Chapter 11 Differential Equations 11.1 Use separation of variables to solve the following differential equations with given initial conditions. (a) = 2ty, y(0) = 10 (b) = y(1 y), y(0) = 0.5, (Hint: 1 y(y 1) = 1 y 1 1 y ). 11.2 Consider the differential equation and initial condition = 1 y, y(0) = 0.5. (a) Solve for y as a function of t using separation of variables. (b) Use the initial condition to find the value of C. 11.3 Consider the differential equation and initial condition = 1 + y2, y(0) = 0.5. Repeat the process described in problem 11.2 and find the solution, i.e. determine the function y(t) that satisfies this differential equation. v.2005.1 - December 16, 2008 1
11.4 Consider the differential equation and initial condition = 1 y2, y(0) = 0.5. Solve using separation of variables (Hint: use partial fractions.) 11.5 Solve the following differential equation and initial condition, assuming that k > 0 is a constant: 11.6 = ky2/3 y(0) = y 0 A certain cylindrical water tank has a hole in the bottom, out of which water flows. The height of water in the tank, h(t), can be described by the differential equation dh = k h where k is a positive constant. If the height of the water is initially h 0, determine how much time elapses before the tank is empty. 11.7 The position of a particle is described by dx = 1 x 2 0 t π 2 (a) Find the position as a function of time given that the partical starts at x = 0 initially. (b) Where is the particle when t = π 2? (c) At which position(s) is the particle moving the fastest? The slowest? 11.8 In an experiment involving yeast cells, it was determined that mortality of the cells increased at a linear rate, i.e. that at time t after the beginning of the experiment the mortality rate was m(t) = m 0 + rt. v.2005.1 - December 16, 2008 2
The experiment was started with N o cells at time t = 0. Let N(t) represent the population size of the cells at time t. (a) If no birth occurs, then dn(t)/ = m(t)n(t). Solve this differential equation by separation of variables, i.e find N(t) as a function of time. (b) Suppose cells are born at a constant rate b, so that the differential equation is dn(t)/ = m(t)n(t) + bn(t). Determine how this affects the population size at time t. 11.9 Muscle cells are known to be powered by filaments of the protein actin which slide past one another. The filaments are moved by cross-bridges of myosin, that act like little motors, which attach, pull the filaments, and then detach. Let n be the fraction of myosin cross-bridges that are attached at time t. A model for cross-bridge attachment is: dn = k 1(1 n) k 2 n, n(0) = n 0 where k 1 > 0, k 2 > 0 are constants. (a) Solve this differential equation and determine the fraction of attached cross-bridges n(t) as a function of time t. (b) What value does n(t) approach after a long time? (c) Suppose k 1 = 1.0 and k 2 = 0.2. Starting from n(0) = 0, how long does it take for 50% of the cross-bridges to become attached? 11.10 The velocity of an object falling under the effect of gravity with air resistance is given by: dv = f(v) = g kv, v(0) = v 0, where g > 0 is acceleration due to gravity and k > 0 is a frictional coefficient (both constant). (a) Sketch the function f(v) as a function of v. Identify a value of v for which no change occurs, i.e., for which f(v) = 0. This is called a stea state value of the velocity or a fixed point. Interpret what this value represents. (b) Explain what happens if the initial velocity is larger or smaller than this stea state value. Will the velocity increase or decrease? (Recall that the sign of the derivative dv/ tells us whether v(t) is an increasing or decreasing function of t. (c) Use separation of variables to find the function v(t). v.2005.1 - December 16, 2008 3
11.11 A model for the velocity of a sky diver (slightly different from the model we have alrea studied) is dv = 9 v2 (a) What is this skydiver s terminal velocity ; that is, near what velocity will the sky diver eventually stabilize? (b) Starting from rest, how long will it take to reach half of this velocity? 11.12 Newton s Law of Cooling Consider the differential equation for Newton s Law of Cooling dt = k(t E) and the initial condition: T(0) = T 0. Solve this differential equation by the method of separation of variables i.e. find T(t). Interpret your result. 11.13 Let V (t) be the volume of a spherical cell that is expanding by absorbing water from its surface area. Suppose that the rate of increase of volume is simply proportional to the surface area of the sphere, and that, initially, the volume is V 0. Find a differential equation that describes the way that V changes. Use the connection between surface area and volume in a sphere to rewrite your differential equation in terms of the volume alone. You should get an equation in the form dv = kv 2/3, where k is some constant. Solve the differential equation to show how the volume changes as a function of the time. (Hint: recall that for a sphere, V = (4/3)πr 3, S = 4πr 2.) 11.14 In problems 11.2, 11.3, and 11.4 above, your method was to calculate a formula for the unknown function y(t). Now you will use qualitative methods for these same problems instead: Sketch a graph of the expression / versus y in each case. (E.g., in problem 11.2 you will be sketching the straight line f(y) = 1 y). In each case, determine the values of y at which / = 0 (stea states, or critical points). v.2005.1 - December 16, 2008 4
11.15 Find the fixed points (stea states, critical points) of the following differential equations and classify their stability. (a) (b) (c) (d) = y2 4 = 4 y2 = y(y 1)(y + 1) = cos(2πy) 11.16 For each part of problem 11.15, sketch the qualitative behavior, i.e. the flows along the y axis. 11.17 Determine the stability type of the equilibrium solution (fixed point) in the sky-diving equation dv = g kv. Interpret your result in the context of the terminal velocity of the sky-diver. 11.18 Newton s Law of Cooling states that the temperature of an object T(t) changes at a rate that depends on the difference between that temperature and the temperature of the environment, E: = k(t E), T(0) = T 0. Assume that E, k are positive constants. dt (a) Sketch the expression dt = k(t E) as a function of T. Identify points on your sketch that correspond to value(s) of the temperature that would not change (i.e. at which dt/ = 0). v.2005.1 - December 16, 2008 5
(b) Use your sketch to identify ranges of the temperature T for which dt/ is negative or positive, i.e. for which the temperature is increasing or decreasing. Use arrows on the T axis to indicate the flow. (c) Interpret your sketch. Give a verbal description of what Newton s Law of Cooling is saying about the temperature T(t) of the bo as the time t increases. 11.19 Find a differential equation that has exactly three stea states, at y = 0, 2, 4 and for which only y = 2 is an unstable stea state. How would you change this to an equation for which y = 2 is stable and the other two states are unstable? 11.20 Consider the differential equation = y2 r, where r is some positive constant. Find the fixed points of this equation and determine their stability. What happens as r is decreased? What happens when r = 0? The Linear Stability Condition fails (WHY?). But would you describe the fixed point as stable or unstable in that case? 11.21 You are a resource ecologist, hired to manage the population of fish in some lakes. Records indicate that in some lakes, the population of fish stabilizes at a very low level, and that in other lakes, the population stabilizes at a high level. The outcome seems to depend on the starting density of fish in the lake. You are asked to model the density of the fish population y(t) in a lake by a differential equation: = f(y) (a) You are asked to predict whether any major changes in the behavior of the fish population would occur if the lake was continuously stocked (i.e. new fish were continuously brought from a hatchery and added to the lake). How would the model change? Use a diagram and explain clearly what you expect would happen, emphasizing any abrupt transitions (bifurcations) that might occur. (b) Repeat part (b) but assume instead that the lake is used for fishing, and that fish are continuously harvested at some constant rate (and no new fish are added). What happens in this case? v.2005.1 - December 16, 2008 6
11.22 (a) Consider the differential equation = sin(y). By plotting / versus y, identify values of y which would not change (stea states), and regions for which y is increasing or decreasing. Use this diagram to summarize all the possible outcomes (behaviors of y(t)) for various initial values of y. (b) Now consider a related equation, = sin(y) + A where A is some positive constant. How does the behavior that you described in (a) change as the constant A is increased gradually? For what value(s) of A does the behavior change dramatically? Such changes are called bifurcations. 11.23 Find the curve passing through the point (1,2) and orthogonal to the family of curves x 2 y = k. 11.24 Sketch the family of curves x 2 9 + y2 = k 2 for k constant. Find the orthogonal trajectories by setting up and solving the appropriate differential equation. Sketch these on the same picture. 11.25 Find the family of curves that are orthogonal to the parabola y = Cx 2 Important remark about a possible pitfall: C is a constant only for a given parabola, not for the whole family. For this reason, we need to eliminate it from the general expression for the slope of the parabola (/dx = 2Cx). This can be done by plugging in C = y/x 2 in place of the constant C, (i.e getting /dx = 2y/x, a relationship that holds for all the parabola). Proceed from this point to find the orthogonal curves. 11.26 Sketch the family of curves xy = k v.2005.1 - December 16, 2008 7
for k constant. Find the orthogonal trajectories by setting up and solving the appropriate differential equation. Sketch these on the same picture. 11.27 According to Klaassen and Lindstrom (1996) J theor Biol 183:29-34, the fuel load (nectar) carried by a hummingbird, F(t) depends on the rate of intake (from flowers) and the rate of consumption due to metabolism. They assume that intake takes place at a constant rate α. They also assume that consumption increases when the bird is heavier (carrying more fuel). Suppose that fuel is consumed at a rate proportional to the amount of fuel being carried (with proportionality constant β). (a) What features are being neglected or simplified in this model? (b) Write down the differential equation model for F(t). (c) Find F(t) as a function of time t. (d) Klaassen and Lindstrom determined that for a hummingbird, α = 0.48 gm fuel /day and β = 0.09 /day. Determine the stea state level of fuel carried by the bird. 11.28 A biochemical reaction S P is catalysed by an enzyme E. The speed of the reaction depends on the concentration of the substrate S. It is found that the substrate concentration, x changes at the rate dx = rx Kx k + x where r, K, k are positive constants. (The second term is often called Michaelis Menten kinetics in chemistry.) How many stea state concentration values are there? What are these values? Explain the behaviour of the solutions x(t) for very large and for very small values of r. You may use sketches of the flow along the x axis. 11.29 Harvesting 1 Consider the Logistic equation with harvesting, dn = rn ( 1 N K ) H (a) Explain the distinction between this term and another possible term that might have been used, µy to represent a mortality or removal rate. v.2005.1 - December 16, 2008 8
(b) Show that by defining a new variable, y = N/K, the equation in part can get rewritten in the form: = ry(1 y) h What is the new constant h? (c) Find the stea states of this equation (all values of y for which / = 0) and describe the stability properties of each one. There may be several cases to consider, and you are asked to list the possible outcomes and how they depend on parameters. (d) Use a sketch of / versus y to explain what happens to the population for very low, intermediate, and high starting population sizes. (e) What harvesting level is reasonable in this model? What happens if the harvesting is increased to unreasonable levels? At what value of h does this transition take place? 11.30 Harvesting 2 (a) The Logistic Equation with harvesting discussed in a problem above has some unrealistic behaviour. Explain why its behaviour for y = 0 is not biologically realistic. (b) As a fix for this difficulty, we might consider a corrected equation of the form = ry(1 y) h ay a + y where now the harvesting rate is h ay. Explain what feature of this new equation makes it a+y more convincing. Explain this new dependence of harvesting on the population level. (i.e. what is being assumed about harvesting when the population is very low? when it is very high?) It will be helpful to sketch how the harvesting depends on the population level. (c) Determine the stea states of this new model equation by finding values of y for which / = 0. Describe the stability properties of each one. There may be several cases to consider, and you are asked to list the possible outcomes and how they depend on the parameters. (d) It is slightly awkward to try to graph / versus y since the expression on the RHS (right hand side) of the equation is now a bit nasty. However, it is easy to plot each of the terms in the equation, i.e. f 1 (y) = ry(1 y), f 2 (y) = hay/(a+y) on the same graph and ask where one term is larger than the other. (This would depend on the values of r and h. You might want to pick values for these for your plot. Then ask yourself how your plot changes if these values were larger or smaller.) By so doing, you can identify both stea states (intersections) and regions where / is positive or negative. Use this idea to find a qualitative description of the behaviour of y in this modified model. v.2005.1 - December 16, 2008 9
11.31 Mouse Population Type 1 A population of mice reproduces at a per capita rate r and is preyed upon by a cat. The predation rate is proportional to the density of the mice. Explain the following simple model for the population of mice: = ry py Determine what happens to the mice. Show that the outcome depends on the relative values of the parameters r and p. 11.32 Mouse population Type 2 Consider a mouse and cat model in which the cat (predator) has a so-called Type II response, i.e. in which the differential equation is = ry P y max a + y. Explain the differences in this version of the model. What is being assumed about the predator? One stea state of this equation is y=0. Find the other stea state. (Does the other stea state always exist? What has to be true for this stea state to be biologically meaningful?) Use the Linear Stability Condition to show that this second stea state is unstable whenever it exists. Interpret your findings. 11.33 Mouse population Type 3 Consider a population of rodents (density y(t)) that grow at an exponential rate (/ = ry), and suppose that a cat is brought in to control the population. The cat is a predator with a Holling Type III response, i.e. it is not very good at catching a small number of mice, it gets better at finding prey when there are quite a few of them, but if there are too many, it can t keep catching more and more of them. A model for the rodent population with predation of this type might be: = ry P y2 1 + y 2 Explain this model. (You may want to use graphical arguments). Use qualitative methods to determine the predictions for the outcome of predation on the prey. Pay particular attention to the way that these predictions depend on the parameter r. v.2005.1 - December 16, 2008 10