Chapter8 - 8.1 I Solving Differential Equations 403...

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Unformatted text preview: 8.1 I Solving Differential Equations 403 Equation (8.38) allows us to eliminate C. Solving (8.38) for C, we find that C = B V"°‘794 and, therefore, 3% = BV‘0'794(O.794)V°‘794‘1 = (0.794)B V’l Rearranging terms yields dB — (0 794)dv B “ ' V Dividing both sides by d t, we get 1 dB 1 d V —— = 0.794 —— . B dz ( ) V dt which is the same as (8.39). I ' EXAMPLE 9- Homeostasis The nutrient content of a consumer (e.g., the percent nitrogen of the consumer’s biomass) can range from reflecting the nutrient content of its food to being constant. The former is referred to as absence of homeostasis, the latter as strict homeostasis. A model for homeostatic regulation is provided in Sterner and Elser (2002). The model relates a consumer’s nutrient content (denoted by y) to its food’s nutrient content (denoted by x) as ’ dy 1 y dx _ 9 x where 9 2 1 is a constant. Solve the differential equation and relate 8 to absence of homeostasis and strict homeostasis (8.40) Solution We can solve (8.40) by separation of variables: i dy__1 dx y 9 x Integrating and simplifying yields 1 lnlyl = 51n|x|+C1 m = eu/e) In lx|+C1 ill, lyl = lxll/‘gec1 1:; y = :izeclxm =.‘ Since x and y are positive (they denote nutrient contents), it follows that 1/9 y“; y=Cx where C is a positive constant. Absence of homeostasis means that the consumer reflects the food’s nutrient content. This occurs when y = C x and thus when 6 = 1. Strict homeostasis means that the nutrient content of the consumer is independent of the nutrient content of I the food; that is, y = C; this occurs in the limit as 6 —-> 00. I Section 8; 1' Problems. fly d in l 8'1" 2. l = e“3-‘, where ya = 10 for x0 = 0 3-1 In Problems 1—8, solve each pure-time differential equation. dx d l l. 51-y— =x+sinx,wherey0=0forxo=0 3. —y = —,whereyo=0whenxo=l ' i‘ dx dx x ‘ 404 Chapter8 I Differential Equations 4. 5% = l4{flywhereyo = 1 whenxo = 0 . 1 5. dz; = 1—:—;,wherex(0) = 2 6. = cos(27r(t — 3)), where x(3) = 1 7. 5:5: 3! + 1,wheres(0) =1 8. 3:: = 5 — 16t2, where h(3) = ~11 9. Suppose that the volume V(t) of a cell at time t changes according to (W —_=1+cost dt with V(0) = 5 Find V(t). 10. Suppose that the amount of phosphorus in a lake at time t, denoted by P(r), follows the equation 31—}: =3t+1 with P(O) =0 dt Find the amount of phosphorus at time t = 10. I 8.1.2 In Problems 11—16, solve the given autonomous differential equations. 11. Q- : 3y, where ya = 2 for x0 = 0 dx 12. 3—): = 20— y), where ya = 2 for x0 = 0 x 13. E = ~2x, where x(1) = 5 dt 14. :4: = 1 — 3x, where x(-—-1) = —2 dh 15. ZS- : 2h + 1, where h(0) = 4 16. (ii—1:, = 5 - N, where N(2) = 3 17. Suppose that a population, whose size at time t is denoted by N (t), grows according to (ii—1:, = 0.3N(t)' with N(O) = 20 Solve this differential equation, and find the size of the population at time t = S. 18. Suppose that you follow the size of a population over time. When you plot the size of the population versus time on a semilog plot (i.e., the horizontal axis, representing time, is on a linear scale, whereas the vertical axis, representing the size of the population. is on a logarithmic scale), you find that your data fit a straight line which intercepts the vertical axis at 1 (on the log scale) and has slope —O.43. Find a differential equation that relates the growth rate of the population at time t to the size of the population at time t. 19. Suppose that a population, whose size at time t is denoted by N (t), grows according to (8.41) where r is a constant. (a) Solve (8.41). (b) Transform your solution in (a) appropriately so that the resulting graph is a straight line. How can you determine the constant r from your graph? (c) Suppose now that, over time, you followed a population which evolved according to (8.41). Describe how you would determine r from your data. 20. Assume that W(t) denotes the amount of radioactive material in a substance at time t. Radioactive decay is then described by the differential equation Sith = —).W(t) with W(0) = W0 where A is a positive constant called the decay constant. (a) Solve (8.42). (b) Assume that W(0) = 123 gr and W(5) = 20 gr and that time is measured in minutes. Find the decay constant A and determine the half-life of the radioactive substance. 21. Suppose that a population, whose size at time t is given by N (t), grows according to (8.42) with N (0) = 10 (8.43) (a) Solve (8.43). (1)) Graph N (t) as a function oft for O 5 t < 10. What happens as t —> 10? Explain in words what this means. 22. Denote by L(t) the length of a fish at time t, and assume that the fish grows according to the von Bertalanffy equation dL Et— = k(34 — L(t)) with L(O) = 2 (a) Solve (8.44). (b) Use your solution in (a) to determine k under the assumption that L(4) = 10. Sketch the graph of L(t) for this value of k. (c) Find the length of the fish when t = 10. ((1) Find the asymptotic length of the fish; that is, find 1im,_,m L(t). 23. Denote by L(t) the length of a certain fish at time t, and assume that this fish grows according to the von Bertalanffy equation (8.44) dL . a? = 14L“, - L(t)) With L(O) = 1 (8.45) where k and Lao are positive constants. A study showed that the asymptotic length is equal to 123 in and that it takes this fish 27 months to reach half its asymptotic length. (a) Use this information to determine the constants k and Lon in (8.45). [Hintz Solve (845).] (b) Determine the length of the fish after 10 months. (c) How long will it take until the fish reaches 90% of its asymptotic length? 24. Let N (I) denote the size of a population at time t. Assume that the population exhibits exponential growth. (a) If you plot log N(t) versus 1, what kind of graph do you get? (b) Find a differential equation that describes the growth of this population and sketch possible solution curves. 25. Use the partial-fraction method to solve dy —— = 1 dx y( +y) where yo : 2 for to = 0. ,. .4__.~—v~z _._ 26. Use the partial-fraction method to solve dv .._'... = _ dx where ya = 2 for x0 = 0. 27. Use the partial-fraction method to solve d l = y(y - 5) (ix where yo = 1 for x0 = 0. 28. Use the partial—fraction method to solve dy_ _ ‘— d—x-—(y 1)(y 2) where yo = O for x0 = 0. 29. Use the partial-fraction method to solve dy __=2 3.. dx y( y) where yo = 5 for x0 = 1. 30. Use the partial-fraction method to solve (1y 1 2 _.=_ _2 d: 2y y where yo = —3 for to = 0. In Problems 31—34, solve the given differential equations. d d 31.—¥=y(1+y) 32. —y=(1+y)2 dx dx d d 33. %=(1+y)3 34. —-y—=(3—y)(2+y) 35. (a) Use partial fractions to show that dx f du _11“ uZ—az—Za (b) Use your result in (a) to find a solution of u—a u+a +C that passes through (i) (0. 0), (ii) (0, 2), and (iii) (0. 4). 36. Find a solution of dy _.__2 4 dx Y+ that passes through (0, 2). 37. Suppose that the size of a population at time t is denoted by N(t) and that N(t) satisfies the differential equation dN N —— = .34 — —— ‘th 0 2 50 dt 0 N (I 200) wr N( ) Solve this differential equation, and determine the size of the population in the long run; that is. find limHm N(t). '38. Assume that the size of a population, denoted by N(t). evolves according to the logistic equation. Find the intrinsic rate of growth if the carrying capacity is 100. N(O) = 10. and NH) = 20. 39. Suppose that N(t) denotes the size of a population at time t und that (IN N —-— =1. N 1— —— (It 5 50) 8.1 L! Solving Differential Equations 405 (a) Solve this differential equation when N (0) = 10. (b) Solve this differential equation when N(O) = 90. (c) Graph your solutions in (a) and (b) in the same coordinate system. (d) Find lim,_,00 N(t) for your solutions in (a) and (b). 40. Suppose that the size of a population, denoted by N(t), satisfies dN N —= .N 1~— .4 d! 07 ( 35) (s 6) (2) Determine all equilibria by solving dN/a't = 0. (b) Solve (8.46) for (i) N(O) = 10, (ii) N(O) = 35, (iii) N(O) = 50. and (iv) N(O) = 0. Find lim,_,°° N(t) for each of the four initial conditions. (c) Compare your answer in (a) with the limiting values you found in (b). 41. Let N(t) denote the size of a population at time t. Assume that the population evolves according to the logistic equation. Assume also that the intrinsic growth rate is 5 and that the carrying capacity is 30. (a) Find a differential equation that describes the growth of this population. (b) Without solving the differential equation in (a), sketch solution curves of N(t) as a function oft when (i) N(O) = 10, (ii) N (O) = 20, and (iii) N (O) = 40. 42. Logistic growth is described by the differential equation The solution of this differential equation with initial condition N (0) = N0 is given by N(t) = -—KK——— (8.47) 1+ (,7; —1)e-~ (a) Show that 1 K —— N0 1 N(t) by solving (8.47) for r. (b) Equation (8.48) can be used to estimate r. Suppose we follow a population that grows according to the logistic equation and find that N(O) = 10, N(S) = 22, N(lOO) = 30, and N(ZOO) = 30. Estimate r. 43. Selection at a Single Locus We consider one locus with two alleles, A1 and A2, in a randomly mating diploid population. That is, each individual in the population is either of type AlAl, A.A2, or A2A2. We denote by p(t) the frequency of the A. allele and by q(t) the frequency ofthe A3 allele in the population at time t. Note that p(t) +q(t) = 1. We denote the fitness of the AA] type by 11),,- and assume that w” =1,w12 21— 5/2, and w; = 1—5, where s is a nonnegative constant less than or equal to 1. That is, the fitness of the heterozygote A 1 A3 is halfway between the fitness of the two homozygotes. and the type A1 A. is the fittest. If s is small, we can show that, approximately. 'I 1 L1: = Esp“ — p) with ])(0) = p” 8.49 (It ( > (a) Use separation of variables and partial fractions to find the solution of (8.49). (b) Suppose p“ = 0.1 and s = 0.01: how long will take until p(t) = 0.5? (c) Find lim,_m p(r). Explain in words what this limit means. 406 Chapter8 I Differential Equations l 8.1.3 In Problems 44—52, solve each differential equation with the given initial condition. 44, fl:2£,wlthyo=1lfxo=1 dx x 45, 1X=x+1,wnhyo=2trxo=o dx y dy y . . ,_= ,wrth =11fx =0 46 dx x+l Yo 0 dy -3 - - _ 47, 2—=(y+1)e ,w1thyo=21fxo—0 x 48, d—y =x2y2,withyo=1ifxo=l dx 1 . 49. 2:” ,withyo=51fxo=2 dx x—l du sin! . . 50, E=u2+1,w1thuo=3lfto=0 51, d—r- = re", With 70 = 1 lfto =0 dt dx 1x 5 ._=-—,withx =2if =3 2. dy 2y 0 Yo 53. (Adapted from Reiss, 1989) In a case study by Taylor et al. (1980) in which the maximal rate of oxygen consumption (in mls“) for nine species of wild African mammals was plotted against body mass (in kg) on a log—log plot, it was found that the data points fall on a straight line with slope approximately equal to 0.8. Find a differential equation that relates maximal oxygen consumption to body mass 54. Consider the following differential equation, which is impor- tant in population genetics: 1 d a(x)g(x) « Ed—x [b(x)g(X)l = 0 Here, b(x) > 0. I 8.2 Equilibria and Their Stability (a) Define y = b(x)g(x), and show that y satisfies (8.50) (b) Separate variables in (8.50), and show that if y > 0, then y=Cexp[2/g-E:—;dx] 55. When phosphorus content in Daphm‘a was plotted against phosphorus content of its algal food on a log—log plot, a straight line with slope 1/7.7 resulted. (See Sterner and Elser, 2002; data from DeMott et al., 1998.) Find a differential equation that relates the phosphorus content of Daphnia to the phosphorus content of its algal food. 56. This problem addresses Malthus's concerns. Assume that a population size grows exponentially according to N (t) = 1000e' and the food supply grows linearly according to F0) = 3! (it) Write a differential equation for each of N (t) and F(t). (b) What assumptions do you need to make to be able to compare whether and, if so, when food supply will be insufficient? Does exponential growth eventually overtake linear growth? Explain. (c) Do a Web search to determine whether food supply has grown linearly, as claimed by Malthus 57. At the beginning of this section, we modified the exponential~ growth equation to include oscillations in the per capita growth rate. Solve the differential equation we obtained, namely, 1:3]. = 2 (1 + sin(27rt)) N(t) with N(O) = 5. In Subsection 8.1.21, we learned how to solve autonomous differential equations and graphed their solutions as functions of the independent variable for given initial conditions. For instance, logistic growth d—iy— = rN(1— 5) (8.51) with initial condition N (0) = No has the solution given in (8.33) and graphed in Figure 8.10 for different initial values The solution of a differential equation can inform us about long-term behavior. as we saw in the case of logistic growth. In particular, if No > 0, then N (t) -> K, the carrying capacity, as t —> co, and if N0 = 0, then N(t) = 0 for all t > 0. Also, if N0 = K, then N(t) = K for all! > 0. What is so special about N0 = K or N0 = 0? We see from Equation (8.51) that if N = K or N that N (t) is constant. = 0, then dN/dt = 0, implying Constant solutions form a very special class of solutions of autonomous differen- tial equations. These solutions are called point equilibria or, simply, equilibria. The constant solutions N = K and N = O are point equilibria of the logistic equation. \fixiw ————_\l 8.2 I Equilibria and Their Stability 417 A graph of g (N) is shown in Figure 8.233. Differentiating g(N) yields 4 2 31V2 g’(N) = r (2N + TEN — 7(— —a) = %(2NK +2aN — 31v2 — aK) We can compute the eigenvalue g’(1\7) associated with the equilibrium 19: if 10 = 0, then g’(0) = ~Ir—<—(—aK) < 0 if A? = a, then g'(a) = %a(K — a) > 0 a ifN ll K, then g’(K) = %K(a ~ K) < 0 As we continue, you should compare the results from the eigenvalue method with the graph of g(N). Since g’(0) < 0, it follows that N = 0 is locally stable. Likewise, since g’ (K) < 0, it follows that I“! = K is locally stable. The equilibrium N = a is unstable, because g’ (a) > 0. This instability is also evident from Figure 8.23a. The Allee effect is an example in which both stable equilibria are locally, but not globally, stable. ‘ y 7 06 .5 (IN U) W %?=g(N)—— g5 .54 23 3 0 - tie-:2 a N l 0 012345678910): Time Figure 8.233 The graph ofg(N) Figure 8.23b Solution curves when r = 0.5, illustrating the Alice effect. a = 2, and K = 5. When the initial condition N (0) is between 0 and 2, the solution curve approaches the locally stable equilibrium N = 0. When the initial condition N (0) is greater than 2, the solution curve approaches the locally stable equilibrium 1?] = K = 5. The approach is from below when 2 < N (0) < 5 and from above when N (0) > 5. We see from Figures 8.23a and 8.23b that if 0 _<_ N (0) < a, then N (t) ——> Gas t —-> oo.Ifa < N(O‘) _<_ K or N(O) z K,then N(t) —> K ast —) 00. To interpret our results, we observe that if the initial population N (0) is too small [i.e., N (0) < a], then the population goes extinct, and if the initial population is large enough [i.e., N (0) > a], then the population persists. That is, the parameter a is a threshold level. The recruitment rate is large enough only when the population size exceeds this level. . Section 8.2 Problems; I 8.2.1 (c) Compute the eigenvalues associated with each equilibrium, 1. Suppose that and discuss the stability of the equilibria. 32’, = ya _ y) 2. Suppose that x (2) Find the equilibria of this differential equation. 3? = (4 .— y)(5 . y) (b) Graph dy/dx as a function of y, and use your graph to discuss x the stability of the equilibria. (in) Find the equilibria of this differential equation. . rev-:9- r 418 Chapter8 I Differential Equations I (b) Graph dy/dx as a function of y, and use your graph to discuss the stability of the equilibria. (c) Compute the eigenvalues associated with each equilibrium, and discuss the stability of the equilibria. 3. Suppose that dy a; =y(y-1)(y-2) (a) find the equilibria of this differential equation. (b) Graph dy/dx as a function of y, and use your graph to discuss the stability of the equilibria. (c) Compute the eigenvalues associated with each equilibrium, and discuss the stability of the equilibria. 4. Suppose that dy __ __ _ fi-m y)(y 3) (it) Find the equilibria of this differential equation. (b) Graph dy/dx as a function of y, and use your graph to discuss the stability of the equilibria. (c) Compute the eigenvalues associated with each equilibrium, and discuss the stability of the equilibria. 5. Logistic Equation Assume that the size of a population evolves according to the logistic equation with intrinsic rate of growth r = 1.5. Assume that the carrying capacity K = 100. (a) Find the differential equation that describes the rate of growth of this population. (b) Find all equilibria, and, using the graphical approach, discuss the stability of the equilibria. (c) Find the eigenvalues associated with the equilibria, and use the eigenvalues to determine the stability of the equilibria. Compare your answers with your results in (b). 6. A Simple Model of Predation Suppose that N (t) denotes the size of a population at time t. The population evolves according to the logistic equation, but, in addition, predation reduces the size of the population so that the rate of change is given by dN N 9N . __ = 1.. _ — —— 8.65 dt N( ) ( ) The first term on the right-hand side describes the logistic growth; the second term describes the effect of predation. (a) Set N 9N 8(N)_N(1_§6)_5+N and graph g(N). (b) Find all equilibria of (8.65). (c) Use your graph in (a) to determine the stability of the equilibria you found in (b). (d) Use the method of eigenvalues to determine the stability of the equilibria you found in (b). 7. Logistic Equation Assume that the size of a population evolves according to the logistic equation with intrinsic rate of growth r = 2. Assume that N(O) = 10. (:1) Determine the carrying capacity K if the population grows fastest when the population size is 1000. (Hint: Show that the graph ode/dt as a function of N has a maximum at K/2.) (b) ifN(0) = 10, how long will it take the population size to reach 1000? (c) Find limH00 N0). 8. Logistic Equation The logistic curve N (t) is an S-shaped curve that satisfies (1 N A = I'N “ Wlil'l = No when No < K. (a) Use the differential equation (8.66) to show that the inflection point of the logistic curve is at exactly half the saturation value of the curve. [Hintz Do not solve (8.66); instead, differentiate the right-hand side with respect to L] (b) The solution N (t) of (8.66) can be defined for all t e R. Show that N (t) is symmetric about the inflection point and that N (O) = No. That is, first use the solution of (8.66) that is given in (8.33), and find the time to so that N00) = K/2 (i.e., the inflection point) is at t = to. Compute N(t0 + h) and N00 - h) for h > 0 and show that t K Nito+h)— ‘5‘ K . —- "2- — N00 - Use a sketch of the graph of N (t) to explain why the preceding equation shows that N (t) is symmetric about the inflection point (to, N 00))- 9. Suppose that a fish population evolves according to the logistic equation and that a fixed number of fish per unit time are removed. That is, Assume that r = 2 and K = 1000. (a) Find possible equilibria, and discuss their stability when H = 100. (b) What is the maximal harvesting rate that maintains a positive population size? 10. Suppose that a fish population evolves according to a logistic equation and that fish are harvested at a rate proportional to the population size. if N (t) denotes the population size at time t, then dN N -——.rN( —7{—)—hN Assume that r = 2 and K = 1000. (a) Find possible equilibria, use the graphical approach to discuss their stability when h = 0.1, and find the maximal harvesting rate that maintains a positive population size. (b) Show that ifh < r = 2, then there is a nontrivial equilibrium. Find the equilibrium. (c) Use (i) the eigenvalue approach and (ii) the graphical approach to analyze the stability of the equilibrium found in (b). I 8.2.2 11. Assume the single—compartment model defined in Subsection 8.2.2: If C(t) is the concentration of the solute at time t, then dC/dt is given by (8.57); that is, dC -1 _ dt ‘V(C’ C) where q, V. and C, are defined as in Subsection 8.2.2. Use the graphical approach to discuss the stability of the equilibrium C" = C,. 12. Assume the single—compartment model defined in Subsection 8.2.2; that is, denote the concentration of the solute at time t by C0), and assume that 5—C— 2 3(20 — C(r)) for I z 0 (1! (8.67) mm“— “Wwfi "u (a) Solve (8.67) when C(O) = 5. (b) Find lim,_,°o C(t). (c) Use your answer in (a) to determine t so that C (t) = 10. 13. Assume the single-compartment model defined in Subsection 8.2.2; that is, denote the concentration of the solution at time t by C (t), and assume that the concentration of the incoming solution is 3 g liter‘1 and the rate at which mass enters is 0.2 liter 5". Assume, further, that the volume of the compartment V = 400 liters. (in) Find the differential equation for the rate of change of the concentration at time t. (b) Solve the differential equation in (a) when C (0) = 0, and find lim C(t). ’ (c) Find all equilibria of the differential equation and discuss their stability. 14. Suppose that a tank holds 1000 liters of water, and 2 kg of salt is poured into the tank. (2) Compute the concentration of salt in g liter—1. [Wm (b) Assume now that you want to redude the salt concentration. One method would be to remove a certain amount of the salt water from the tank and then replace it by pure water. How much salt water do you have to replace by pure water to obtain a salt concentration of 1 g liter-1? (c) Another method for reducing the salt concentration would be to hook up an overflow pipe and pump pure water into the tank. That way, the salt concentration would be gradually reduced. Assume that you have two pumps, one that pumps water at a rate of lliter s", the other at a rate of 21iter s‘l. For each pump, find out how long it would take to reduce the salt concentration from the original concentration to 1 gliter‘l and how much pure water is needed in each case. (Note that the rate at which water enters the tank is equal to the rate at which water leaves the tank.) Compare the amount of water needed using the pumps with the amount of water needed in part (b). 15. Assume the single~compartment model introduced in Subsec- tion 8.2.2. Denote the concentration at time t by C(t), measured in mg/L, and assume that (Id-’6: = 0.37(254 mg/L —— C(t)) fort z 0 (8) Find the equilibrium concentration. (b) Assume that the concentration is suddenly increased from the, equilibrium concentration to 400 mg/L. Find the return time to equilibrium, denoted by TR, which is the amount of time until the: initial difference is reduced to a fraction e”. (c) Repeat (b) for the case when the concentration is suddenly increased from the equilibrium concentration to 800 mg/L. (d) Are the values for TR computed in (b) and (c) different? 16. Assume the compartment model as in Subsection 8.2.2. Suppose that the equilibrium concentration is C, and the initial concentration is Co. Express the time it takes until the initial deviation C0 — C, is reduced to a fraction p in terms of TR. 17. Assume the compartment model as in Subsection 8.2.2. Suppose that the equilibrium concentration is C,. The time TR has an integral representation that can be generalized to systems with more than one compartment. Show that 0°C(t)-C, T: -—————-dt " fa C(o)—C, 8.2 I Equilibria and ThelrStability 419 [Hinu Use (8.58) to show that M = e—<q/V)n C(O) —- C, and integrate both sides with respect tot from 0 to 00.] 18. Use the compartment model defined in Subsection 8.2.2 to investigate how the size of a lake influences nutrient dynamics in the lakeafter a perturbation. Mary Lake and Elizabeth Lake are two fictitious lakes in the North Woods that are used as experimental lakes to study nutrient dynamics. Mary Lake has a volume of 6.8 x 103 m3, and Elizabeth Lake has twice that volume, or 13.6 x 103 m3. Both lakes have the same inflow/outflow rate q = 170 liter 5“. Because both lakes share the same drainage area, the concentration C, of the incoming solute is the same for both lakes, namely, C, = 0.7 mgliter“. Assume that at the beginning of the experiment both lakes are in equilibrium; that is, the concentration of the solution in both lakes is 0.7 mg liter‘l. Your experiment consists of increasing the concentration of the solution by 10% in each lake at time 0 and then watching how the concentration of the solution in each lake changes with time. Assume the single-compartment model to make predictions about how the concentration of the solution will evolve. (Note that Im3 of water corresponds to 1000 liters of water.) (8) Find the initial concentration Co of the solution in each lake at time 0 (Le, immediately after the 10% increase in concentration of the solution). (b) Use Equation (8.58) to determine how the concentration of the solution changes over time in each lake. Graph your results. (c) Which lake returns to equilibrium faster? Compute the return time to equilibrium, TR, for each lake, and explain how it is related to the eigenvalues corresponding to the equilibrium concentration C, for each lake. 19. Use the single-compartment model defined in Subsection 8.2.2 to investigate the effect of an increase in the input concentration C, on the nutrient concentration in a lake. Suppose a lake in a pristine environment has an equilibrium phosphorus concentration of 0.3 mg". The volume V of the lake is 12.3 X 10° m3, and the inflow/outflow rate q is equal to 220 liter s". Conversion of land in the drainage area of the lake to agricultural use has increased the input concentration from 0.3 mgliter‘l to 1.1mg1iter‘1. Assume that this increase happened instantaneously. Compute the return time to the new equilibrium, denoted by TR, in days, and find the nutrient concentration in the lake TR units of time after the change in input concentration. (Note that 1 m3 of water corresponds to about 1000 liters of water.) El 8.2.3 20. Levins Model Denote by p = p(t) the fraction of occupied patches in a metapopulation model, and assume that d a? = 2p(1— p) — p for t 2 0 (8.68) (a) Set gov) = 2110— p) — p- Graph 8(P) for p e[0.11- (b) Find all equilibria in (8.68) that are in [0, 1]. Use your graph in (a) to determine their stability. (c) Use the eigenvalue approach to analyze the stability of the equilibria that you found in (b). 21. Levins Model Denote by p = p(t) the fraction of occupied patches in a metapopulation model, and assume that d 3:3 = O.5p(1- p) — 1.5p fort a 0 (8.69) (a) Set g(p) = 0.5p(1—- p) - 1.5p. Graph g(p) for p 6 [0,1]. “13...; “Balm: .. .1 a... “v 420 Chapter8 I Differential Equations (b) Find all equilibria of (8.69) that are in [0, 1]. Use your graph in (a) to determine their stability. (c) Use the eigenvalue approach to analyze the stability of the equilibria that you found in (b). 22. A Metapopuiation Model with Density-Dependent Extinc- tion Denote by p = p(t) the fraction of occupied patches in a metapopulation model. and assume that 'd —”=cp(1—p)—p2 dt where c > 0. The term p2 deseribes the density-dependent extinction of patches; that is. the per-patch extinction rate is p, and a fraction p of patches are occupied, resulting in an extinction rate of p2. The colonization of vacant patches is the same as in the Levins model. (a) Set g(p) = cp(1 — p) -— p2 and sketch the graph of g(p). (b) Find all equilibria of (8.70) in [0, 1]. and determine their stability. (c) Is there a nontrivial equilibrium when c > 0? Contrast your findings with the corresponding results in the Levins model. 23. Habitat Destruction In Subsection 8.2.3. we introduced the Levins model. To study the effects of habitat destruction on a single species, we modify equation (8.63) in the following way: We assume that a fraction D of patches is permanently destroyed. Consequently, only patches that are vacant and undestroyed can be successfully colonized. These patches have frequency 1 — p(t)— D if p(!) denotes the fraction of occupied patches at time t. Then rip .27=cp(1_—p-D)—mp (a) Explain in words the meaning of the different terms in (8.71). (b) Show that there are two possible equilibria: the trivial equilibrium p1 = Oand the nontrivial equilibrium p2 = 1—D—- Sketch the graph of p2 as a function of D. (c) Assume that m < c such that the nontrivial equilibrium is stable when D = 0. Find a condition for D such that the nontrivial equilibrium is between 0 and 1, and investigate the stability of both the nontrivial equilibrium and the trivial equilibrium under that condition. (d) Assume the condition that you derived in (c); that is, the nontrivial equilibrium is between 0 and 1. Show that when the system is in equilibrium, the fraction of patches that are for r 2 0 (8.70) (8.71) vacant and undestroyed—that is, the sites that are available for colonization—is independent of D. Show that the effective colonization rate in equilibrium—that is, 0 times the fraction of available patches—is equal to the extinction rate. This equality shows that the effective birth rate of new colonies balances their extinction rate at equilibrium. I 8.2.4 24. Alice Effect Denote the size of a population at time t by N (I), and assume that fl=2N(N—10)<l—i) fort20 d! 100 (8.72) (a) Find all equilibria of (8.72). (b) Use the eigenvalue approach to determine the stability of the equilibria you found in (a). ‘ (c) Set N g(N) = 2N(N - 10) (1 — for N 2 0, and graph g(N). Identify the equilibria of (8.72) on your graph. and use the graph to determine the stability of the equilibria. Compare your results with your findings in (b). Use your graph to give a graphical interpretation of the eigenvalues associated with the equilibria. 25. Alice Effect Denote the size of a population at time t by N (t). and assume that dN —— = O.3N(N — 17) (1 - l) for r a o (8.73) d t 200 (a) Find all equilibria of (8.73). (b) Use the eigenvalue approach to determine the stability of the equilibria you found in (a). (c) Set N N =0.3NN—17 1—-—- g( ) ( ) ( 200) for N 2 0, and graph g(N). Identify the equilibria of (8.73) on your graph, and use the graph to determine the stability of the equilibria. Compare your results with your findings in (b). Use your graph to give a graphical interpretation of the eigenvalues associated with the equilibria. I 8.3 Systems of Autonomous Equations (Optional) In the preceding two sections, we discussed models that could be described by a single differential equation. If we wish to describe models in which several quantities interact, such as a competition model in which various species interact, more than one differential equation is needed. We call this model a system of differential equations. We will restrict ourselves again to autonomous systems—that is, systems whose dynamics do not depend explicitly on the independent variable (which typically is time). This section is a preview of Chapter 11, in which we will discuss systems of differential equations in detail. A thorough analysis of such systems requires a fair amount of theory, which we will develop in Chapters 9 and 10. Since we are not yet equipped with the right tools to analyze systems of differential equations, this section will be rather informal. As with movie previews, you will not know the full story after you finish reading the section, but reading it will (hopefully) convince you that systems of differential equations provide a rich tool for modeling biological systems. i j i i i i i l i We assume that p1 == 131 1 — [31. Hence, __ ~‘V if Since (1 p2/ d t > 0 when species 1 is in equilibrium and species 2 has a low abundance it follows that species 2 can invade. We conclude that species 1 and 2 can coexist when 2 C2 > C1. 8.3 I Systems of Autonomous Equations (Optional) 427 = 1 —1/c1 and that p2 is very small. Then 1 — 13, -— p2 a 1 C2 [72 cz——1—C1+1=p2 --Cl >0 c1 Cl 02 2 ———-cI>0, or 02>c1 ’ This mechanism of coexistence is referred to as the competition—colonization trade-off. That is, the weaker competitor (species 2) can compensate for its inferior competitiveness by being a superior colonizer (c; > cf). 'Section.8‘.3i Problems Ll 8.3.1 In Problems 14, we will investigate the classical Kermack— McKendrick model for the spread of an infectious disease in a population of fixed size N. (This model was introduced in Subsection 8.3.1, and you should refer to that subsection when working out the problems.) If S(t) denotes the number of susceptibles at time t, l (t) the number of infectives at time t, and R(t) the number of immune individuals at time t, then dS .. = _ 1 dt bS d1 _ =bSI -— 1 dt a and R(t) = N —— S(t) — [(0. 1. Determine, in each of the following cases, whether or not the disease can spread (Hint: Compute Rm): (9) 5(0) = 1000, a = 200, b = 0.3 (b) 5(0) = 1000,a =200,b = 0.1 2. Assume that a = 100 and b = 0.2. The critical number of susceptibles Sc(0) at time 0 for the spread of a disease that is introduced into a population at time 0 is defined as the minimum number of susceptibles for which the disease can spread. Find $40). 3. Suppose that a = 100, b = 0.01, and N = 10. 000. Can the disease spread if, at time 0, there is one infected individual? 4. Refer to the simple model of epidemics in Subsection 8.3.1. (3) Divide (8.75) by (8.74) to show that when l > 0, d1 a1 dS~bS Also, show that when R(0) = 0, [(0) = 10, and S(O) = So, the solution of (8.84) satisfies — 1 (8.84) so a l(t)=N—S(t)+gln so where I (t) denotes the number of infectives, N the total number of individuals in the population, and S(t) the number of susceptibles at time t. (b) Since [(2) gives the number of infectives at time t and dI/dt = bSl -— a], if S(O) > a/b, then dI/dt > 0 at time t = 0. Also, since lim,_,°° [(0 = 0, there is a time t > 0 at which I (t) is maximal. Show that the number of susceptibles when I (t) is maximal is given by S =: a/b. [Hintz When I (t) attains a maximum, the derivative of I (t) with respect to t, d] /dt, is equal to 0.] (c) In (a), you expressed I (t) as a function of S (t). Use your result in (b) to show that the maximal number of infectives is given by N—g+gln(flé) Imax = b b so (d) Use your result in (c) to show that Irm is a decreasing function of the parameter a /b for a/b < So (i.e., in the case in which the infection can spread). Use the latter statement to explain how a and b determine the severity (as measured by (max) of the disease. Does this make sense? I 8.3.2 5. Assume the compartment model of Subsection 8.3.2, with a = 5,!) = 0.02,m =1,andc = 1. (11) Find the system of differential equations that corresponds to these values. (b) Determine which values of N, result in a nontrivial equilibrium, and find the equilibrium values for both the autotroph and the nutrient pool. 6. Assume the compartment model of Subsection 8.2.3, with a = 1,17 =0.01,m =2,c = 1, and N, = 500. (in) Find the system of differential equations that corresponds to these values. (b) Plot the zero isoclines corresponding to this system. (c) Use your graph in (b) to determine whether the system has a nontrivial equilibrium. 7. Assume the compartment model of Subsection 8.3.2, with a = 1,17 = 0.01,m = 2.c =1 and-N, = 200. (9) Find the system of differential equations that corresponds to these values. (b) Plot the zero isoclines corresponding to this system. (c) Use your graph in (b) to determine whether the system has a nontrivial equilibrium. 428 Chapter8 I Differential Equations I 8.3.3 8. Assume the hierarchical competition model introduced in Subsection 8.3.3, and assume that the model describes two species. Specifically, assume that dPi ._ =2 1.. .. d! [M P1) Pt (1' “£3 = 5P2(1" P1 — P2) — P2 ‘“ 2P1P2 (:1) Find all equilibria. (b) Determine whether species 2 can invade a monoculture of species 1. (Assume that species 1 is in equilibrium.) 9. Assume the hierarchical competition model introduced in Subsection 8.3.3, and assume that the model describes two species Specifically, assume that (1P1 ____=2 1.. _ dz P1( P1) P1 d £1 = 3172(1' P1 — P2) “ P2 '2P1P2 (a) find all equilibria. (b) Determine whether species 2 can invade a monoculture equilibrium of species 1. . 10. Assume the hierarchical competition model introduced in Subsection 8.3.3, and assume that the model describes two species Specifically, assume that dPi ._ = 2 1.. .. [1’ P1( P1) P1 (1 ~53— = 6P2(1 — p1 - p2) - p2 — 2pm: (8) Use the zero-isocline approach to find all equilibria graphi- cally. (b) Determine the numerical values of all equilibria. 1]. Assume the hierarchical competition model introduced in Subsection 8.3.3, and assume that the model describes two species. Specifically, assume dPl _.=3 1.. _. d! P1( P1) Pt (1 732' = 5172(1 - Pl" P2) " P2 - 3P1P2 (a) Use the zero—isocline approach to find all equilibria graphi~ cally. (b) Determine the numerical values of all equilibria. 12. (Adapted from Crawley, 1997) Denote plant biomass by V, and herbivore number by N. The plant—herbivore interaction is modeled as dV V . __.=_. V __ _ dt a (1 K) bVN fl=cVN—dN dt (:1) Suppose the herbivore number is equal to 0. What differential equation describes the dynamics of the plant biomass? Can you explain the resulting equation? Determine the plant biomass equilibrium in the absence of herbivores. (b) Now assume that herbivores are present. Describe the effect of herbivores on plant biomass; that is, explain the term -bVN in the first equation. Describe the dynamics of the herbivores— that is, how their population size increases and what contributes to decreases in their population size. (c) Determine the equilibria (1) by solving —-— = and —- = 0 and (2) graphically. Explain why this model implies that “plant abundance is determined solely by attributes of the herbivore,” as stated in Crawley (1997). Chapter 8 Key Terms. . Discuss the following definitions and concepts: 1. Differential equation 2. Separable differential equation 3. Solution of a differential equation 4. Pure-time differential equation 5. Autonomous differential equation 10. Equilibrium 11. Stability 12. Eigenvalue 6. Exponential growth 7. Von Bertalanffy equation 8. Logistic equation 9. Allometric growth 13. Single-compartment model 14. Levins model 15. Alice effect 16. Kermack—McKendrick model 17. Zero isocline 18. Hierarchical competition model Chapter 8" Review Problems , l. Newton’s Law of Cooling Suppose that an object has temperature T and is brought into a room that is kept at a constant temperature 7),. Newton's law of cooling states that the rate of temperature change of the object is proportional to the difference between the temperature of the object and the surrounding medium. (a) Denote the temperature at time t by T0). and explain why is the differential equation that expresses Newton‘s law of cooling. (b) Suppose that it takes the object 20 min to cool from 30°C to 28°C in a room whose temperature is 21°C. How long will it take the object to cool to 25°C if it is at 30°C when it is brought into the room? [Hintz Solve the differential equation in (a) with the initial condition T(0) = 30°C and with Ta = 21°C. Use T(20) = 28°C to determine the constant k.] 2. (Adapted from Cain et al., 1995) In this problem, we discuss a model for clonal growth in the white clover Trifolium repens. 'II rcpens is a widespread perennial clonal plant species that spreads through stolon growth. (A stolon is a horizontal stern.) By mapping the shape of a clone over time, Cain et al. estimated ’MMWW g 7 stolon elongation and dieback rates as follows. Denote by S(t) tlle stolon length of the clone at time t. Cain et al. observed that the change in stolon length was proportional to the stolon length; that IS, dS —— S dz‘X Introducing the proportionality constant r, called the net growth rate, we find that dS . — = S 8.85: dt r ( ) (a) Suppose that Sf and So are the final and the initial stolon lengths, respectively, and that T denotes the period of observation. Use (8.85) to show that r, the net growth rate, can be estimated from S. 1 r = — In --L T So (Him: Solve the differential equation (8.85) with initial condition S(O) = S0, and use the fact that S(T) = S,.] (b) The net growth rate r is the difference between the stolon elongation rate b and the stolon dieback rate m; that is, r=b-m Let B be the total amount of stolon elongation and D be the total amount of stolon dieback over the observation period of length T. Show that T s B =/ mod: = Erie” - 1) 0 T D =/ mS(t)dt = Effie” ~— 1) 0 (c) Show that B - D = S f — So, and rearrange the equations for B and D in (b) so that you can estimate b and m from r, B, and D; that is, show that r3 r8 Sf—So B—D rD rD Sf—So B—D m: ((1) Explain how B and r can be estimated if S f, S0, and D are known from field measurements. Use your result in (c) to explain how you would then find estimates for b and m. 3. Diversification of Life (Adapted from Benton, 1997, and Walker, 1985) Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models for diversification. They are analogous to models for population growth; however, the quantities involved have a different interpretation. We denote by N (t) the diversification function, which counts the number of taxa as a function of time, and by r the intrinsic rate of diversification. Chapter8 I Review Problems 429 (a) (Exponential Model) This model is described by -—- = r,N (8.86) Solve (8.86) with the initial condition N (0) at time 0, and show that r, can be estimated from [Hint To find (8.87), solve for r in the solution of (886).] (b) (Logistic Growth) This model is described by 51-]! = r,1v(1— (8.88) where K is the equilibrium value. Solve (8.88) with the initial condition N (0) at time 0. and show that r, can be estimated from __ 1 K —- N(O) 1 N(t) for N(t) < K. (c) Assume that N (0) _= 1 and N (10) = 1000. Estimate r, and r, for both K = 1001 and K = 10, 000. ((1) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of [r] to represent underestimates of exponential rates, because some radiation will have followed distinctly sigmoid paths during the interval evaluated. (e) Explain why the exponential model is a good approximation to the logistic model when N / K is small compared with 1. 4. A Simple Model for Photosynthesis of Individual Leaves (Adapted from Horn, 1971) Photosynthesis is a complex mecha- nism; the following model is a very simplified caricature: Suppose that a leaf contains a number of traps that can capture light. If a trap captures light, the trap becomes energized. The energy in the trap can then be used to produce sugar, which causes the energized trap to become unenergized. The number of traps that can become energized is proportional to the number of unenergized traps and the intensity of the light. Denote by T the total number of traps (unenergized and energized) in a leaf, by I the light intensity, and by x the number of energized traps. Then the following differential equation describes how the number of energized traps changes over time: dx 2‘; = —x)1 - [(2): Here, k, and k2 are positive constants. Find all equilibria, and use the eigenvalue approach to study their stability. 5. Gompertz Growth Model This model is sometimes used to study the growth of a population for which the per capita growth rate is density dependent. Denote the size of a population at time t by N(t); then, for N z 0, (171-11! = kN(ln K — In N) with N(O) = No (8.90) (a) Show that ___ __ E ~kt N(t) - Kexp[ (ln No)e ] is a solution of (8.90). To do this, differentiate N (t) with respect to t and show that the derivaive can be written in the form (8.90). Don’t forget to show that N (0) = No. Use a graphing calculator to sketch the graph of N(t) for N0 = 100, k = 2, and K = [000. The function N (t) is called the Gompertz growth curve. 430 Chapter8 I Differential Equations '(b) Use l’Hospital's rule to show that lim NlnN = 0 N-90 and use this equation to show that lim,.,_,0 dN/dt = 0. Are there any other values of N where dN/dt = 0? (c) Sketch the graph of dN/dt as a function of N for k = 2 and K = 1000. Find the equilibria, and use your graph to and discuss their stability. Explain the meaning of K. 6. Island Blogeography Preston (1962) and MacArthur and Wilson (1963) investigated the effect of area on species diversity in oceanic islands. It is assumed that species can immigrate to an island from a species pool of size P and that species on the island can go extinct. We denote the immigration rate by I (S) and the extinction rate by E (S), where S is the number of species on the island. Then the change in species diversity over time is if. = I(S)—E(S) 8.91 d, ( ) For a fixed island, the simplest functional forms for I (S) and E (S) are 1(3) — 1 S (8 92) _ c P . E (S) S (8 93) = m— r P where c, m, and P are positive constants. (in) Find the equilibrium species diversity 3' of (8.91) with I (S) and E (S) given in (8.92) and (8.93). (b) It is reasonable to assume that the extinction rate is a decreasing function of island size. That is, we assume that if A denotes the area of the island, then m is a function of the island size A, with dm/dA < 0. Furthermore, we assume that the immigration rate I does not depend on the size of the island. Use these assumptions to investigate how the equilibrium species diversity changes with island size. (c) Assume that 5(0) = So. Solve (8.91) with I (S) and E(S) as given in (8.92) and (8.93), respectively. (d) Assume that S0 = 0. That is, the island is initially void of species. The time constant T for the system is defined as 5(7) = (1 — (1)5w Show that, under the assumption So = 0, P c+m T: (e) Use the assumptions in (b) and your answer in (d) to investigate the effect of island size on the time constant T; that is, determine whether T(A) is an increasing or decreasing function of A. 7. Chemostat A chemostat is an apparatus for growing bacteria in a medium in which all nutrients but one are available in excess One nutrient. whose concentration can be controlled, is held at a concentration that limits the growth of bacteria. The growth chamber of the chemostat is continually flushed by adding nutrients dissolved in liquid at a constant rate and allowing the liquid in the growth chamber, which contains bacteria, to leave the chamber at the same rate. If X denotes the number of bacteria in the growth chamber, then the growth dynamics of the bacteria are given by dX E— =r(N)X —qX (8.94) where r(N) is the growth rate depending on the nutrient concentration N and q is the input and output flow rate. The equation for the nutrient flow is given by (IN 7; —qNo—4N-V(N)X Note that (8.94) is (8.79) with m = 0, N, = qNo, and a = e = q and that (8.95) is (8.78) with m = 0. (3) Explain in words the meaning of the terms in (8.94) and (8.95). (b) Assume that r(N) is given by the Monod growth function N N =b_—__. r” k+N. where k and b are positive constants. Draw the zero isoclines in the N—X plane, and explain how to find the equilibria (N, X) graphically. (c) Show that a nontrivial equilibrium (an equilibrium for which N and I? are both positive) satisfies (8.95) r(N)—q =0 qNo—qN —-r(N)}? =0 (8.96) (8.97) Show also that (8.96) has a positive solution N if q < b, and find an expression for N. Use this expression and (8.97) to find )2. (d) Assume that q < b. Use your results in (c) to show that J? > 0 A ifN < N0 and N < No ifq < bNo/(k + N0). Furthermore, show A that N is an increasing function of q for q < b. (e) Use your results in (d) to explain why the following is true: As we increase the flow rate q from 0 to bND/(k + N0), the nutrient concentration N increases until it reaches the value No and the number of bacteria decreases to 0. 8. (Adapted from Nee and May, 1992, and Tilman, 1994) In Subsection 8.3.3, we introduced a hierarchical competition model. We will use this model to investigate the effects of habitat destruction on coexistence. We assume that a fraction D of the sites is permanently destroyed. Furthermore, we restrict our discussion to two species and assume that species 1 is the superior and species 2 the inferior competitor. In the case in which both species have the same mortality (m1 = ml). which we set equal to 1, the dynamics are described by (1171 I = CiPiU " Pt - D) — P1 (398) sz W = czp2(1‘ .01- P2 — D) -‘ P2 — CiPipz (3-99) where phi = 1. 2, is the fraction of sites occupied by species 1'. (a) Explain in words the meanings of the different terms in (8.98) and (8.99). (b) Show that A 1 [71 = 1 -' - - D Ci is an equilibrium for species 1, which is in (O, 1), and is stable if D <l~l/c.andc, >1. (c) Assume thatc, > land D < l—-1/c). Show that speciesann invade the nontrivial equilibrium of species 1 [computed in (b)] if (:2 > cfa — D) i l l (d) Assume that c1 = 2 and c; = 5. Then species 1 can survive as long as D < 1/2. Show that the fraction of sites that are occupieczl by species 1 is then 1 é—D forOfiDsi 131= 1 0 for—stl 2 Show also that ‘ 1+D f 0<D<1 =— ~ or - pz 10 “ “2 For D > 1/2, species 1 can no longer persist. Explain why the: 5—“ Chapter8 l Review Problems 431 dynamics for species 2 reduce to in this case. Show, in addition, that the nontrivial equilibrium is of the form 1 1 fi2=1—-1-—Dfor—$D51—- 5 2 5 Plot 131 and p; as functions of D in the same coordinate system. What happens for D > 1 -— 1/5? Use the plot to explain in words how each species is affected by habitat destruction. (e) Repeat (d) for 01 = 2 and c2 = 3. M— ——-————-—_—__________ ...
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