DYNAMIC COMPLEXITY IN DISCRETE-TIME POPULATION MOD

DYNAMICCOMPLEXITYINDISCRETE-TIMEPOPULATIONMODELSJanicaYlikarjulaThesisforthedegreeofLicentiateofTechnologyHelsinki,January1999HELSINKIUNIVERSITYOFTECHNOLOGYDepartmentofEngineeringPhysicsandMathematicsSystemsAnalysisLaboratoryChapter11.1IntroductionThemainaimofpopulationecologyistoexaminetheabundancesanddistributionsoforganismsandtheprocesseswhichdeterminethem(Begonetal.1996).Mathe-maticalmodellingisapowerfultooltoachievethisgoalbecauseittakesintoaccountonlythemainaspectsinthebiologicalsystemsandexcludestheunimportantfac-tors.Mathematicallanguagerequiresalsoexactformulation,whichforcesustomakealltheassumptionsclearlywhenwebuildamathematicalmodel.Inaddition,theabstractionobtainedbycreatingmathematicalmodelsismucheasiertodealwiththanthecomplexityofthesurroundingnature.Withtheoryandmathematicalmodellingwecanmakeexperimentsindierentconditionsandsituationsthantheonesweareexperiencingandconductforexamplesensitivityanalysis,whichhelpsustounderstandthedynamicalprocessesbetter.Moreimportantlyitmayarousenewquestionsandgivefreshideasforexperiments.Atthesametimeitsupportsandrationalisestheexperimentalresearchbyaddressingtheexperimetstothepotentiallymostinterestingcases.Inthisthesischangesinpopulationsizesareapproachedbymathematicalmod-elling.Themainfocusofthestudiesofpopulationdynamicshastraditionallybeenonlocalstabilityanalysis{searchingtheequilibriumpointsanddeterminingthenatureoftheirstability.However,thiskindofstudyisnotadequatewiththenon-linearsystemsbecauseoftheirirregularbehaviour,e.g.,longorchaotictransients1(Kaitalaetal.1999)andintermittency(Ylikarjula&Kaitala1998;Kaitalaetal.1999).Inthepopulationmodelsthenonlineareectsarestillrealistic,becausemanyecologicalphenomenaarenaturallynonlinear,e.g.,Alleeeect(evidencefortheAlleeeectinnaturalpopulationsseeKuussaarietal.1998)andthesatura-tionofthepredationrate.Includingalltherelevantparametersinlargesimulationmodelsinordertounderstandlong-termecologicalchangeswillprobablyworkverypoorly.Alternatively,wecanstudyqualitativelythemainfeaturesofthepopulationbehaviour(Vandermeer1993).TheseminalworkbyRobertMay(1974,1976)arousedthequestionwhethercom-plexdynamicsarewidespreadinnature.Hediscoveredthatalsosimplerst-orderdierenceequations,frequently-usedinecologicalmodels,elicitverycomplexdy-namics.Incorporatingecologicallyrealisticfeaturesintopopulationmodelsoftenleadstocomplexdynamics.Suchfeaturesareforexamplesimpleinteractionsamongspecies(e.g.,Ives1991).Ifthesemultidimensionalsystemsarefalselyanalysedinfewerdimensions,thecomplexityofthedynamicsmaybehidden(Turchin&Taylor1992).Therehasbeenavividdiscussionamongecologistsconcerningthecomplexityinnaturalecosystems(e.g.,Pimm1984;Godfray&Blythe1990;Turchin&Taylor1992).Thelaboratoryexperimentswithourbeetle(Triboliumcastaneum)donebyR.F.Costantinoandhisco-workershavegivennewinsightintothisquestion(Costantinoetal.1995,1997,1998;Dennisetal.1995;Desharnaisetal.1997).Theauthorsshowedexperimentallythatchangesindemographicparameters(e.g.,fecundityandsurvival)maycausequalitativechangesinthebehaviourofthepopu-lationasasimplenonlinearage-structuredmodelpredicts.Thishasgiveninsuringevidenceoftheimportanceofnonlinearmathematicsinpopulationbiologyandtheabilityoffeasiblemathematicalmodelstopredictchangesinthedynamicsofecologicalsystems(Rohani&Miramontes1996;Rohani&Earn1997).Inecologyspecialinteresthasrelatedtotheconceptofchaos(Schaer&Kot1985;Berryman&Millstein1989;Ives1991;Hastingsetal.1993;Scheuring&Janosi21996).Inshort,chaosisthestateofthesystemwherethebehaviourisaperiodicandinpracticeunpredictable(Piirila&Seppanen1990).MorepreciselychaoticsystemcanbedenedasasystemwhichhasatleastonepositiveaverageLyapunovexponent(Hilborn1994;forcalculationofLyapunovexponentofdiscretesystemsseee.g.,vonBremenetal.1997).Oneofthebasicphenomenarelatedtochaosissensitivitytoinitialvaluessometimesreferredtoasthebutteryeect:smallchangesininitialconditionscanaltersubstantiallythebehaviourofthesystem.Thismeansthatalthoughthesystemisdeterministic,thecalculationsbecomeverysoonquiteinaccuratebecauseoftheniteprecisionofanynumericalcalculations.Thereareseveralproblemsrelatedtodiscoveringchaoticdynamicsinnature(e.g.,Godfray&Blythe1990).Twomajordicultiesaretheshortnessandthenoisinessofbiologicaltimeseries.Long-termirregularchangesinnaturalpopulationdensitiescanbeasignofchaosbutalsoforexamplestochasticenvironmentalnoisecausedbychangesinenvironmentcanmakeotherwiseperiodicpopulationbehaveseeminglyasachaoticpopulation.Problemsmayalsoariseforexampleinlaboratoryexperimentsifthereistoolittleknowledgeoftheconsequencesofdierentstartingconditionsand,thus,oftransientbehaviour.Furthermore,changingselectionpressuresmaycomplicatetheidenticationofpopulationdynamics.Thepracticalandtheoreticaloptionsfordetectingcomplicateddynamicsinpopulationdataarestillamajorchallengeinecologicalresearch(Farmer&Sidorowich1987;Bartlett1990,Sugihara&May1990;Sugiharaetal.1990;Casdagli1991;Cazelles&Ferriere1992;Stone1992;Tsonis&Elsner1992;Sugihara1994;Stoneetal.1996).Theincreasingnumberofecologicalpaper