> #OK to post #Tim Nasralla, hw24, 11/29/21 First Written: Nov. 2021 This is DMB.txt, A Maple package to explore Dynamical models in Biology (both discrete and continuous) accompanying the class Dynamical Models in Biology, Rutgers University. Taught by Dr. Z. (Doron Zeilbeger) The most current version is available on WWW at: http://sites.math.rutgers.edu/~zeilberg/tokhniot/DMB.txt . Please report all bugs to: DoronZeil at gmail dot com . For general help, and a list of the MAIN functions, type "Help();". For specific help type "Help(procedure_name);" ------------------------------ For a list of the supporting functions type: Help1(); For help with any of them type: Help(ProcedureName); ------------------------------ For a list of the functions that give examples of Discrete-time dynamical systems (some famous), type: HelpDDM(); For help with any of them type: Help(ProcedureName); ------------------------------ For a list of the functions continuous-time dynamical systems (some famous) type: HelpCDM(); For help with any of them type: Help(ProcedureName); ------------------------------ > #Question 3ii: Find when the differential equation (found in pdf) reaches the > ground using dsolve > dsolve({D(D(x))(t)-2*D(x)(t)+9.81=0,x(0)=100,D(x)(0)=0},x(t)) 981 981 40981 x(t) = - --- exp(2 t) + --- t + ----- 400 200 400 > evalf(solve(0 = -(981*exp(2*t))/400 + (981*t)/200 + 40981/400, t)) -20.88735984, 1.90989362 > #Since time cannot be negative, the solution for when the ball reaches the > ground would approximately be 1.91 seconds > #Question 5: Using the Orb function, find stable fixed points of the given > discrete time dynamical systems. #i, f(x) = x+1 / x+2 > Orb([(x+1)/(x+2)], [x], [0.5], 1000, 1010) [[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888]] > Orb([(x+1)/(x+2)], [x], [0.7], 1000, 1010) [[0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888], [0.6180339888]] > Orb([(x+1)/(x+2)], [x], [0], 1000, 1010) [[4224696333392304878706725602341482782579852840250681098010280137314308584370\ 1307072241235996391415110884460875389096036076401947116435960292719833125987\ 3732625355580260699158591522949245390499872225679531698287448247299226390183\ 3716778060607011615497886719879858311468870876264597369086722884023654422295\ 2433479644801395153495629720876526560695298064998419774487201556128026654045\ 54171717881930324025204312082516817125/6835702259575806647045396549170580107\ 0554080293655245654075533677980824544080540149545343189531138027266037267695\ 2344747823819219271452667793994333830610140510541481970566409090181363729645\ 3767095528104868264704914433529355579148731044685634135487735897954629842516\ 9471014942535758696998934009765395457402148198191519520850895384229545651467\ 20383752121972115725761141759114990448978941370030912401573418221496592822626 ], [110603985929681115257521221515120628896352608696162056634178335051123910\ 3877818472217865791859225531381504981430843305108587838690435812270721192665\ 0904838731358970622312655676817043129750358765817784900185247579396906521619\ 4809824478227462411471032337846745097008284159723705181732387866162850001939\ 6803545816778363209160043910139504221780278991355862181409317448129737191765\ 5853533113087912842725598622533579109639751/17896100852543918172797518700682\ 6429966906688989817302288253868729104734931862387371331922375453691165416535\ 4107795649856411657909707264938515186998921094013646438544201831976771885676\ 7046812532913313005053512284311340050975060131178867431875282590969682629139\ 5433453630738647717491084865096859767335137756729876027840436855286398179967\ 8294951029731074378620890024243913103264630251205445794375512717204075507570\ 2462377], [28956499445512029698549640852194705886325929768597935892243220378\ 0228645319644234593118501561376244303567033553863895496499949660014307720923\ 6379664011577886782335606433097544453589989679717129873109790523875986370824\ 6572594541113626690178116429694203467303649244173779046235289922347273125970\ 9769274818111311553864161352859677412130390007523002108693656003020747237365\ 02950302156045167545856597852770663288654812102128/4685260029805594787134715\ 9552877348883016598667579666121068607250933338025150662196445042393682993546\ 8983568964643460482141115450985034214775156666293267190042877415063492952122\ 5475666384398383164441091029227214801958662356960124480555760999171228517314\ 9932788787519142120100061671455759635656953660995586804142989200178971496381\ 0310357837018105081801093865109749661756339829484585572220038003529799427040\ 43730514564505], [7580909974356797756989680040507205476934252843617760201331\ 1827628956202557115085655756892549820617977255060251850735597864106511099934\ 1935698794632694424978721110975706802706567906565606411551303755088155310320\ 1172783319616414235843224778810814197937661723643803169292116633535159380303\ 2761627930588477397935298375616314257464122244074784454110719049474986813049\ 689912136933250614602389549656950832713367332385326666633/122661700041623925\ 4412439599579494036523591271037572681343804348798895405822657478522019349435\ 0361152415341714831508164607821805619843761504739512989876921687639883907702\ 9565869045413224485138962019919725823304159747419819733743603237805398099854\ 2645497671692268192120632864354132652587923972848842494729847394413648164932\ 2896050327511056815592122722958437332402465608777091619907315961155345730381\ 2656071376115841231138], [19847079978519190301114076036302145842170165553993\ 4870147692262508845743139380833507958827493324229501408477400165817243942324\ 5672983718086172745931682117147485099366477098365258360697851262940923954285\ 4135433617147525301589788596167005318620799624392638440566485090498445277076\ 4859061825158912814837950382674739740432807486424625519185352610031946279059\ 360137074345999907849449687762001103114254645369438708501167897771/321132499\ 8268158284523847203209708620740607826436921382820726973887352837216465813601\ 6076243682784102556189454848089889002054262349680942366466972306698093162490\ 8775724739402394881483009611433054415348267177640331222672835631629564907858\ 5843078505084761515748916701170477392061781243206175561976990874233674141811\ 0524930071538512879429592076595317350864373346109900208767876564876092161265\ 6571558458025510084617009128909], [51960329961200773146352548068399232049576\ 2438183627008429764959897581026861027414868119589930152070526970371948646716\ 1339628671907951812322819443162351926463734187123724492389207175527947377271\ 4681077680853197650269792585152951552657791177051584675240253598055652102203\ 2191976942983382442715110513925373750088920845682108201809754313481273375985\ 119787703093598173348087586615098448683613759685813103394948793118177026680/ 8407357994388235599159102010049631825698232208273191467118376572863163105826\ 7399622828035236697991155253226649712761502398340981429199065594661403930217\ 3577998327488097188641315599035804349160201226125075709616833920598687161145\ 0914857703548236972608786875554481911390799311831211077030602713082130127971\ 1750310195093140891719488310777719414193829329634682705927235017526538074720\ 9605226416257371561420458877735186155589], [13603390990508312913794356816889\ 5550306558565901094615514160261718389733744370141109639994229713198207950263\ 8445774331157946277005087171888228558355537366224371746200469637880236316588\ 5990868873480369018842415933366185245386906606180636821253395440132812235360\ 0471216111212316006409108550298641872693817086759202279661351711900463742125\ 8467517923413084049920657445698262851995845658288840175943184664815407670853\ 363182269/220107489848965485129534588269391868563540887983826530185344027447\ 0213648026375407324680294664111893632034904942901946181929686819379162544175\ 1723948395398023700736885668265215519156244034360475492630269599512101705391\ 2322585180570954945248016324127415991109145290330019205434318519878856325772\ 6939950967985095124747544926036199520529035661659861706380396747716718048438\ 117376592867894066592200556226235866548588549337858], [356141399754048614267\ 4781564382874188700994538849211456995042891654110985470076818421080236961243\ 8757115375433886762773398759638244663344324037307503769060267418198890364644\ 0178823221300252293489729992884419280350715764776454246632761313460550278528\ 7441134627457615461304177503249289874066244145666889138852687147544158443155\ 2041579502941291777851194644466683741637467009693724385261829067681437408910\ 51274219441912520127/5762488896030140993970127447076792874336403418687476758\ 8484831661243246334964522257457605316253557693435724483315782235218056506438\ 4549697657890314521644582911189357760329092334014783740595898244656295580414\ 4013677696770990394272037162587085666026561601045718910518761496231846434488\ 6626295018726068401068377822722917033691914036784793297908376468258448434160\ 90881795139086749031394076973360344297117287140768030461857985], [9323902893\ 5706271366449090114596670630373979575366882158435260577784356189665290441668\ 4076858659964505510999172025450086168161446831183140898263389559335185585371\ 3466639755325128380050408481917343862884648336817184854418754936738364775721\ 6911688118490421803463681342228004093496837785366957460182737292456904694198\ 3586181234660783642962392001543194530929008446504554184005612146992025988012\ 8488038008338414987472374378112/15086391789600768130615036458536459937373801\ 3762241649746920092239027602524629812699126013002119554143986824400518327243\ 8348726511215732838556153704080979768496560704424304624846852788781444089979\ 0425840452480830862551189745331010401938308776834838410643226065278652984296\ 6411961182671993252478908805693525282516921275651815380115149089537108530787\ 71154133500555927336979142870501314336853488832335125625555755502836236097], [24410294683171395267259945469996127000411199333760853190535535281681195871429510314079442068798555059453792431772087225245168879580469159794544170936403149540819320510882801573596907938222922817134288725100817648047405608500267748766714030468003650259685406411646787207097050545802045736020993909154298598218721111963426993884619351338577630868510716463423585020972878819198991971234596733617320373133963970742975210614209/39496686472772163397874981928532586937785000709985018165227544505583956123892491583992043369010510473852474871823919949628656144692626488180105707977212947225784927555125847822065435826037363716924714565553298478909956798245598759168652339244838488670328632476925440191393691741920312935346241800034867950746972804090992175422630866247531341721589487617557085576900215798341862472548933587106152708259589526498478046850306 ], [639069811559435586651349273985287139381962000437458713557630797872651519\ 9532200189807148543780906553330626730359600717487382502427309564797464987891\ 3616096766604248066008649395662343764260286534059003290654116126957362406745\ 8665079353663697128421389300140388885722273984907422877223586713672357091891\ 6654896569391605441916930725021758610897259010020408098067059787309461754085\ 4443783530320723473081393553497241453257464515/10340366762871572206300990932\ 7061300875981200753730889520990624292849108119214493482063528806819576007158\ 7421754199271245024811689657221361547555868908290439923891756211344972177277\ 7959029765025098371785620741460586731920499146526710401870895768062760034267\ 1365497667589884434029642671606713477509224034499712666720145411344729881083\ 8336403143116896916985377561747733104158827169163324639078296257896531430237\ 39931304314821], [1673106487846592807281448367255900148141774007974767608767\ 5370408011426011453649538013501424462864154046500947901593429937630619323881\ 7784129405465804445140758993423687143146613390123354557936785042721146861530\ 7328246816117373317750393850786705227665303567102540698949883751763173650302\ 7808071321841320104867836063619983051403713130141974928690178989577951842677\ 2646405033423571360115994228553098871046696520981384561779336/27071431641337\ 5002791154746052651315690158601551207650397744328372963368233750988862198543\ 0514482175476237516544358614238787873622045399202841610526952741847513825993\ 0827764383111790294485558703602643900306894533869200081672879704214340378762\ 8203394130699381619567562578259610347007701884794190727637235548391027356345\ 2418587670123852533896012134795874780561829474197154493062882764484581363827\ 24660699839544721315866094157]] > #Seemingly, for function i, the only fixed point is 0.618. > #ii, f(x) = 5/2*x - 5/2*x^2 > Orb([5/2*x-5/2*x^2], [x], [0], 1000, 1010) [[0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0]] > Orb([5/2*x-5/2*x^2], [x], [0.01], 1000, 1010) [[0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995]] > Orb([5/2*x-5/2*x^2], [x], [0.5], 1000, 1010) [[0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005], [0.5999999995], [0.6000000005]] > Orb([5/2*x-5/2*x^2], [x], [0.65], 1000, 1010) [[0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990], [0.6000000010], [0.5999999990]] > Orb([5/2*x-5/2*x^2], [x], [-0.5], 1000, 1010) [[-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)]] > #The only stable fixed point is seemingly x= 0.6, or 3/5s. > #iii, f(x) = 7/2*x - 7/2*x^2 > Orb([7/2*x-7/2*x^2], [x], [0], 1000, 1010) [[0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0]] > Orb([7/2*x-7/2*x^2], [x], [0.1], 1000, 1010) [[0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683]] > Orb([7/2*x-7/2*x^2], [x], [0.5], 1000, 1010) [[0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060]] > Orb([7/2*x-7/2*x^2], [x], [0.8269407060], 1000, 1010) [[0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683]] > Orb([7/2*x-7/2*x^2], [x], [0.7], 1000, 1010) [[0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637], [0.382819683], [0.8269407060], [0.500884212], [0.8749972637]] > Orb([7/2*x-7/2*x^2], [x], [-0.5], 1000, 1010) [[-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)], [-Float(infinity)]] > #There are seemingly no stable fixed points for this function.