12. Exponential growth simulation.

Transcription

1 1. Exponenial growh simulaion. Mos people hink of exponenial growh as being growh ha is very fas. However, exponenial growh has a precise meaning a populaion grows exponenially when is growh rae is proporional o is sie. For example a populaion which growh by 3% each year is growing exponenially. The growh rae in his case is proporional o sie wih consan of proporionaliy.3. Suppose we wan o rack his populaion over many years saring a sie () = A. Then afer one year i has sie: (1) = A +.3A = (1.3)A. Noice he expression on he righ. Tha simple piece of algebra hides a wonderful concepual jump. I says ha o add 3% every year is he same as muliplying by 1.3. The reason ha s so grea is ha i makes ieraion easy. In year we muliply by 1.3 again: And again and again: () = (1.3)(1.3)A = (1.3) A. () = (1.3) A and ha gives us a formula for a any ime. The graph of agains saring a A = 1 is ploed a he righ. In 5 years he populaion grows o nearly 5. Okay. Do you expec a real populaion of squirrels recenly arrived in a new fores o grow according o ha nice smooh curve? Cerainly no. Real life is full of uncerainies, boh good and bad, and we expec considerable variaion. Even wih an average growh rae of 3%, he populaion will someimes, hrough ill luck, decrease. Cerainly i will oscillae above and below he heoreical curve A simulaion We each our sudens abou exponenial growh bu how many of hem have acually observed such growh? Our objecive here is o consruc such a process in he classroom, no one of hose smooh exponenial curve ype populaions, bu one ha has o cope wih he real life uncerainies of birh and deah. To simulae his sochasic behaviour, we flip coins. The game is bes played wih a large class. To ge he daa repored here I combined hree grade 1 classes The model This sochasic aspec of populaion growh can be modeled wih a large class of sudens each having 3 coins. A any ime he populaion consiss of hose individuals who are sanding. Sar by having a few (e.g 5) sudens sand. To move one ime sep (one year), each sanding individual flips his/her 3 coins. There are hree possible oucomes. H 1T T 1H 3H or 3T one offspring no offspring and no deah deah Afer everyone sanding has flipped I ask all hose in he las caegory (3H or 3T) o si down. [They are now ou of he populaion, a leas unil hey are invied back in as someone else s offspring.] Then all hose in he firs caegory (H 1T) say sanding and ge o choose one siing individual o sand. All hose in he middle caegory (T 1H) simply say sanding. Thus, if here are x individuals in he firs caegory and y individuals in he las, he populaion sie increases by x y. 1 exponenial simulaion 1

2 The daa A he righ is a daa se I generaed wih a class of 1 sudens. For example, of he 5 sanding on he firs flip, rolled H&1T and rolled 3H for a ne increase of =. I s really nice o have a compuer in he classroom o plo he daa as hey are generaed. The graph below cerainly shows an increasing populaion wih an acceleraing growh rae, bu here is considerable variaion especially a he early sage when he sie is small. I ask he class wheher his daa belongs o an exponenial populaion. Is he growh rae proporional o he sie, a leas on average? How would we deermine ha? How do you deermine wheher any daa se follows an exponenial curve? One suden ells me ha you fi an exponenial rend line and see how good i is. Tha s no a bad answer bu i s a bi waffly. My favorie answer o his quesion is ha you look a he log plo, because if is exponenial, hen log() will be linear. And we are raher good a recogniing wheher a bunch of poins lies in a sraigh line. [Why is ha? because ligh ravels in a sraigh line?] If is exponenial hen log() is linear Here s he algebra. Sar wih he equaion: Take logs: = Ar log() = log(ar ) = log(a) + log(r ) = log(a) + log(r) Wha does ha say? ha he graph of log() agains is a sraigh line wih slope log(r) and y-inercep log(a) Analysis of he daa Check i ou. A he righ I plo log() agains. Is i a sraigh line? Well, yes and no. There s cerainly considerable scaer, especially a he beginning, bu he daa has a srong overall linear characer. I s worh noing ha we would expec more scaer a he beginning when he populaion is small. When here are a large number of individuals, he bad luck of some will be more likely o balance he good luck of ohers, and he ne resul will be close o expecaion. Anyway, i is he linear characer of he log plo which ells us wheher he original daa is exponenial. And we seem here o have an exponenially growing populaion log exponenial simulaion

3 How do we ge a bes-fi exponenial equaion for he populaion? One way is o draw a bes-fi line for he log daa and find he slope of is y-inercep. We can do his wih pencil and ruler, or if we have a compuer or calculaor we can fi a rend-line. A he righ I used an excel spreadshee o ge he rend line: log() = How do we ge he exponenial equaion from ha? We exponeniae i! log( ) = 1 = 1 = 1 1 = 1 = 5.34 (1.165)..775 Our bes esimae from he daa, is ha he populaion sared wih 5.34 individuals and grew by 16.5% per year..665 ( 1 ) The calculaion above is one ha I migh have made when I was a suden back in he 6 s. Wih he echnological power sudens have oday, hey could have bypassed he logarihm suff alogeher and fied an exponenial rend-line o he original daa. Such a line (which is acually a curve) is ploed a he righ (again wih excel). [I s worh poining ou ha his curve is exacly he same as he log line, bu ploed on he - axes. Indeed, e.1531 = ] However, I make my sudens go he logarihm roue as well. One reason for ha is ha I wan hem o gain some familiariy in working wih he logarihm, bu anoher, and even more imporan one, is ha I wan hem o see ha sraigh line, because ha s he bes proof ha he daa se is exponenial. The heoreical curve. Wha sor of populaion growh curve did we expec? Afer all, we have a quie precise coin-flipping rouine ha we used o generae he daa wha sor of heoreical curve ough ha o have given us? Did we expec i o be exponenial, and if so, wih wha parameers? Here s he argumen. The able a he righ gives he probabiliies of he differen oucomes. I ells us ha if we ake 8 ypical individuals hen we expec 3 of hem o have an offspring and of hem o die, giving us a ne change of 3 = 1. Thus, each ime sep, we ge (on average) an increase of 1 for every 8 individuals, giving us an average growh rae of 1 in 8 or 1.5%. In paricular, he growh is cerainly exponenial (on average) because he expeced increase is proporional o he sie log y =.665x y = e.1531x Wha kind of curve did we expec? This is an excellen quesion o pose o he class. We have o look carefully a he coin-flipping rouine. Is he growh rae (on average) proporional o he populaion sie? If so, wha is he consan of proporionaliy? prob oucome effec 1/8 HHH die 3/8 HHT HTH offspring THH 3/8 HTT THT no change TTH 1/8 TTT die 1 exponenial simulaion 3

4 A growh rae of 1/8 means a muliplier of 9/8. Saring wih sie 5, afer years, we expec sie () = 5 (9/8). A he righ we plo his formula as a curve and compare i wih he daa. The curve lies markedly below he daa is final heigh (a =) is 53, well below 98, he final heigh of he daa poins. I doesn seem like a paricularly good fi a all. Wha happened? Sudying he daa we see ha we did seem o have a lucky surge around generaion 1. Tha carried he daa well above he curve, and once up here, i would be ap o say ha way. In fac ha brings us o he quesion of jus wha sor of variaion we migh expec. Randomness How close a fi could we expec beween he daa generaed from his experimen and he heoreical curve? Quesions like his can be quie challenging and hey are sudied by heoreical saisics. Bu one hing I hough of doing, jus o see wha happened, was o repea he experimen many imes, say 5 imes, and see how big a range of daa ses I go Does he curve provide a good fi o he daa? Apparenly no. Bu do we expec a good fi? Le s look a hings a bi more closely. No waning o wear hin he generous paience of my sudens, I hough I d ry o ge he compuer o do all he work. My programming skills are raher modes, bu I do know my way around a few MAPLE commands, and, happily enough, I managed o consruc a passable MAPLE program (which I include a he end). I did 5 runs of years, each run saring a =5. Well, 4 of hose runs wen exinc he populaion fell o ero a some poin and hen, of course, sayed here. The remaining 1 runs showed grea variaion in he sie afer years. The lowes (ha didn go exinc) reached a sie of only =14 afer years, whereas he highes go up o =166. These runs (he high run and he low run) are ploed a he righ along wih he heoreical curve. [Noe ha o accommodae he larger populaion I have exended he verical scale.] So is he heoreical curve he average, in some sense, of he high run and he low run? Le s see he final heigh of he heoreical curve is 53 and he average of 14 and 166 is: = 9. Hmm 9 is definiely no close o 53. We could already see his from he graph he heoreical curve is no halfway beween he high daa se and he low daa se, bu is much closer o he low daa se. Wha are we o make of his? The idea here is ha hese wo "exreme" daa ses serve as boundaries. I should be relaively rare for a daa se o fall ouside of hem. Of course, if we had aken he bes and he wors ou of 1 simulaions, we'd expec hese boundaries o be a bi wider bu no much. 1 exponenial simulaion 4

5 I urns ou ha we don really expec he heoreical curve o be he average of he high run and he low run. A leas, no if we use he convenional average. The average we ve calculaed above is called he arihmeic mean (AM): AM(14, 166) = = 9 bu here are oher averages around. In fac he correc average o use in his case urns ou o be he geomeric mean (GM) defined as he square roo of he produc: GM(14, 166) = = 48. and indeed ha s graifyingly close o 53. The poin is ha he law of growh of hese populaions is muliplicaive, and i follows ha he appropriae average o use is he GM, ha being a muliplicaive average. Anoher way o look a his is o observe ha aking he GM of wo numbers is really he same as aking he AM of heir logarihms. More precisely he log of he GM of wo numbers is he AM of he logs of he numbers: log(gm(a, b)) = AM(log(a), log(b)) [Verify his i s a good exercise!] So he fac ha he heoreical curve is close o he GM of he low run and he high run ells us ha if we plo he runs on a log scale, he heoreical curve (which will now be a sraigh line) should be close o he AM of he low run and he high run. The graph appears a he righ. On his scale, he heoreical line does appear o be prey much he midpoin (on average) of he high and low runs..5 log Finally, here s he maple program. The # indicaes a commen. > R:=rand(1..8): #defines R as a procedure which generaes a random ineger beween 1 and 8 N:=5:prin(N); #N is he populaion sie. I sars a 5. for j from 1 o #execues he insrucions below for each of years do for i from 1 o N #execues he insrucions below for each individual in he populaion do r:=r(): #r is a random ineger beween 1 and 8 if r<3 hen N:=N-1: #holds wih probabiliy /8 end if: if r>5 hen N:=N+1: #holds wih probabiliy 3/8 end if: end do: > prin(n); > end do: 1 exponenial simulaion 5

6 Problems 1. Here's a game you can play wih a bole of pennies. Sar wih, say, 6 coins in a cup. Shake hem and spill hem on he able and wihou paying aenion o wheher hey are heads or ails, gaher hem in groups of sie 3. Each ime a group has a leas heads, add a coin o i, and each ime a group has 3 ails, ake a coin away. Groups ha have neiher condiion are jus lef he way hey are. Then gaher up all he coins and repea. When he number of coins is no a muliple of 3, he one or wo exra coins can be jus pu o he side and recycled ino he nex generaion (hey neiher give birh nor die). Collec a daa se for 5 generaions. Esimae wha you hink he sie of he populaion "ough" o be afer his ime.. Suppose we have a populaion whose sie changes in accordance wih he general rules we formulaed a he beginning: ha in each ime sep here are hree possibiliies for each individual: reproduce and make one offspring probabiliy r die probabiliy d neiher of he above probabiliy 1 r d Now suppose ha we don' know wha r and d are bu wha we do have is he daa from he firs run (abulaed on he firs page). The problem is o use he daa o ge some idea of he sie of r and d. Now because birh and deah all happen wihin a single generaion, i's ap o be hard o ease r and d apar, bu here's some informaion abou he pair of hem you can surely ge. An esimae? An average? Explain your analysis. 1 exponenial simulaion 6