MATHEMATICAL MODELS FOR HARVESTING OF NATURAL RESOURCES

Abstract

Overtheyears,ordinarydifferentialequations(ODEs)areusedtoformulatethemathematicalmod- els describingthedynamicsofpredatorpreyspecies.Theeconomicworthofoneorboththespecies has leadtoexploitationofthesespeciesintermsofharvesting.Theharvestedpredatorpreysystem exhibitsinterestingandcomplexdynamics.Inthisthesistheroleofharvestingonthepredatorprey system isexplored.Theeffectofcontinuous,piece-wisecontinousandswitchingharvestingisex- amined indistinctpredatorpreysystem.Theworkofthisthesisisorganizedinsevenchapter. The chapter1coversthefundamentalsofpredatorpreysystemandthemathematicalmethodsto analyze thesesystemsintermsoftheirstabilityandbifurcation.Abriefoverviewofdiscontinuous Filippovsystemsandthemethodologytoinvestigatethesesystemsispresented.Inthelastsection, the summaryofthethesisisappended. Chapter 2focusesontheselectiveharvestingofpredatorsinapredator-preysystem.Therefuge factorfortheprotectionofpreypopulationwhentheirdensityfallsbelowacertainthresholdisin- troduced. Atthesametime,additionalfoodisprovidedtothepredators.Theresultingsystemis analyzed usingapiece-wisesmoothFilippovmodel.Thestudyexaminestheexistenceandstability of equilibriumstatesintheindividualsmoothsubsystems.Theequilibriumstatesarecategorizedas regularorvirtualequilibriumstatesbasedonthequalityoftheadditionalfoodwithrespecttothe preypopulation.Itisdeterminedthatthetwointeriorequilibriumpointsareneverregularsimulta- neously inthescenariowhereonlyrefugeisprovidedtothepreywithoutprovidingadditionalfood for thepredators. In chapter3,theFilippovpredatorpreymodelisdevelopedforharvestingthatalternatesbetween the preyandpredator.Thesystemincorporatesanon-linearharvestingfunction.Theboundaryat which theswitchingoccursisdeterminedbyathresholdvaluethatreliesontheratiobetweenthe preyandpredatorpopulation.Thecatchabilitycoefficientofthepreyharvestingsystemplaysacru- cial roleindeterminingthenumberofboundaryequilibriumstatesinthesubsystemthatinvolves preyharvesting.Ontheotherhand,thenumberofinteriorequilibriumstatesinthesubsystemwith predator harvestingprimarilydependsonthecatchabilitycoefficientofthepredator.Theanalysis obtains conditionsonlevelofeffortthatensurestheco-existenceintheformofstablelimitcycle. Chapter 4examinesacontinuouspredatorpreysystemthatincludesselectiveharvestingofpredators. The systemconsiderslogisticpreygrowthandaratio-dependentfunctionalresponse.Adynamically varyingeffortisemployed,forMichaelis-Mententypeharvestingfunction.Toexplorethesystem dynamics aboutnon-singularorigin,itistransformedintoanotherregularsystem.Itisobservedthat the systemcancollapseevenwhenstartingwithpositiveinitialconditions.TheoccurrenceofHopf bifurcation aboutthecoexistenceequilibriumstateisestablishedbyconsideringtheparameter m, representing thefractionofpredatorsavailableforharvestingasbifurcatingparameter.Bytreating the fraction m as acontrolvariable,theoptimalharvestingpolicyisderivedusingPontraygin’smax- imum principle. Chapter 5investigatestheimpactoffearwithinthepreypopulationduetothepresenceofpreda- tor population.Thestudyspecificallyexaminesthestagestructureofpredators,assumingthatonly adult predatorsfeedonprey.ThemodelincorporatesaHollingtypeIIfunctionalresponse.The existenceofdistinctequilibriumstatesandtheirstabilitybehaviorisanalyzed.Thehighlevelof fear destabilizesthesystemintermsofnon-existenceofinteriorequilibriumstate.Also,thestudy establishes thesignificantinfluenceofthefearlevelontheoccurrenceofsaddlenodebifurcation. The chapter6scrutinizesthesysteminvolvingstagestructurewithinpreyspeciesandintraspecific competition betweenjuvenileandadultpreyinapredator-preysystem.Theanalysisincorporatesa Holling typeIIfunctionalresponse,assumingthatthejuvenilepreyarethepreferredfoodsourcefor the generalistpredator.Thesurvivalofthepredatorspeciesisensuredbysupplyingalternativefood to thespecies.Consideringeconomicworthofadultprey,proportionalharvestingofadultpreyis incorporated. Analgebraicequationisintroducedtoanalyzetheeconomicbenefitsresultingfrom the harvestingofadultprey.Thesingularity-inducedbifurcation(SIB)isdeducedforthecoexisting equilibrium stateofthedifferential-algebraicsystem,specificallyat v = 0, where v represents the profit orlossfromharvesting.ToeliminatetheSIB,astatefeedbackcontrollerisrecommendedfor the differential-algebraicsystem. Finally, Chapter 7 presents conclusions,summaryoffindingsofthethesisandfutureresearchdi- rections inpredatorpreysystems.