Class Note for ECE 449 at UA
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Date Created: 02/06/15
A I we 449549 ontinnons System mutualitth Population Dynamics II o In this class we shall analyze behavioral patterns of ecosystems in which more than two species interact with each other 0 Such systems frequently exhibit chaotic behavior 0 Chaotic models shall be analyzed and discussed November 24 2003 start Presentation EAEE I we 449549 Glontinnnns System mutualitth Table of Contents Generalization of ecological models Gilpin model Chaotic motion Bifurcations Structural vs behavioral complexity Limits to complexity Forces of creation November 24 2003 start Presentation A I we 449549 ontinnous System mnaml Generalization of Ecological Models 0 What happens when eg three species compete for the same food source The previously used model then needs to be extended as follows There never show up terms such as as such a term would indicate that eg the competition between x2 and x3 disappears when x1 dies out November 24 2003 start Presentation A I we 449549 Gluntinnonx System mutualitml The Gilpin Model I Michael Gilpin analyzed the following threespecies ecosystem model 0 A single predator x3 feeds on two different species of prey x1 and x2 both of which furthermore compete for the same food source and suffer from crowding effects 0 The initial populations for all three species were arbitrarily set to 100 animals each We simulate over 5000 time units November 24 2003 start Presentation A I we 449549 ontiouous System mnoml The Gilpin Model II 0 The model was coded in Modelica 149 51 y no Bert IIEEI ilil l l JEJ Real 31 StartIUU Real x2 l3tarcIDU Real 83 startIUU Ll equatmn darx1 x1 U UUl xlAZ U UU1x1x2 UVUlxlx3 darx2 x2 U UUlSxlz12 U UUl IZAZ UUUlI12x3 derx3 x3 UUUSxlx3 D UUUSX2X3 J Declarations 01 class November 24 2003 Start Presentation I we 449549 Gluntionnns System oodinggl The Gilpin Model III 0 We use simulation control as follows Simulaliuu Inwrvzl 4mm 5mm Iquot Textual data lormat Slap time lt annnngt l Double precision ow nmw l 7 Store r mum n l39j mix I Numbermlnlzmls 50quot 17 Output varlahles l7 Auxillavy variables ilnlenluliun Argnrluun Dassl selecliun uidislant time grid Tolerant m tore variables at events Fixed Imegwmrsw n F Output debug Information November 24 2003 Start Presentation A l 434243 449549 Giontinuuus system mudmul The Gilpin Model IV x 1 sun 7 4m 7 u sun H mm 2mm anun ADDEI n v2 Sun 7 ACID 7 n an a mum 2mm 3mm man an 36 2mm mu7 n H mm mm auun mun sum November 24 2003 megabit 5 655mm gt ill ALE l 449549 ontinnnn pstm1 whalingl The Gilpin Model V Most of the time there are plenty of x2 animals around Once in a while the predator x3 population explodes in a pattern similar to that of the Latka Volterra model The predator then heavily decimates the x2 population The x1 population is usually hampered by strong competition from the x2 population for the common food source Thus when the x2 population is decimated the x1 population can thrive for a short while However the x2 population recovers quickly depriving again the x1 population of their food November 24 2003 Start Presentation 5w 449549 manna system monaml The Gilpin Model VI 0 Yet the behavioral pattern of each cycle is slightly different from that of the previous one This can be better seen in phase portraits mm mm sun Ennr ADD 2m m H mm mm 0 sun mun November 24 2003 Start Presentation ZAL cm 449549 osmium stem Mbdmgl The Gilpin Model VII In a limit cycle the phase portrait would show a single orbit The observed behavior is called chaotic Each orbit is slightly different from the last If the simulation were to proceed over an infinite time period the orbits would cover an entire region of the phase plane Chaotic behavior is caused here because the two preys can coexist at different equilibrium levels ie the predator can be fed equally well by eating animals of the x1 kind as of the x2 kind One prey can substitute the other In continuoustime systems chaos can only exist in 3rd and higher order systems November 24 2003 Start Presentation A I we 449549 ontinnons System mutualitth The DiscreteTime Logistic Equation I 0 In the case of discretetime systems chaos can already exist in 1 t order systems 0 To study chaos in its purest form we shall analyze the behavioral patterns of the diseretet ime logistic model 0 a is a parameter that shall be varied as part of the experiment November 24 2003 start Presentation EASE I we 449549 Glontinnnns System mutualitth The DiscreteTime Logistic Equation II Discrete Time Logistic Equation we nd graphically the interSCCti0ns m between the two functlons m 03 611 v9 o 04 02 n2 m a o 0 06 u m o m 94 M m a In the range a E 00 10 there ls amp 04 a Z 075 only a single solution x 00 mu m M M As a approaches a value of a 10 the two curves become more and more parallel Consequently it takes more and more iterations before the steady 39o v2 9 v5 v5 10 39a 92 w mu 9 10 state valueis reached a 09 n 0999 39 m ou DJ 04 02 12 November 24 2003 start Presentation All I 65 449549 Glontinnons System mnaml The DiscreteTime Logistic Equation III 0 Discrete Time Logisatic Equation In the range a E 10 30 there are H M two intersections between the two a a u 5 functions However only one of the two solutions is stable There is still only D 12 OJ DJ 09 LO 39n 02 04 0 LB 1 a 2 L1 20 one steady state solutlon 1 u Ln m 03 The iteration converges rapidly for m 05 intermediate values but as a 9 w approaches either a value of a 10 or 07 u2 alternatively a value of a 30 the Iteranon converges more and more a 275 a 2999 Slowly November 24 2003 start Presentation I 65 449549 Glontinnons System mnaml The DiscreteTime Loglstlc Equatlon IV quota Discrete Time Logi tic Equation In the range a E 3 0 35 a limit M M cycle is observed 0 u a 4 M For a 305 and a 33 the discrete M M limit cycle has aperiad afZ 0390 02 01 05 18 Ln 00 32 D 115 08 LE 5 305 a 35 For a 345 and a 35 the discrete 12 La limit cycle has aperiad 0f4 o a na a B 05 04 DA V 02 02 00 02 04 13 DJ 10 lLl 02 04 0 08 10 a 345 a 35 November 24 2003 start Presentation A cw 449549 toutian 51mm whalingl The DiscreteTime Logistic Equation V DiscreteiTime Logistic Equation In the range a E 3395i 40 the quot0 observed behavioral patterns become increasingly bizarre 05 05 05 05 m m For a 356 a discrete limit cycle With a period of 8 is being observed al 0 m M 09 iu 39a 02 0 ma ma iu 33955 g 3396 For a 3 6 the behavior is chaotic For a 384 a discrete limit cycle with a period of 3 is being observed For a 399 the behavior is again chaotic o or 02 v4 M a ia n 364 For a gt 4 0 the system is unstable November 24 2003 Start Presentation law 449549 ontinuuns system mmml The DiscreteTime Logistic Equation VI We can plot the stable steadystate solutions as a function of the parameter a Bifurcation Map of Discrete Logistic Model Lo The dark region in the plot to the left is the chaotic region yet even within the chaotic region there are a few mm chaotic islands such as in the vicinity of a 3 84 09 05 State Variable x oocoooo Hmugmmu 0 29 3 0 3 32 33 34 35 35 37 38 39 40 Parameter a November 24 2003 Start Presentation A I we 449549 untinnons System mutualitth Bifurcations I 0 How can the bifurcation points of the discretetime logistic model be determined 0 A simple algorithm is presented below 0 We start with an assumption of a xed steady state 0 We know that this assumption applies to the parameter range a e 10 3 0 0 Thus we shall try to compute these two boundaries November 24 2003 start Presentation EA I we 449549 Glontinnnns System mutualitth Bifurcations II 0 The equation has two solutions x 0 0 and x2 a 1 0a We know that the second solution x2 is stable 0 We move the stable solution to the origin using the transformation 0 This generates the difference equation November 24 2003 start Presentation A I we 449549 untinnons System mutualitth Bifurcations III 0 We linearize this difference equation around the origin and find 0 This difference equation is marginally stable for a 10 and a 3 0 0 We now proceed assuming a stable limit cycle with a discrete period of 2 thus November 24 2003 start Presentation EA I we 449549 Glontinnnns System mutualitth Bifurcations IV 0 We evaluate this equation recursively until xk2 has become a function of xk only 0 This leaves us with a 4th order polynomial in xk The previously found two solutions must also satisfy this new polynomial ie we can divide by these two solutions and again obtain a 2nd order polynomial in xk This new polynomial has again two solutions One of them is a 30 the other provides us with the next bifurcation point November 24 2003 start Presentation 10 I cut 449mm Gluntimmns System illobeliugl The Gilpin Model VIII Let us now look once more at the Gilpin model We shall treat the competition factor k as the parameter to be varied in the experiment The nominal value of k is k 10 We shall vary k around its nominal value We shall display only the x1 population November 24 2003 start Presentation AL I an 449549 aluminum system long The Gilpin Model IX For k 098 we observe a limit cycle with peaks reaching each time the same level Chaos in Gilpliwrol39s Model e moo NHL i m For k 099 we observe a 11m1t cycle where the peaks toggle between two V discrete levels Only looking at the 393 oomamueeomuaonn 39Zouauemamueemu Peaks we could say that we haVe a k 098 k 099 lImlt cycle w1th a dlscrete perlod of 2 m 1 2m a moo moo saw u i am For k 0995 we have a limit cycle m N am with a discrete period of 3 0 m For k 1 0 the behavior is chaotic zen I 1 2m 000 l V 000 l I wan 1100004500 005000 45w mu mm men an snow k 0995 k 10 November 24 2003 start Presentation 11 A an 449549 Luminous System mohng The Gilpin Model X r t v For k 10025 k 1005 and k Chaos m Gupliwrti S Medei 1 0075 the behavior remains chaotic won 1 non m m uuuv The behav1or remains chaot1c for values of k lt 1 0089 um mu I m anu t u o no uu om woo mo 4500 me new no we man no mo me k 10025 k 3 1005 For k gt 10089 such as k 101 the 6 x1 population quickly dies out man eon mm um m v no o cocoa man mm mm we won 5mm asou won mm we was 5000 k 10075 k 101 November 24 2003 start Presentation ltIgt l 6126 449549 Summons 51mm lohelingl True Behavior or Numerical Artifact I In the case of the discretetime logistic model we were able to analyze the observed behavior analytically and verify that chaos indeed occurs In the case of the Gilpin model this is no longer as easy The question thus needs to be raised whether what we have observed is indeed the true behavior of the system or whether we fell prey to a numerical artifact November 24 2003 start Presentation 12 A I we 449549 ontinnons System mnemgl True Behavior or Numerical Artifact II 0 To this end I propose to apply a logaritth transformation on the Gilpin model 0 The modified Gilpin model presents itself as follows The analytical results of the two models must be identical yet their numerical properties are very different Start Presentation lt3 November 24 2003 gt EA I we 449549 Gluntinnons System mutualitml True Behavior or Numerical Artifact III 0 We can now plot the discrete bifurcation maps of the two models If they are the same then chaos is indeed for real also in this model Bifurcation Maps of Gilpin39s Model 1000 1000 son 900 am II sea 3 lmnqgi39j Him 5 ml g 4 Wm 4 Lynn a Original 9 Modi ed 200 zoo Gilpin model Gilpin model 500 000 005 097 09 La 095 097 099 01 Comp etition k Competiticn k Start Presentation November 24 2003 lt2Igt l3 A I we 449549 ontinnons System whalingl Structural vs Behavioral Complexity I 0 We have seen that simple deterministic differential equations can lead to incredibly complex behavioral patterns in the solution space gt The behavioral comp lexfi of a system is generally much greater than its structural comglexity November 24 2003 Start Presentation lt2JEIgt EAEE I we 449549 Glontinnnns System whalingl Structural vs Behavioral Complexity II 0 Looking at the Gilpin model we may reach the conclusion that chaotic behavior is the exception to the rule that it occurs rarely and is rather fragile Nothing could be farther from the truth 0 As the structural complexity the order of a differential equation model increases the chaotic regions grow larger and larger In fact they quickly dominate the overall system behaVior It is thus utterly surprising that noone recognized chaos for what it is until the 1960s Before then chaotic behaVior was always interpreted as a result of impurity November 24 2003 Start Presentation ltnrgt 14 A I we 449549 ontinnons System mmml Chaos in Mechanical Systems Jnmqlulmnd quotmm Ammanmn Winlaw HE E mm 5M An malinn etup ab Tvme mm November 24 2003 Start Presentation ltncgt I we 449549 Glontinnnns System mutualitml Limits to Complexity 0 We may thus ask ourselves what limits complexity in our universe How come that trees in uni SOn grow leaves in the spring and shed them again in the fall How come thatWe can still recognize structure at all among this maddening complexityvthat39 the laws of nature present us with 0 There are three mechanisms jtlia39ti mif39COmpleXity 3 Physical construingquot 39Whenquot Connecting two subsystems the combined degrees of freedom are usually lower than the sum of the individual degrees of freedom 3 Control mechanisms Controllers in a system abundant in nature tend to restrict the possible modes of behavior of a system 3 Energy The laws of thermodynamics state that each system sheds as much energy as it can This also limits complexity November 24 2003 Start Presentation lt2Igt 15 A I we 449549 untinnons System mutualitth The Forces of Creation I Chaos provides nature with a great mechanism for constant innovation We are used to viewing Murphy s law as something negative what can go wrong will go wrong However Murphy s law can also be interpreted as something highly positive what can grow eventually will grow Chaos is the great innovator It brings any and every system constantly to the greatest degree of disorder that it can be in Chaos is built into the very fabric of our universe At the molecular level the molecules move around like the balls on the pool table in total chaos This is what we measure as entropy Entropy is being maximized November 24 2003 Start Presentation lt2JEIgt A I we 449549 Glontinnnns System mutualitth The Forces of Creation 11 Yet chaos alone would leave us with a universe that is just an accumulation of random white noise No structure would be retained For structure to be preserved we also need the opposite force the great organizer a force that fosters order that sifts through the different possibilities discards the bad ones and only preserves those that look mo st promising Three such mechanisms were outlined before The most powerful among them Energy is being minimized November 24 2003 Start Presentation ltnrgt l6 A we 449549 ontinnons System mutualitth Conclusions 0 In the last two lectures we have looked at predominantly inductive techniques for modeling population dynamics 0 Yet these techniques have failed to e g provide us with a satisfactory model that could help us understand the mechanisms that lead to the oscillatory behavior of the larch bud moth zeiraphera diniana In the next lecture we shall come up with an improved methodology to deal with these types of systems November 24 2003 start Presentation EASE em 449549 Glontinnnns System mutualitth References Cellier FE 1991 Continuous System Modeling SpringerVerlag New York Chapter 10 Gilpin ME 1979 Spiral chaos in a predatorprey model The American Naturalist 113 pp 306308 November 24 2003 start Presentation 17
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