9.4 The Logistic Equation

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1 0 C H P TE R 9 INTRODUCTION TO DIFFERENTIL EQUTIONS 6. y./; y D y C t, y./ D, h D 00 With t 0 D, y 0 D, F.t;y/ D y C t,andh D 00 we compute n t n y n y 0 C hf.t 0 C h=; y 0 C.h=/F.t 0 ;y 0 // D 68.0 y C hf.t C h=; y C.h=/F.t ;y // D y C hf.t C h=; y C.h=/F.t ;y // D y C hf.t C h=; y C.h=/F.t ;y // D y 4 C hf.t 4 C h=; y 4 C.h=/F.t 4 ;y 4 // D y C hf.t C h=; y C.h=/F.t ;y // D ssume that f.t/ is continuous on Œa; b. ShowthatEuler sidpointethodappliedto y D f.t/ with initial condition y.a/ D 0 and time step h D.b a/=n for N steps yields the N th midpoint approximation to Z b y.b/ D f.u/du a For a differential equation of the form y D f.t/,theequationsforeuler smidpointmethodreduceto m k D f t k C h and y k D y k C hf t k C h With a step size of h D.b a/=n, y.b/ D y N.Startingfromy 0 D 0,wecompute y D y 0 C hf t 0 C h D hf t 0 C h y D y C hf t C h D h f t 0 C h C f t C h y D y C hf t C h D h f t 0 C h C f t C h C f t C h y N D y N C hf NX D h f kd0 t N C h D h f t 0 C h C f t C h C f t C h C C f t N C h t k C h Z b Observe this last expression is exactly the N th midpoint approximation to y.b/ D f.u/du. a 9.4 The Logistic Equation Preliminary Questions. Which of the following differential equations is a logistic differential equation? (a) y D y. y / (b) y D y y (c) y D y t (d) y D y. y/ 4 The differential equations in (b) and (d) are logistic equations. The equation in (a) is not a logistic equation because of the y term inside the parentheses on the right-hand side; the equation in (c) is not a logistic equation because of the presence of the independent variable on the right-hand side.

2 SECTION 9.4 The Logistic Equation. Is the logistic equation a linear differential equation? No, the logistic equation is not linear. y D ky y can be rewritten y D ky k y and we see that a term involving y occurs.. Is the logistic equation separable? Yes, the logistic equation is a separable differential equation. Exercises. Find the general solution of the logistic equation y D y y Then find the particular solution satisfying y.0/ D. y D y. y=/ is a logistic equation with k D and D ; therefore,thegeneralsolutionis y D e t =C The initial condition y.0/ D allows us to determine the value of C The particular solution is then D =C I C D I so C D y D. Find the solution of y D y. y/, y.0/ D 0. By rewriting 0 C D e t C e t y. y/ as 6y y ; we identifythegivendifferentialequationasalogisticequationwithk D 6 and D. Thegeneralsolutionistherefore y D e 6t =C The initial condition y.0/ D 0 allows us to determine the value of C The particular solution is then 0 D =C I C D 0 I so C D 0 7 y D D e 6t 0 7e 6t. Let y.t/ be a solution of y D 0y. 0y/ such that y.0/ D 4. Determine lim y.t/ without finding y.t/ explicitly. t This is a logistic equation with k D and D, sothecarryingcapacityis. Thustherequiredlimitis. 4. Let y.t/ be a solution of y D y. y=/.statewhethery.t/ is increasing, decreasing, or constant in the following cases (a) y.0/ D (b) y.0/ D (c) y.0/ D 8 This is a logistic equation with k D D. (a) 0<y.0/<,soy.t/ is increasing and approaches asymptotically. (b) y.0/ D ; thisrepresentsastableequilibriumandy.t/ is constant. (c) y.0/ >,soy.t/ is decreasing and approaches asymptotically.. populationofsquirrelslivesinaforestwithacarryingcapacityof000.ssumelogisticgrowthwithgrowthconstant k D 06 yr. (a) Find a formula for the squirrel population P.t/,assuminganinitialpopulationof00squirrels. (b) How long will it take for the squirrel population to double?

3 C H P TE R 9 INTRODUCTION TO DIFFERENTIL EQUTIONS (a) Since k D 06 and the carrying capacity is D 000, thepopulationp.t/ of the squirrels satisfies the differential equation with general solution P 0.t/ D 06P.t/. P.t/=000/; P.t/ D 000 e 06t =C The initial condition P.0/ D 00 allows us to determine the value of C The formula for the population is then 00 D 000 =C I C D 4I so C D P.t/ D 000 C e 06t (b) The squirrel population will have doubled at the time t where P.t/ D 000. Thisgives 000 D 000 C e 06t I C e 06t D I so t D ln 8 It therefore takes approximately.8 years for the squirrel population to double. 6. The population P.t/of mosquito larvae growing in a tree hole increases according to the logistic equation with growth constant k D 0 day and carrying capacity D 00. (a) Find a formula for the larvae population P.t/,assuminganinitialpopulationofP 0 D 0 larvae. (b) fter how many days will the larvae population reach 00? (a) Since k D 0 and D 00, thepopulationofthelarvaesatisfiesthedifferentialequation with general solution P 0.t/ D 0P.t/. P.t/=00/; P.t/ D 00 e 0t =C The initial condition P.0/ D 0 allows us to determine the value of C The particular solution is then 0 D 00 =C I C D 0I so C D 9 P.t/ D 00 C 9e 0t (b) The population will reach 00 after t days, where P.t/ D 00. Thisgives 00 D 00 C 9e 0t I C 9e 0t D I so t D 0 ln 6 97 It therefore takes approximately.97 days for the larvae to reach 00 in number. 7. Sunset Lake is stocked with 000 rainbow trout, and after year the population has grown to 400. ssuming logistic growth with a carrying capacity of 0,000, find the growth constant k (specify the units) and determine when the population will increase to 0,000. Since D 0;000, thetroutpopulationp.t/satisfies the logistic equation P 0.t/ D kp.t/. P.t/=0;000/; with general solution P.t/ D 0;000 e kt =C

4 SECTION 9.4 The Logistic Equation The initial condition P.0/ D 000 allows us to determine the value of C 000 D 0;000 =C I C D 0I so C D 9 fter one year, we know the population has grown to 400. Let s measure time in years. Then 400 D 0;000 C 9e k C 9e k D 40 9 e k D 8 k D ln years The population will increase to 0,000 at time t where P.t/ D 0;000. Thisgives 0;000 D C 9e 0960t D 0;000 C 9e 0960t e 0960t D 9 t D ln 9 9 years Spread of a Rumor rumorspreadsthroughasmalltown.lety.t/ be the fraction of the population that has heard the rumor at timet and assume that the rate at which the rumor spreads is proportional to the product of the fraction y of the population that has heard the rumor and the fraction y that has not yet heard the rumor. (a) Write down the differential equation satisfied by y in terms of a proportionality factor k. (b) Find k (in units of day ), assuming that 0% of the population knows the rumor at t D 0 and 40% knows it at t D days. (c) Using the assumptions of part (b), determine when 7% of the population will know the rumor. (a) y 0.t/ is the rate at which the rumor is spreading, in percentage of the population per day. By the description given, the rate satisfies y 0.t/ D ky. y/; where k is a constant of proportionality. (b) The equation in part (a) is a logistic equation with constant k and capacity (no more than 00% ofthepopulationcanhear the rumor). Thus, y takes the form e kt =C The initial condition y.0/ D 0 allows us to determine the value of C 0 D =C I C D 0I so C D 9 The condition y./ D now allows us to determine the value of k D C 9e k I C 9e k D I so k D ln days The particular solution of the differential equation for y is then (c) If 7% ofthepopulationknowstherumorattimet,wehave C 9e 0896t 4 D C 9e 0896t C 9e 0896t D 4

5 4 C H P TE R 9 INTRODUCTION TO DIFFERENTIL EQUTIONS t D ln Thus, 7% of the population knows the rumor after approximately.67 days. 9. rumorspreadsthroughaschoolwith000students.t8,80studentshaveheardtherumor,andbynoon,halftheschool has heard it. Using the logistic model of Exercise 8, determine when 90% of the students will have heard the rumor. Let y.t/ be the proportion of students that have heard the rumor at a time t hours after 8.Inthelogisticmodelof Exercise 8, we have a capacity of D (00% ofstudents)andanunknowngrowthfactorofk.hence, e kt =C The initial condition y.0/ D 008 allows us to determine the value of C sothat D =C I C D I so C D The condition y.4/ D 0 now allows us to determine the value of k C e kt D C e 4k I C e 4k D 4I so k D 4 ln 0606 hours 90% ofthestudentshaveheardtherumorwhen 09. Thus 9 0 D C e 0606t C e 0606t D 0 9 t D 07 ln 76 hours 0606 Thus, 90% of the students have heard the rumor after 7.6 hours, or at 6 P. 0. simplermodelforthespreadofarumorassumesthattherateatwhichtherumorspreadsisproportional(withfactor k)tothefractionofthepopulationthathasnotyetheardtherumor. (a) Compute the solutions to this model and the model of Exercise 8 with the values k D 09 and y 0 D 0. (b) Graph the two solutions on the same axis. (c) Which model seems more realistic? Why? (a) Let y.t/ denote the fraction of a population that has heard a rumor, and suppose the rumor spreads at a rate proportional to the fraction of the population that has not yet heard the rumor. Then y 0 D k. y/; for some constant of proportionality k.separatingvariablesandintegratingbothsidesyields Thus, dy y D k dt ln j yj Dkt C C e kt ; where D e C is an arbitrary constant. The initial condition y.0/ D 0 allows us to determine the value of 0 D so D 09 With k D 09, wehave 09e 09t. Using the model from Exercise 8 with k D 09 and y.0/ D 0, wefind C 9e 09t

6 SECTION 9.4 The Logistic Equation (b) The figure below shows the solutions from part (a) the solid curve corresponds to the model presented in this exercise while the dashed curve corresponds to the model from Exercise (c) The model from Exercise 8 seems more realistic because it predicts the rumor starts spreading slowly, picks up speed and then levels off as we near the time when the entire population has heard the rumor.. Let k D and D in the logistic equation. (a) Find the solutions satisfying y.0/ D 0 and y.0/ D. (b) Find the time t when y.t/ D. (c) When does y.t/ become infinite? The general solution of the logistic equation with k D and D is (a) Given y.0/ D 0, wefindc D 0 9, and y.t/ D On the other hand, given y.0/ D,wefindC D, and e t =C 0 0 D 9 e t 0 9e t (b) From part (a), we have Thus, y.t/ D when (c) From part (a), we have Thus, y.t/ becomes infinite when D y.t/ D y.t/ D e t 0 0 9e t 0 0 9e t I 0 9e t D I so t D ln 9 8 y.t/ D e t e t D 0 or t D ln. tissueculturegrowsuntilithasamaximumareaof cm.thearea.t/ of the culture at time t may be modeled by the differential equation D k p 7 where k isagrowthconstant. (a) Show that if we set D u,then u D k u Then find the general solution using separation of variables. (b) Show that the general solution to Eq. (7) is.t/ D p Ce.k= /t Ce.k=p/t C

7 6 C H P TE R 9 INTRODUCTION TO DIFFERENTIL EQUTIONS (a) Let D u.thisgives D u u,sothateq.(7)becomes u u D ku u D k u u Now, rewrite du dt D k u as du u = D k dt The partial fraction decomposition for the term on the left-hand side is p u = D p C p ; C u u soafterintegratingbothsides,weobtain p p C u ln p D ˇ u ˇˇˇˇˇ kt C C Thus, p C u p u D Ce.k=p /t u.ce.k=p /t C / D p.ce.k=p /t / and u D p p Ce.k= /t Ce.k=p/t C (b) Recall D u.therefore,.t/ D p Ce.k= /t Ce.k=p/t C. In the model of Exercise, let.t/ be the area at time t (hours) of a growing tissue culture with initial size.0/ D cm,assumingthatthemaximumareais D 6 cm and the growth constant is k D 0. (a) Find a formula for.t/. Note The initial condition is satisfied for two values of the constant C.ChoosethevalueofC for which.t/ is increasing. (b) Determine the area of the culture at t D 0 hours. (c) Graph the solution using a graphing utility. (a) From the values for and k we have.t/ D 6 Ce t=40 Ce t=40 C and the initial condition then gives us.0/ D D 6 Ce 0=40 Ce 0=40 C so, simplifying, D 6 C ) C C C C D 6C C C 6 ) C 4C C D 0 C C

8 SECTION 9.4 The Logistic Equation 7 and thus C D or C D. The derivative of.t/ is 0.t/ D 6Cet=40.Ce t=40 C /.Cet=40 / For C D =, 0.t/ can be negative, while for C D =, itisalwayspositive.soletc D =. (b) From part (a), we have and.0/. (c).t/ D 6 y et=40 et=40 C t Show that if a tissue culture grows according to Eq. (7), then the growth rate reaches a maximum when D =. ccording to Eq. (7), the growth rate of the tissue culture is k p. /. Therefore d k p D d k = k= = D k = D 0 when D =. Becausethegrowthrateiszerofor D 0 and for D and is positive for 0<<,itfollowsthatthe maximum growth rate occurs when D =.. In 7, Benjamin Franklin predicted that the U.S. population P.t/ would increase with growth constant k D 008 year. ccording to the census, the U.S. population was million in 800 and 76 million in 900. ssuming logistic growth with k D 008, findthepredictedcarryingcapacityfortheu.s.population.hint Use Eqs. () and (4) to show that P.t/ P.t/ D P 0 P 0 ekt ssuming the population grows according to the logistic equation, P.t/ P.t/ D Cekt But so C D P 0 P 0 ; P.t/ P.t/ D P 0 P 0 ekt Now, let t D 0 correspond to the year 800. Then the year 900 corresponds to t D 00, and withk D 008, we have Solving for, wefind D e.008/.00/ D.e8 / 76 e Thus, the predicted carrying capacity for the U.S. population is approximately 94 million.

9 8 C H P TE R 9 INTRODUCTION TO DIFFERENTIL EQUTIONS 6. Reverse Logistic Equation Consider the following logistic equation (with k; B > 0) dp dt D kp P B 8 (a) Sketch the slope field of this equation. (b) The general solution is P.t/ D B=. e kt =C/, wherec is a nonzero constant. Show that P.0/ > B if C > and 0<P.0/<Bif C<0. (c) Show that Eq. (8) models an extinction explosion population. That is, P.t/ tends to zero if the initial population satisfies 0<P.0/<B,andittendsto after a finite amount of time if P.0/ > B. (d) Show that P D 0 is a stable equilibrium and P D B an unstable equilibrium. (a) The slope field of this equation is shown below. 0 0 y t (b) Suppose that C>0.Then C <, C >, and P.0/ D B C > B On the other hand, if C<0,then C >, 0< C <, and 0<P.0/D B C < B (c) From part (b), 0<P.0/<Bwhen C<0.Inthiscase, e kt =C is never zero, but ekt C as t.thus,p.t/ 0 as t.ontheotherhand,p.0/ > B when C>0.Inthiscase e kt =C D 0 when t D k ln C. Thus, P.t/ as t ln C k (d) Let F.P/ D kp P B Then, F 0.P / D kc kp B. Thus, F 0.0/ D k<0,andf 0.B/ D kck D k>0,sop D 0 is a stable equilibrium and P D B is an unstable equilibrium. Further Insights and Challenges InExercises7and8,lety.t/ be a solution of the logistic equation where >0and k>0. dy dt D ky y 9

10 SECTION 9.4 The Logistic Equation 9 7. (a) Differentiate Eq. (9) with respect to t and use the Chain Rule to show that d y dt D k y y y (b) Show that y.t/ is concave up if 0<y<=and concave down if = < y <. (c) Show that if 0<y.0/<=,theny.t/ has a point of inflection at y D = (Figure ). (d) ssume that 0<y.0/<=.Findthetimet when y.t/ reaches the inflection point. y Inflection point y(0) FIGURE ninflectionpointoccursaty D = in the logistic curve. t (a) The derivative of Eq. (9) with respect to t is y 00 D ky 0 kyy0 D ky0 y D k y ky y D k y y y (b) If 0<y<=, y and y are both positive, so y00 >0.Therefore,y is concave up. If = < y <, y > 0,but y < 0,soy00 <0,soy is concave down. (c) If y 0 <, y grows and lim. If0<y<=, y is concave up at first. Once y passes =, y becomes concave t down, so y has an inflection point at y D =. (d) The general solution to Eq. (9) is thus, y D = when Now, C D y 0 =.y 0 /,so y D e kt =C D e kt =C I D e kt =C t D k ln. C/ t D k ln y 0 y 0 D k ln y 0 y 0 8. Let y D e kt =C be the general nonequilibrium Eq. (9). If y.t/ has a vertical asymptote at t D t b,thatis,if lim,wesaythatthesolution blowsup att D t b. tt b (a) Show that if 0<y.0/<,theny does not blow up at any time t b. (b) Show that if y.0/ >,theny blows up at a time t b,whichisnegative(andhencedoesnotcorrespondtoarealtime). (c) Show that y blows up at some positive time t b if and only if y.0/ < 0 (and hence does not correspond to a real population). (a) Let y.0/ D y 0.Fromthegeneralsolution,wefind y 0 D =C I C D y 0 I so C D y 0 y 0 If y 0 <,thenc<0,andthedenominatorinthegeneralsolution, e kt =C,isalwayspositive.Thus,when0<y.0/<, y does not blow up at any time.

11 0 C H P TE R 9 INTRODUCTION TO DIFFERENTIL EQUTIONS (b) e kt =C D 0 when C D e kt.solvingfort we find t D ln C k Because C D y 0 y 0 and y 0 >,itfollowsthatc>,andthus,lnc>0.therefore,y blows up at a time which is negative. (c) Suppose that y blows up at some t b >0.Frompart(b),weknowthat t b D ln C k Thus, in order for t b to be positive, we must have ln C<0,whichrequiresC<.Now, so t b >0if and only if This last inequality holds if and only if y 0 D y.0/ < 0. C D y 0 y 0 ; y 0 y 0 < or equivalently y 0 D > y 0 y 0 9. First-Order Linear Equations Preliminary Questions. Which of the following are first-order linear equations? (a) y 0 C x y D (b) y 0 C xy D (c) x y 0 C y D e x (d) x y 0 C y D e y The equations in (a) and (c) are first-order linear differential equations. The equation in (b) is not linear because of the y factor in the second term on the left-hand side of the equation; the equation in (d) is not linear because of the e y term on the right-hand side of the equation.. If.x/ is an integrating factor for y 0 C.x/y D B.x/,then 0.x/ is equal to (choose the correct answer) (a) B.x/ (b).x/.x/ (c).x/ 0.x/ (d).x/b.x/ The correct answer is (b).x/.x/. Exercises. Consider y 0 C x y D x. (a) Verify that.x/ D x is an integrating factor. (b) Show that when multiplied by.x/,thedifferentialequationcanbewritten.xy/ 0 D x 4. (c) Conclude that xy is an antiderivative of x 4 and use this information to find the general solution. (d) Find the particular solution satisfying y./ D 0. (a) The equation is of the form y 0 C.x/y D B.x/ for.x/ D x and B.x/ D x.bytheorem,.x/ is defined by (b) When multiplied by.x/,theequationbecomes Now, xy 0 C y D xy 0 C.x/ 0 y D.xy/ 0,so.x/ D er.x/ dx D e ln x D x xy 0 C y D x 4.xy/ 0 D x 4