Mah 337 - Elemenary Differenial Equaions Joseph M. Mahaffy, jmahaffy@sdsu.edu Deparmen of Mahemaics and Saisics Dynamical Sysems Group Compuaional Sciences Research Cener San Diego Sae Universiy San Diego, CA 92182-772 hp://jmahaffy.sdsu.edu Spring 219 Ouline 1 2 3 4 5 1/5) 2/5) Previously showed ha for Malhusian growh or Radioacive decay he linear differenial equaion: has he soluion: = a y wih y) = y, y) = y e a. More generally, we have he following soluion: Mehod General Soluion o Linear Growh and Decay Models) Consider The soluion is = a y wih y ) = y. y) = y e a ). Example: Linear Decay Model Example: Linear Decay Model: Consider The soluion is =.3 y wih y4) = 12 y) = 12 e.3 4) This soluion shows a subsance decaying a a rae k =.3 saring wih 12 unis of subsance y. However, he soluion is shifed horizonally) by 4 unis of ime. 3/5) 4/5)
1 A diagram of he Modeling Process Real-world phenomenon Deermine dependen and independen variables Assign symbols o variables, Choose sensible unis of measuremen for variables Apply principles, laws and assumpions Mahemaical represenaion : Afer a murder or deah by oher causes), he forensic scienis akes he emperaure of he bo DATA observaions, experimens, and measuremens. Refine model Analysis Compuing Graphics Laer he emperaure of he bo is aken again o find he rae a which he bo is cooling Two or more) daa poins are used o exrapolae back o when he murder occurred Compare model predicions wih daa. Qualiaive predicions Quaniaive predicions Mahemaical inferences This propery is known as 5/5) 6/5) 2 saes ha he rae of change in emperaure of a cooling bo is proporional o he difference beween he emperaure of he bo and he surrounding environmenal emperaure If T ) is he emperaure of he bo, hen i saisfies he differenial equaion dt = kt ) T e) wih T ) = T The parameer k is dependen on he specific properies of he paricular objec bo in his case) T e is he environmenal emperaure T is he iniial emperaure of he objec Murder Example 1 Murder Example Suppose ha a murder vicim is found a 8:3 am The emperaure of he bo a ha ime is 3 C Assume ha he room in which he murder vicim lay was a consan 22 C Suppose ha an hour laer he emperaure of he bo is 28 C Normal emperaure of a human bo when i is alive is 37 C Use his informaion o deermine he approximae ime ha he murder occurred 7/5) 8/5)
Murder Example 2 Soluion: From he model for and he informaion ha is given, if we se = o be 8:3 am, hen we solve he iniial value problem dt = kt ) 22) wih T ) = 3 Make a change of variables z) = T ) 22 Then z ) = T ), so he differenial equaion above becomes dz = kz), wih z) = T ) 22 = 8 This is he radioacive decay problem ha we solved The soluion is z) = 8 e k Murder Example 3 Soluion con): From he soluion z) = 8 e k, we have z) = T ) 22, so T ) = z) + 22 T ) = 22 + 8 e k One hour laer he bo emperaure is 28 C Solving Thus, k = ln 4 3) =.2877 T 1) = 28 = 22 + 8 e k 6 = 8 e k or e k = 4 3 9/5) 1/5) Murder Example 4 Soluion con): I only remains o find ou when he murder occurred A he ime of deah, d, he bo emperaure is 37 C T d ) = 37 = 22 + 8 e k d Murder Example 5 Graph of Bo Temperaure over ime 4 38 36 Bo Temperaure Thus, 8 e k d = 37 22 = 15 or e k d = 15 8 = 1.875 This gives k d = ln1.875) or d = ln1.875) k = 2.19 The murder occurred abou 2 hours 11 minues before he bo was found, which places he ime of deah around 6:19 am 11/5) T) 34 32 3 28 26 24 22 Time of deah, d Room Temperaure 2 8 6 4 2 2 4 6 8 1 12 12/5)
Soluion of General Linear Model 1 Soluion of General Linear Model: Consider he Linear Model Rewrie equaion as = a y + b wih y ) = y = a y + b ) a Make he subsiuion z) = y) + b dz a, so = and z ) = y + b a I follows ha dz = a z wih z ) = y + b a Soluion of General Linear Model 2 The linear growh model given by dz = a z wih z ) = y + b a, has been solved by our previous mehod. The soluion is: z) = y + b ) e a ) = y) + b a a. I follows ha he soluion, y) is y) = y + b ) e a ) b a a. 13/5) 14/5) Soluion of General Linear Model 3 The linear differenial equaion saisfies: = a y + b = a y + b ) a Mehod Soluion of General ) Consider he linear differenial equaion = a y + b ) wih y ) = y. a Wih he subsiuion z) = y) + b a, we obain he soluion: y) = y + b ) e a ) b a a. Example of Linear Model 1 Example of Linear Model Consider he Linear Model Rewrie equaion as = 5.2 y wih y3) = 7 =.2y 25) Make he subsiuion z) = y) 25, so dz = dz =.2 z wih z3) = 18 and z3) = 18 This mehod produces a verical shif of he soluion. 15/5) 16/5)
Example of Linear Model 2 Example of Linear Model The subsiued model is dz =.2 z wih z3) = 18 Example of Linear Model 3 The linear differenial equaion was ransformed ino he IVP: =.2y 25), wih y3) = 7 The graph is given by 3 Thus, z) = 18 e.2 3) = y) 25 2 The soluion is y) = 25 18 e.2 3) y) 1 17/5) 1 1 2 3 4 5 18/5) How do we make he previous graph? MaLab is a powerful sofware for mahemaics, engineering, and he sciences MaLab sands for Marix Laboraory Designed for easy managing of vecors, marices, and graphics Valuable subrouines and packages for specialy applicaions I is a necessary ool for anyone in Applied Mahemaics Auonomous Differenial Equaion The general firs order differenial equaion saisfies = f, y). A very imporan se of DEs ha we su are called Auonomous Differenial Equaions Definiion Auonomous Differenial Equaion) A firs order auonomous differenial equaion has he form = fy). The funcion, f, depends only on he dependen variable. 19/5) 2/5)
Classificaion of Equilibria The firs sep of any qualiaive analysis is finding equilibrium soluions There are a variey of local behaviors near an equilibrium, y e Definiion Equilibrium Soluions) Consider auonomous DE = fy). If y) = c is a consan soluion or equilibrium soluion o his DE, hen =. Therefore he consan c is a soluion of he algebraic equaion fy) =. Equilibrium soluions are also referred o as fixed poins, saionary poins, or criical poins. 21/5) 1 An asympoically sable equilibrium, ofen referred o as an aracor or sink has any nearby soluion approach y e as 2 An unsable equilibrium, ofen referred o as a repeller or source has any nearby soluion leave a region abou y e as 3 A neurally sable equilibrium has any soluion say nearby he equilibrium, bu no approach he equilibrium y e as 4 A semi-sable equilibrium in 1D) has soluions on one side of y e approach y e as, while soluions on he oher side of y e diverge away from y e 22/5) Taylor s Theorem Linearizaion Le y e be an equilibrium soluion of he DE so fy e ) =. = fy), The nex sep is finding he local behavior near each of he equilibrium soluions of he DE = fy). Theorem Taylor Series) If for a range abou y e, he funcion, f, has infiniely many derivaives a y e, hen fy) saisfies he Taylor Series fy) = fy e ) + f y e )y y e ) + f y e ) y y e ) 2 +... 2! Since fy e ) =, hen he dominae erm near y e is he linear erm f y e )y y e ). 23/5) Theorem Linearizaion abou an Equilibrium Poin) Le y e be an equilibrium poin of he DE above and assume ha f has a coninuous derivaive near y e. If f y e ) <, hen y e is an asympoically sable equilibrium. If f y e ) >, hen y e is an unsable equilibrium. If f y e ) =, hen more informaion is needed o classify y e. 24/5)
Model 1 Model Consider he logisic growh equaion: dp = fp ) =.5P Equilibria saisfy fp e ) =, so 1 P ) 2 P e =, he exincion equilibrium P e = 2, he carrying capaciy I is easy o compue f P ) =.5.1P 2 Since f ) =.5 >, P e = is an unsable equilibrium or repeller Since f 2) =.5 <, P e = 2 is a sable equilibrium or aracor Model 2 Geomeric Local Analysis: Equilibria are P e = and P e = 2 The graph of fp ) gives more informaion To he lef of P e =, fp ) < Since dp = fp ) <, P ) is decreasing Noe ha his region is ouside he region of biological significance For < P < 2, fp ) > Since dp = fp ) >, P ) is increasing Populaion monoonically growing in his area For P > 2, fp ) < Since dp = fp ) <, P ) is decreasing Populaion monoonically decreasing in his region 25/5) 26/5) Model 3 Phase Porrai Use he above informaion o draw a Phase Porrai of he behavior of his differenial equaion along he P -axis The behavior of he differenial equaion is denoed by arrows along he P -axis When fp ) <, P ) is decreasing and we draw an arrow o he lef When fp ) >, P ) is increasing and we draw an arrow o he righ Equilibria A solid do represens an equilibrium ha soluions approach or sable equilibrium An open do represens an equilibrium ha soluions go away from or unsable equilibrium Model 4 Phase Porrai: Consiss of P -axis, arrows, and equilibria. fp) 3 25 2 15 1 5 5 < > > > > < 5 5 1 15 2 25 populaion P) 27/5) 28/5)
Model 5 Model 7 Diagram of Soluions for Logisic Growh Model 25 2 15 P) 1 5 Logisic Growh Model 5 1 15 2 29/5) Summary of Qualiaive Analysis Graph shows soluions eiher moving away from he equilibrium a P e = or moving oward P e = 2 Soluions are increasing mos rapidly where fp ) is a a maximum Phase porrai shows direcion of flow of he soluions wihou solving he differenial equaion Soluions canno cross in he P -plane Phase Porrai analysis Behavior of a scalar DE found by jus graphing funcion Equilibria are zeros of funcion Direcion of flow/arrows from sign of funcion Sabiliy of equilibria from wheher arrows poin oward or away from he equilibria 3/5) 1 2 Consider he differenial equaion: Find all equilibria dx = 2 sinπx) Deermine he sabiliy of he equilibria Skech he phase porrai Show ypical soluions 31/5) For he sine funcion below: dx = 2 sinπx) The equilibria saisfy 2 sinπx e ) = Thus, x e = n, where n is any ineger The sine funcion passes from negaive o posiive hrough x e =, so soluions move away from his equilibrium The sine funcion passes from posiive o negaive hrough x e = 1, so soluions move oward his equilibrium From he funcion behavior near equilibria All equilibria wih x e = 2n even ineger) are unsable All equilibria wih x e = 2n + 1 odd ineger) are sable 32/5)
3 Phase Porrai: Since 2 sinπx) alernaes sign beween inegers, he phase porrai follows below: 4 Diagram of Soluions for Sine Model 5 2 4 2sinπx) 1 > < > < > x) 3 2 1 1 2 1 2 3 4 5 x 33/5).1.2.3.4.5.6 34/5) : Inroducion 1 The shell of a snail exhibis chiraliy, lef-handed sinisral) or righ-handed dexral) coil relaive o he cenral axis The Indian conch shell, Turbinella pyrum, is primarily a righ-handed gasropod [1] The lef-handed shells are exceedingly rare The Indians view he rare shells as very holy The Hindu god Vishnu, in he form of his mos celebraed avaar, Krishna, blows his sacred conch shell o call he army of Arjuna ino bale So why does naure favor snails wih one paricular handedness? Gould noes ha he vas majoriy of snails grow he dexral form. Clifford Henry Taubes [2] gives a simple mahemaical model o predic he bias of eiher he dexral or sinisral forms for a given species Assume ha he probabiliy of a dexral snail breeding wih a sinisral snail is proporional o he produc of he number of dexral snails imes sinisral snails Assume ha wo sinisral snails always produce a sinisral snail and wo dexral snails produce a dexral snail Assume ha a dexral-sinisral pair produce dexral and sinisral offspring wih equal probabiliy By he firs assumpion, a dexral snail is wice as likely o choose a dexral snail han a sinisral snail Could use real experimenal verificaion of he assumpions [1] S. J. Gould, Lef Snails and Righ Minds, Naural Hisory, April 1995, 1-18, and in he compilaion Dinosaur in a Haysack 1996) 35/5) [2] C. H. Taubes, Modeling Differenial Equaions in Biology, Prenice Hall, 21. 36/5)
2 3 Taubes Snail Model Le p) be he probabiliy ha a snail is dexral A model ha qualiaively exhibis he behavior described on previous slide: dp = αp1 p) p 1 ), p 1, 2 where α is some posiive consan Wha is he behavior of his differenial equaion? Wha does is soluions predic abou he chiraliy of populaions of snails? 37/5) Taubes Snail Model This differenial equaion is no easy o solve exacly Qualiaive analysis echniques for his differenial equaion are relaively easily o show why snails are likely o be in eiher he dexral or sinisral forms The snail model: dp = fp) = αp1 p) p 1 ), p 1, 2 Equilibria are p e =, 1 2, 1 fp) < for < p < 1 2, so soluions decrease fp) > for 1 2 < p < 1, so soluions increase The equilibrium a p e = 1 2 is unsable The equilibria a p e = and 1 are sable 38/5) 4 Phase Porrai: dp.1 = αp1 p) p 1 ) 2 Phase Porrai for Snail Model α =1.5) 4 Diagram of Soluions for Snail Model 1.8 Snail Model αp1 p)p 1/2).5 < < < > > >.6 p).4.5.2.1.2.4.6.8 1 p 39/5) 2 4 6 8 1 4/5)
5 1 Snail Model - Summary Figures show he soluions end oward one of he sable equilibria, p e = or 1 When he soluion ends oward p e =, hen he probabiliy of a dexral snail being found drops o zero, so he populaion of snails all have he sinisral form When he soluion ends oward p e = 1, hen he populaion of snails virually all have he dexral form This is wha is observed in naure suggesing ha his model exhibis he behavior of he evoluion of snails This does no mean ha he model is a good model! I simply means ha he model exhibis he basic behavior observed experimenally from he biological experimens 41/5) Thick-Billed Parro: Rhynchopsia pachyrhycha 42/5) 2 3 Thick-Billed Parro: Rhynchopsia pachyrhycha A gregarious monane bird ha feeds largely on conifer seeds, using is large beak o break open pine cones for he seeds These birds used o fly in huge flocks in he mounainous regions of Mexico and Souhwesern U. S. Largely because of habia loss, hese birds have los much of heir original range and have dropped o only abou 15 breeding pairs in a few large colonies in he mounains of Mexico The pressures o log heir habia pus his populaion a exreme risk for exincion Thick-Billed Parro: Rhynchopsia pachyrhycha The populaions of hese birds appear o exhibi a propery known in ecology as he Allee effec These parros congregae in large social groups for almos all of heir aciviies The large group allows he birds many more eyes o wach ou for predaors When he populaion drops below a cerain number, hen hese birds become easy arges for predaors, primarily hawks, which adversely affecs heir abiliy o susain a breeding colony 43/5) 44/5)
4 5 : Suppose ha a populaion su on hick-billed parros in a paricular region finds ha he populaion, N), of he parros saisfies he differenial equaion: dn = N r an b) 2), where r =.4, a = 1 8, and b = 22 Find he equilibria for his differenial equaion Deermine he sabiliy of he equilibria Draw a phase porrai for he behavior of his model Describe wha happens o various saring populaions of he parros as prediced by his model 45/5) Equilibria: Se he righ side of he differenial equaion equal o zero: N e r ane b) 2) = One soluion is he rivial or exincion equilibrium, N e = When r an e b) 2) =, hen N e b) 2 = r r or N e = b ± a a Three disinc equilibria unless r = or b = r/a Wih he parameers r =.4, a = 1 8, and b = 22, he equilibria are N e = N e = 2 42 46/5) 6 7 Phase Porrai: Graph of righ hand side of differenial equaion showing equilibria and heir sabiliy Soluions: For dn = N r an b) 2) 1 1 zoom near origin) 5 dn/ 5 5 > > > < dn/.5.5 > < < > 1 2 3 N 4 3 N) 2 1 3 2 1 N) 1 2 zoom near origin) 2 4 6 8 1 2 3 4 5 N 2 4 6 8 1 47/5) 48/5)
8 Maple Commands for Direcion Fields Inerpreaion: Model of From he phase porrai, he equilibria a 42 and are sable The hreshold equilibrium a 2 is unsable If he populaion is above 2, i approaches he carrying capaciy of his region wih he sable populaion of 42 If he populaion falls below 2, he model predics exincion, N e = This agrees wih he descripion for hese social birds, which require a criical number of birds o avoid predaion Below his criical number, he predaion increases above reproducion, and he populaion of parros goes o exincion If he parro populaion is larger han 42, hen heir numbers will be reduced by sarvaion and predaion) o he carrying capaciy, N e = 42 49/5) wihdeools): de := diffp ), ) =.5 P ) 1 1 2 P )) ; DEplode, P ), =..1, P =..25, [[P ) = ], [P ) = 1], [P ) = 2], [P ) = 25]], color = blue, linecolor = ); 5/5)