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mathematics
calculus
Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
Suppose that types a and b interact with a fixed population of 500 of the other (set b = 500 in the differential equation for a).If each state variable in a system of autonomous differential equations does not respond to changes in the value of the other, but depends only on a constant value, the
Start from a = 750 and b = 500. Take two steps, with a step length of Δt = 0.1.Apply Euler's method to the competition equationsstarting from the given initial conditions. Assume that μ = 2.0, λ = 2.0, Ka = 1000, and Kb = 500.
Start from a = 250 and b = 500. Take two steps, with a step length of Δt = 0.1.Apply Euler's method to the competition equationsstarting from the given initial conditions. Assume that μ = 2.0, λ = 2.0, Ka = 1000, and Kb = 500.
Start from a = 750 and b = 500. Take four steps, with a step length of Δt = 0.05. How do your results compare with those in Exercise 7?Apply Euler's method to the competition equationsstarting from the given initial conditions. Assume that μ = 2.0, λ = 2.0, Ka
Finding equilibria of coupled differential equations requires solving simultaneous equations. The following are linear equations, where the only possibilities are no solutions, one solution, or a whole line of solutions. For each of the following pairs, solve each equation for y in terms of x, set
Finding equilibria of nonlinear coupled differential equations requires solving nonlinear simultaneous equations that can have any number of solutions. For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both
Graph the null clines in the phase plane and find the equilibria of the following.Predator-prey model(Equation 5.5.1) with λ = 1.0, δ = 3.0, ˆˆ = 0.002, η = 0.005.
Graph the null clines in the phase plane and find the equilibria of the following. Predator-prey model with λ = 1.0, δ = 3.0, ∈ = 0.005, η = 0.002.
Graph the null clines in the phase plane and find the equilibria of the following.Newton's law of coolingwith α = 0.01 and α2 = 0.1.
Graph the null clines in the phase plane and find the equilibria of the following. Newton's law of cooling with α = 0.01 and α2 = 0.5.
Graph the null clines in the phase plane and find the equilibria of the following.Competition model(Equation 5.5.4) with λ = 2.0, μ = 1.0, Ka = 106, Kb = 107.
Graph the null clines in the phase plane and find the equilibria of the following. Competition model with λ = 1.0, μ = 2.0, Ka = 106, Kb = 107. How do the results compare with those in Exercise 15? Why?
The equations in Exercise 11.Redraw the phase planes for the above problem, but make the other choice for the vertical variable. Check that you get the same equilibrium.Exercise 11Predator-prey model(Equation 5.5.1) with λ = 1.0, δ = 3.0, ˆˆ = 0.002, η = 0.005.
The equations in Exercise 12. Redraw the phase planes for the above problem, but make the other choice for the vertical variable. Check that you get the same equilibrium. Exercise 12 Predator-prey model with λ = 1.0, δ = 3.0, ∈ = 0.005, η = 0.002.
The equations in Exercise 15.Redraw the phase planes for the above problem, but make the other choice for the vertical variable. Check that you get the same equilibrium.Exercise 15Competition model(Equation 5.5.4) with λ = 2.0, μ = 1.0, Ka = 106, Kb = 107.
Finding equilibria of coupled differential equations requires solving simultaneous equations. The following are linear equations, where the only possibilities are no solutions, one solution, or a whole line of solutions. For each of the following pairs, solve each equation for y in terms of x, set
The equations in Exercise 16. Redraw the phase planes for the above problem, but make the other choice for the vertical variable. Check that you get the same equilibrium. Exercise 16 Competition model with λ = 1.0, μ = 2.0, Ka = 106, Kb = 107.
If each state variable in a system of autonomous differential equations does not respond to changes in the value of the other, but depends only on a constant value, the two equations can be considered separately. In this case, the phase plane is particularly simple. Find the null clines and
Find the null clines and equilibria for the following predator-prey models.1. The model in Section 5.5, Exercise 23.2. The model in Section 5.5, Exercise 24.3. The model in Section 5.5, Exercise 25.4. The model in Section 5.5, Exercise 26.
Find and graph the null clines, and find the equilibria for the following models.1. The model found in Section 5.5, Exercise 27.2. The model found in Section 5.5, Exercise 28.3. The model found in Section 5.5, Exercise 29.4. The model found in Section 5.5, Exercise 30.
Finding equilibria of coupled differential equations requires solving simultaneous equations. The following are linear equations, where the only possibilities are no solutions, one solution, or a whole line of solutions. For each of the following pairs, solve each equation for y in terms of x, set
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.The models of diffusion derived in Section 5.5, Exercises 31 and 32 assume that the membrane between the vessels is equally permeable in
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. The models of diffusion derived in Section 5.5, Exercises 31 and 32 assume that the membrane between the vessels is equally permeable in both
Suppose that individuals of type b reduce the per capita growth rate of type a by half as much as individuals of type a, and that individuals of type a reduce the per capita type b growth rate by twice as much as individuals of type b. The equations areIn our model of competition, the per capita
Suppose that individuals of type b reduce the per capita type a growth rate by half as much as individuals of type a, and that individuals of type a reduce the per capita type b growth rate by half as much as individuals of type b. The equations are(There should be four equilibria.)In our model of
The model in Section 5.5, Exercise 35. Find the null clines and equilibria of this model when α = 2.0, μ = 1.0, and k = 1.0. Draw the null clines and find equilibria of the above extensions of the basic disease model.
The model in Section 5.5, Exercise 36. Find the null clines and equilibria of this model when α = 2.0 and μ = 1.0. Draw the null clines and find equilibria of the above extensions of the basic disease model.
The model in Section 5.5, Exercise 37. Find the null clines and equilibria of this model when α = 2.0, μ = 1.0, and k = 0.5. Draw the null clines and find equilibria of the above extensions of the basic disease model.
The model in Section 5.5, Exercise 38. Find the null clines and equilibria of this model when α = 2.0, μ = 1.0, and k = 4.0. Draw the null clines and find equilibria of the above extensions of the basic disease model.
The model in Section 5.5, Exercise 39. Find the null clines and equilibria of this model when α = 2.0, μ = 1.0, and b = 2.0. Draw the null clines and find equilibria of the above extensions of the basic disease model.
Finding equilibria of coupled differential equations requires solving simultaneous equations. The following are linear equations, where the only possibilities are no solutions, one solution, or a whole line of solutions. For each of the following pairs, solve each equation for y in terms of x, set
The model in Section 5.5, Exercise 40. Find the null clines and equilibria of this model when α = 2.0, μ = 1.0, and b = 1.0. Draw the null clines and find equilibria of the above extensions of the basic disease model.
Finding equilibria of nonlinear coupled differential equations requires solving nonlinear simultaneous equations that can have any number of solutions. For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both
Finding equilibria of nonlinear coupled differential equations requires solving nonlinear simultaneous equations that can have any number of solutions. For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both
Finding equilibria of nonlinear coupled differential equations requires solving nonlinear simultaneous equations that can have any number of solutions. For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both
Finding equilibria of nonlinear coupled differential equations requires solving nonlinear simultaneous equations that can have any number of solutions. For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both
Finding equilibria of nonlinear coupled differential equations requires solving nonlinear simultaneous equations that can have any number of solutions. For each of the following pairs, solve each equation for y in terms of x, set the two equations for x equal, and solve for x. Check that both
Suppose the following functions are solutions of some differential equation. Graph these as functions of time and as phase-plane trajectories for 0 ≤ t ≤ 2. Mark the position at t = 0, t = 1, and t = 2. x(t) = t, y(t) = 3t.
On the following phase-plane diagrams, use the direction arrows to sketch phase-plane trajectories starting from two different initial conditions.
The diagram in Exercise 9.Use the information in the phase-plane diagram to draw direction arrows on the null clines.
The diagram in Exercise 10.Use the information in the phase-plane diagram to draw direction arrows on the null clines.
Start from a = 750 and b = 500. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 7.
Start from a = 250 and b = 500. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 8.
Suppose α = 0.3 and α2 = 0.1. Start from H = 60 and A = 20. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 11.
Suppose α = 0.3 and α2 = 0.1. Start from H = 0 and A = 20. Take two steps, with a step length of Δt = 0.1, as in Section 5.5, Exercise 12.
Suppose α = 3.0 and α2 = 1.0. Start from H = 60 and A = 20. Take two steps, with a step length of Δt = 0.25, as in Section 5.5, Exercise 13. Does this diagram help explain what went wrong?
Suppose α = 3.0 and α2 = 1.0. Start from H = 0 and A = 20. Take two steps, with a step length of Δt = 0.5, as in Section 5.5, Exercise 14. Does this diagram help explain what went wrong?
Draw the null clines and direction arrows for the following models of springs. Make sure to include positive and negative values for the position x and the velocity v.1. The model in Section 5.5, Exercise 15.2. The model in Section 5.5, Exercise 16.
Suppose the following functions are solutions of some differential equation. Graph these as functions of time and as phase-plane trajectories for 0 ≤ t ≤ 2. Mark the position at t = 0, t = 1, and t = 2. a(t) = 2e-t, b(t) = e-2t.
The solution x(t) = cos(t) (Section 5.5, Exercise 17) of the spring equation in Section 5.5. Exercise 15. Sketch the given solution of the above models of springs first as a pair of functions of time, and then in the phase-plane. Check that the solution follows the arrows.
The solution x(t) = e-t cos(t) (Section 5.5, Exercise 18) of the spring equation with friction in Section 5.5, Exercise 16. Sketch the given solution of the above models of springs first as a pair of functions of time, and then in the phase-plane. Check that the solution follows the arrows.
The model in Section 5.5, Exercise 23. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 24. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 27. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 28. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 29. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 30. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 33. For the above problem, add direction arrows to the phase-plane.
Suppose the following functions are solutions of some differential equation. Graph these as functions of time and as phase-plane trajectories for 0 ≤ t ≤ 2. Mark the position at t = 0, t = 1, and t = 2. f(t) = 1 + t, g(t) = e-t.
The model in Section 5.5, Exercise 34. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 35. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 36. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 37. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 38. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 39. For the above problem, add direction arrows to the phase-plane.
The model in Section 5.5, Exercise 40. For the above problem, add direction arrows to the phase-plane.
The model in Exercise 25 starting from (1500, 200). Is there another path for the solution that is consistent with the direction arrows? For the above problem, use the direction arrows on your phase-plane to sketch a solution starting from the given initial condition.
The model in Exercise 26 starting from (1500, 200).
The model in Exercise 27 starting from (200, 300).
Suppose the following functions are solutions of some differential equation. Graph these as functions of time and as phase-plane trajectories for 0 ≤ t ≤ 2. Mark the position at t = 0, t = 1, and t = 2. x(t) = 1 + t (t - 2), y(t) = t(3 - t).
The model in Exercise 28 starting from (200, 300).
The model in Exercise 31 starting from (0.5, 1).
The model in Exercise 32 starting from (0.5, 1).
The model in Exercise 35 starting from (0.5, 1).
The model in Exercise 36 starting from (0.5, 0.5). Can you be sure that the solution behaves exactly like your picture? For the above problem, use the direction arrows on your phase-plane to sketch a solution starting from the given initial condition.
From the following graphs of solutions of differential equations as functions of time, graph the matching phase-plane trajectory.
From the following graphs of solutions of differential equations as functions of time, graph the matching phase-plane trajectory.
From the following graphs of phase-plane trajectories, graph the matching solutions of. differential equations as functions of time.
From the following graphs of phase-plane trajectories, graph the matching solutions of. Differential equations as functions of time.
On the following phase-plane diagrams, use the direction arrows to sketch phase-plane trajectories starting from two different initial conditions.
Give an equation like Equation 5.8.2 describing a system with two thresholds. The system has a stable equilibrium at a resting potential at 0, but will be pushed to a higher positive equilibrium if the potential is raised above a particular positive threshold, and to a negative equilibrium if the
What happens to the phase plane and the cell if the applied current is negative? Can the cell lose its ability to respond if the applied current is negative and large?
Draw phase-line diagrams for Equation 5.8.2 in the following cases. a. a = 0.01. How might this neuron malfunction? b. a = 0.5. c. a = 0.99. Why might this neuron work poorly?
Use the method of separation of variables to solve(Equation 5.8.3) assuming that v is constant. Show how the dynamics are slowed when ˆˆ is small, but that ˆˆ does not affect the equilibrium.
Assuming that w is constant (perhaps the potassium channels are jammed in a particular state) and a = 0.3, figure out the dynamics of(Equation 5.8.4) thought of as a one-dimensional differential equation. Try it with w = 0.01, w = 0.05, and w = 0.1.
Draw direction arrows on the null clines for the Fitzhugh-Nagumo equations.
Sketch the phase-plane trajectory and solutions of the Fitzhugh-Nagumo equations when the initial stimulus is less than the threshold.
Sketch the phase-plane diagram and solution for the Fitzhugh-Nagumo equations for the following values of ∈. (Think of changing ∈ as changing the direction arrows: the arrows are nearly horizontal when ∈ is small because w, the vertical variable, changes slowly.) a. ∈ very small. b. ∈
With positive applied current, there could be three intersections of the null clines. Draw a phase-plane diagram illustrating this scenario, and take a guess at the dynamics. Try to make sense of the results biologically.
There could also be a single intersection with positive applied current, but on the rightmost decreasing part of the v-null cline. Draw such a phase plane. Assuming that this equilibrium is stable, sketch the dynamics. Why might this cell also be thought of as excitable?
Consider the differential equationdC/dt = 3(Г - C) + 1where C is the concentration of some chemical in a cell, measured in moles per liter, and Г is a constant.a. What kind of differential equation is this? Explain the terms in the equation.b. Draw the phase-line diagram.c. Verify the stability
Consider again the differential equationdC/dt = 3(Г – C) + 1where Г is a constant.a. Solve the equation when C(0) = 0.b. Check your answer.c. Find C(0.4).d. After what time will the solution be within 5% of its limit?
Solve the pure-time differential equation starting from the initial condition p(1) = 1, find p(2), and add the curve to your graph. The above exercise compare the behavior of two similar-looking differential equations, the pure-time differential equation dp/dt = t and the autonomous differential
Check that the solution of the autonomous differential equation starting from the initial condition b(0) = 1 is b(t) = et-1. Find b(2) and sketch the solution. The above exercise compare the behavior of two similar-looking differential equations, the pure-time differential equation dp/dt = t and
Check that the solution of the autonomous differential equation starting from the initial condition b(1) = 1 is b(t) = et-1. Find b(2) and add to the sketch of the solution. The above exercise compare the behavior of two similar-looking differential equations, the pure-time differential equation
Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. Check that x(t) = -1/2 + 3/2e2t is a solution of the differential equation dx/dt = 1 + 2x with initial condition x (0) = 1.
Check the solution of the autonomous differential equation, making sure that it also matches the initial condition.Check that b(t) = 10e3t is a solution of the differential equation db/dt = 3b with initial condition b(0) = 10.
Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. Check that G (t) = 1 + et is a solution of the differential equation dG/dt = G - 1 with initial condition G(0) = 2.
Check the solution of the autonomous differential equation, making sure that it also matches the initial condition. Check that z(t) = 1 + √1 + 2t is a solution of the differential equation dz/dt = 1/z - 1 with initial condition z(0) = 2.
Estimate x(2) if x obeys the differential equation dx/dt = 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.
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