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 I 0 Today we shall look at the problem of modeling population dynamics ie detennining the sizes of populations of biological species as functions of time 0 Such systems are modeled as pure mass flows ie energy conservation laws are not being considered 0 Consequently bond graphs are not suitable for describing these types of models November 21 2003 start Presentation EAEE I we 449549 Glontinnnns System mutualitth Table of Contents 0 Limitations of bond graphs Exponential growth 0 Limits to growth Logistic function Continuoustime vs discretetime Chain letter US census 0 Curve fitting Predatorprey models Larch bud moth Competition and cooperation November 21 2003 start Presentation A I we 449549 Giantinnons System mutualitth Limitations of Bond Graphs I 0 Bond graphs have been designed around the conservation principles of physics energy conservation mass conservation and are therefore only suitable for the description of physical systems 0 Chemistry was a border line case Although it is possible to model chemical reaction dynamics down to the level of physics this is not truly necessary since the reaction rate equations are decoupled from the energy balance equations Hence this is rarely done We did it because the bond graphic interpretation of chemical reactions offered additional insight that we could not have gained easily by other means 0 Yet as the complexity of molecules grows especially in organic chemistry it becomes more and more difficult to know what the elementary step reactions are and at that level chemistry becomes an empirical science the knowledge of which is essentially covered by interpretations of observations alone November 21 2003 start Presentation EA I we 449549 Glontinnnns System mutualitth Limitations of Bond Graphs II 0 As we proceed to systems of ever increasing complexity such as in biochemistry the situation becomes truly hopeless 0 Although we live in a physical universe and although a majority of scientists would agree that the laws of nature are ultimately laws of physics we lack the detailed understanding necessary to e g explain the processes of mitosis and meio sis cell division on the basis of the underlying physics or worse to explain how the genetic code directs the cells to reproduce a functioning living being from its blueprint 0 With this lecture we are taking a giant step bypassing organic chemistry biochemistry molecular biology cell biology genetics etc jumping right to the level of population dynamics ie taking a macroscopic look at how populations of species develop in size over t1me November 21 2003 start Presentation A I we 449549 ontinnons System mutualitth Exponential Growth I 0 The change in population per time unit can obviously be expressed as the difference between birth rate and death rate 0 It is reasonable to assume that both the birth rate and the death rate are proportional to the population and therefore November 21 2003 start Presentation EAEE I we 449549 Glontinnnns System mutualitth Exponential Growth II Populations of all species grow exponentially over time 0 This is also true for human beings gt Every species eventually outgroWs its resources I In the ultimate instance populations are controlled by hunger rather than brains November 21 2003 start Presentation A I out 449549 ontinnons System mnemgl Limits to Growth As food gets scarce ie as soon as all available food is being consumed by the population we can determine the food per capita as the total food divided by the population If not enough food is available the birth rate will decrease and the death rate will increase This is called the crowding effect The most commonly used assumption is that a one species ecosystem can support a xed number of animals of the given species start Presentation November 21 2003 Al I out 449549 Gluntinnons System mutualitth The Logistic Equation The above equation is called the ogltmngoustime logistic equation Continuous Time Logistic Mo del 10cc V sou E n ma 2 5 4m f A A I Ch u 33 2M II t t t 1 Exponential growth s 000 L o 5 m 15 20 25 Time years P 1000 m start Presentation November 21 2003 A I we 449549 ontinnons System mnemgl Continuoustime vs Discretetime Model 0 Applying the forwardEuler integration algorithm 0 to the differential equation dscribing the population It may be justified to use this much cruder model either because the accuracy of our model is not all that great anyway or because a population may reproduce only in spring h 10 November 21 2003 start Presentation EASE I we 449549 Glontinnnns System mutualitth The Chain Letter I 0 Population dynamics modeling techniques may also be applied to macroeconomic modeling Let us consider the model of a chain letter 0 The following rules are set to govern this artificial model 3 A chain letter is received with tWO addresses on it the address of the sender and the address of the sender s sender 3 After receiving the letter a recipient sends 1 to the sender s sender He or she then sends the letter on to 10 other people again with two addresses his or her own as the new sender and the sender s address as the new sender s sender 3 The letter is only mailed within the US v Every recipient answers the letter exactly once When a recipient receives the same letter for a second or subsequent time he or she simply throws it away November 21 2003 start Presentation u I we 449549 ontinnons System mutualitth The Chain Letter II 0 Special rules are needed to provide initial conditions 3 The originator sends the letter to 10 people without sending money to anyone v If a recipient receives the letter with only one address the sender s address he or she sends the letter on to 10 other people with two addresses his or her own as the sender and that of the originator as the sender s sender No money is paid to anyone in this case 0 Every sender has 100 receiver s receivers thus is expected to make 100 0 Except for the first 11 who don t pay anything every sender pays exactly 1 0 Hence this is a wonderful and totally illegal way of making money out of thin air November 21 2003 start Presentation EA I we 449549 Glontinnnns System mutualitth The Chain Letter III 0 We can model the chain letter easily as a discrete system I is the average number of new infections per recipient R the number of new recipients can be computed as the number of new infections per recipient multiplied by the number of recipients one step earlier P the number of already infected people can be computed as the number of people infected previously plus the new recruits November 21 2003 start Presentation A I we 449549 ontinnons System mutualitth The Chain Letter IV 0 We can easily code this model in Modelica dul cm Lull CL I anuali n l BEE emu ap jaggegyg gge Parameter Real E39ch ZBSSSUEUU quotTotal US pupuiancn zuuu census data Real F start1 quotPDDulatlun infected wnh chain letter Real R startlU quotCurrant reclplancs c chaln letter Real RR quotNext recl lenbs c chem lent Erquot Real I quotCurrent average Infectlnn rat E per partlclnant Real II quotNext average Infectlan race liar Participantquot Real H quotAverage amuunt a money recalv d par partlcluantquot 4 gtlJ aquatic A when eempieoi than I 101 e prePP ct R IpreR F minprePR new II 1 1 e PPLQL RR IIeR M 141 7 1 4 rDsclsmmions o1 class November 21 2003 start Presentation I we 449549 Glontinnnns System mutualitth Simulation Results 3 Initially every participant makes exactly 99 as expected 3 However already after seven 0 generations the entire US n population has been infected 35 Thereafter everyone who still 9 n K n participates loses 1 November 21 2003 start Presentation A i 450343 449549 ontinuous gym mammal Simulation Results 1 gourdquot 4n Prosperity Recession u 5 1B 15 2D November 21 2003 3 Ej gl we Start Presentation u i m 449549 Ginntinuuus 5mm mammal Interpretation As long as exponential growth prevails ie as long as the second derivative of the population growth is positive the population is able to borrow money from the future They effectively eat the bread of their children 0 Once the in ection point has been passed the debts made by previous generations have to be paid back November 21 2003 21 1 51 Sta Presentati m A we 449549 ontinnons System mutualitth US Census I 0 In the US population statistics have been collected once ever 10 years since 1850 0 I used Matlub to fit a logistic model to the available census data 0 I then used Modelica to plot the real census data together With the curve fit November 21 2003 start Presentation I we 449549 Glontinnnns System mutualitth US Census 11 L 7 n 7 Parameter Real to 155m 39 Lrat census yearquot Parameter Reel ti znuu quotLast census veerquot Parameter Reel veersus u t 39lu hf quotCensus Parameter Reel ensus16 x 1es 23 19 Real 9 quotUs pupulationquot Reel e quottime in actual years u Estlmated logistic meceemedei Parameter Reel a uu231 Parameter Real 1 5 7473 11 Reel Pe starbcansus1 quotEsblmatsd U 57 populationquot ltl l equatlon c tlma en E Plecew1sext xgr1dveers veerldscensus Estlmeted logistlc mebeemodel dertpe seine hJEPe Z lEeclaretiens e1 Glass November 21 2003 start Presentation A I we 449549 ontinnoux System mnaml US Census III 3E3 Pm P80 We can nu langer rely on an increasing number of 2 children ta payfar bur retirement benefits November 21 2003 Start Presentation A I we 449549 Gluntinnonx System mnaml Curve Fitting I Let us look how the curve tting was done Since we only have measurement data for the population itself not for its derivative we first need to approximate the population gradient To this end we lay a quadratic Polynomial through three neighboring population data points November 21 2003 Start Presentation 10 A I we 449549 ontinnous System mnaml Curve Fitting II 0 In a matrixvector form V V Vandermamle matrix M atlab notation November 21 2003 Start Presentation ASL I we 449549 Gluntinnonx System mutualitml Curve Fitting III 0 Now that we have the coefficient vector we can approximate the population gradient We could equally well have used other interpolation polynomials such as cubic splines or inverse Hermite interpolation November 21 2003 Start Presentation 11 A l we 449549 ontinnoux system mmml Curve Fitting IV 0 We are now ready to curvefit the logistic model We have n equations in the two unknowns a and b November 21 2003 start Presentation EAEh I it 449549 un nuonx System mutualitml Curve Fitting V 0 In a matrixvector form W V Vandermamle matrix November 21 2003 start Presentation 12 IA l we 449549 Summons system mmml Curve Fitting VI 0 In general Navember 21 2003 start Presentation Al I it 449549 un nuonx System mutualitml Curve Fitting VII 0 Therefore PenroseMaare pseudomverse M atlab notation t Navember 21 2003 start Presentation 13 I we 449549 ontinnons System mutualitth PredatorPrey Models I 0 When multiple species interact with each other the simple logistics model no longer suffices A simple twospecies model with one species feeding upon another was proposed by Lotl a and Volterra The Lotka Volterra model makes the assumption that the predator population without prey would die out by exponential decay whereas the prey population would grow beyond all bounds due to an unlimited supply of its own food November 21 2003 start Presentation EA I we 449549 Glontinnnns System mutualitth PredatorPrey Models 11 0 When predator meets prey PPM ley a certain percentage of the energy stored in the prey population is transferred to the predator population 0 The efficiency of the feeding is less than 100 Thus some energy is lost in the process k lt 10 Lotka Volterra models lead to cyclic oscillations as they are indeed frequently observed in nature 0 Especially insect populations such as locust seem to show up in large numbers in fixed time intervals whereas they are ahnost extinct in between November 21 2003 start Presentation A I we 449549 untinnons System mutualitth The Larch Bud Moth I 0 The larch bud moth is an insect that lives in the upper Engiadina Valley of Southeastern Switzerland at altitudes between 1600 7 2000 m 0 Its larvae feed on the needles of the larch trees The population has a cycle time of exactly nine years ie once every nine years the insect population is larger by several orders of magnitude and all the larch trees turn brown because of them 0 Hence the larch bud moth population was curvefitted to the predator population of a Lotka Volterra model November 21 2003 start Presentation EA I we 449549 Glontinnnns System mutualitth The Larch Bud Moth II Larch Bud Moth Population 39 r s 3 300 200 7 7 r r e r Predator Fit mo 00 I 1 254 1957 won mus mun man 1172 i975 197a Time calendar years The curve fit is excellent indeed Does this mean that we now understand the population dynamics of the larch bud moth Unfortunately the answer to this question is a decided no November 21 2003 start Presentation 15 A I we 449549 ontinnons System mutualitth The Larch Bud Moth III 0 The larch bud moth is also plagued by parasites Thus if the insect population is large the chances of spreading the parasites among them grows drastically Thus it may make equally much sense to curvefit the larch bud moth population to the prey population of a Lotka Volterra model 0 This was attempted as well November 21 2003 I start Presentation EASE I we 449549 Glontinnnns System mutualitth The Larch Bud Moth IV 300 Larch Bud Moth Population 39 39 3 g 200 Prey Ht 100 o t 154 1357 mm mna man man quot72 75 1978 Time calendar years The curve fit is equally excellent Thus we cannot conclude from the quality of the curve fit alone that the underlying model represents correctly the causeeffect relationship of the biological system November 21 2003 start Presentation 16 A I we 449549 ontinnons System mutualitth The Dangers of Curve Fitting Curve fitting can only be used for the purpose of interpolation in space and extrapolation in time as long as the predicted variables stay Within their observed ranges Models obtained induCtiyebk by Curve tting a mathematical model to a set of observed data should never be used to explain the internal variables of the model 0 Such a model has no internal validity A better internally valid larch bud moth model shall be presented later November 21 2003 start Presentation ASL I we 449549 Glontinnnns System mutualitth Competition and Cooperation I 0 Two species can also interact With each other in other ways 0 They can eg compete for the same food source or they can cooperate eg in a symbiosis November 21 2003 start Presentation l7 A I we 449549 ontinnons System mutualitth Competition and Cooperation II 0 Animals of a single species can also cooperate eg by protecting each other in a herd grouping or they can suffer from crowding Of course several of these phenomena can take effect simultaneously November 21 2003 start Presentation EAEL we 449549 Glontinnnns System mutualitth Conclusions 0 We have looked at singlespecies ecosystems first We found that these populations always exhibit exponential growth followed by saturation This behavior can be modeled using the continuoustime logistic model We have seen that twospectes ecosystems often exhibit oscillatory behavior This behavior can be modeled using the Lotka Volterra model 0 In the next class we shall look at behavioral patterns exhibited by multispecies ecosystems November 21 2003 start Presentation l8 A I we 449549 ontinnons System mnemgl References Cellier FE 1991 Continuous System Modeling SpringerVerlag New York Chapter 10 Cellier FE 2002 Matlab code to curve t a logistic model to the US census data Cellier FE and A Fischlin 1982 Computerassisted modeling of illdefined systems in Progress in Cybernetics and Systems Research Vol 8 General Systems Methodology Mathematical Systems Theory Fuzzy Sets R Trappl GJ Klir and FR Pichler eds Hemisphere Publishing pp 417429 November 21 2003 Start Presentation
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