer. α = 0.2/min and A = 10?C and H(0)= 40. Estimate H(1) and H(2) using Δt = 1. Compare with Exercise 39. Reference Euler’s method ... ...
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5.1.42 Use Euler’s method to estimate the temperature for the following cases of Newton’s law of cooling. Compare with the exact answer. α = 0.02/min and A = 30?C and H(0) = 40. Estimate H(1) and H(2) using Δt = 1. Compare with Exercise 40. Why is the result so close? Reference Euler’s method ... ...
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5.1.43 Use the solution for Newton’s law of cooling (Exercises 39 and 40) to find the solution expressing the concentration of chemical inside a cell as a function of time in the following examples. Find the concentration after 10 seconds, 20 seconds, and 60 seconds. Sketch your solutions for the first minute. ...
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5.1.44 Use the solution for Newton’s law of cooling (Exercises 39 and 40) to find the solution expressing the concentration of chemical inside a cell as a function of time in the following examples. Find the concentration after 10 seconds, 20 seconds, and 60 seconds. Sketch your solutions for the first minute. ...
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5.1.45 Recall that the solution of the discrete-time dynamical system .... This is closely related to the differential equation .... For what values of ... and r does this solution match b(t)= ... (the solution of the differential equation with λ = 2 and b(0) = 1.0 ×...) for all values of t?
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5.1.46 Recall that the solution of the discrete-time dynamical system .... This is closely related to the differential equation .... For what values of ...and r does this solution match b(t)= ... (the solution of the differential equation with λ=−3 and b(0) = 100) for all values of t?
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5.1.47 Recall that the solution of the discrete-time dynamical system .... This is closely related to the differential equation .... For what values of λ do solutions of the differential equation grow? For what values of r do solutions of the discrete-time dynamical system grow?
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5.1.48 Recall that the solution of the discrete-time dynamical system .... This is closely related to the differential equation .... What is the relation between r and λ? That is, what value of r produces the same growth as a given value of λ?
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5.1.49 Use Euler’s method to estimate the value of p(t) from the selection differential equation (Equation 5.1.4) for the given parameter values. Compare with the exact answer using the equation for the solution. Graph the solution, including the estimates from Euler’s method. Suppose μ= 2.0/h, λ= 1.0/h, and p(0)= 0.1. Estimate the proportion after 2 hours using a time step of Δt = 0.5. Reference Euler’s method ... ...
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5.1.50 Use Euler’s method to estimate the value of p(t) from the selection differential equation (Equation 5.1.4) for the given parameter values. Compare with the exact answer using the equation for the solution. Graph the solution, including the estimates from Euler’s method. Suppose μ= 2.5/h, λ= 3.0/h, and p(0)= 0.6. Estimate the proportion after 1 hour using a time step of Δt = 0.25. Reference Euler’s method ... ...
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5.1.51 Suppose that an endothermic (warm-blooded) animal generates heat at a rate proportional to its metabolic rate with constant ..., and loses heat at a rate proportional to its surface area with constant .... Heat loss is also proportional to the difference between its temperature and the ambient temperature A as in Newton’s Law of Cooling. The temperature T of a conveniently spherical creature of radius r would then follow ... Find the equilibrium temperature as a function of r . For the same values of ..., which animals stay warmest?
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5.1.52 Suppose that an endothermic (warm-blooded) animal generates heat at a rate proportional to its metabolic rate with constant ..., and loses heat at a rate proportional to its surface area with constant .... Heat loss is also proportional to the difference between its temperature and the ambient temperature A as in Newton’s Law of Cooling. The temperature T of a conveniently spherical creature of radius r would then follow Suppose that ... = 1.0 and that A=−20.0°C. Find the value of ... required to maintain an equilibrium temperature of 40.0°C when r =1.0 and when r =2.0. Which organism needs to generate less heat to maintain its temperature?
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5.1.53 Suppose that the ambient temperature oscillates with period T according to ... The differential equation is ... which has a solution with A(0) = 20.0 of ...
a. Use a computer algebra system to check that this answer works.
b. Plot H(t) for five periods using values of T ranging from 0.1 to 10.0 when α =1.0. Compare H(t) with A(t). When does the temperature of the object most closely track the ambient temperature?
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5.1.54 Apply Euler’s method to solve the differential equation ... with the initial condition b(0) = 1.0. Compare with the sum of the solutions of ...Do you think there is a sum rule for differential equations? If your computer has a method for solving differential equations, find the solution and compare with your approximate solution from Euler’s method.
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