简介：Westudyamodelforthelong-termbehaviorofasingle-speciespopulationwithsomedegreeofpollutiontoleranceinapollutedenvironment.Themodelconsistsofthreeordinarydifferentialequations:oneforthepopulationdensity,onefortheamountoftoxicantinsidethelivingorganisms,andonefortheamountoftoxicantintheenvironment.Wederivesufficientconditionsforthepersistenceandtheextinctionofthepopulationdependingontheexogenousinputrateofthetoxicantintotheenvironmentandthelevelofpollutiontoleranceoftheorganisms.Numericalsimulationsarecarriedouttoillustrateourmainresults.

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简介：Anon-autonomouscompetingsystemisinvestigatedinthispaper,wherethespeciesxcandiffusebetweentwopatchesofaheterogeneousenvironmentwithbarriersbetweenpatches,butforspeciesy,thediffusiondoesnotinvolveabarrierbetweenpatches,furtheritisassumedthatalltheparametersaretimedependent.Itisshownthatthesystemcanbemadepersistentundersomeappropriateconditions.Moreover,sufficientconditionsthatguaranteetheexistenceofauniquepositiveperiodicorbitwhichisgloballyasymptoticstablearederived.

简介：Apredator-preysystemwithaconstantproportionofpreyrefugeandstage-structureforpreyspeciesisproposedandstudiedinthispaper.Asetofconditionsforthepermanenceofthesystemisobtained.Thelocalstabilityofthesystemisdiscussedbythesignofeigenvalues.Furthermore,byusingtheiterativemethod,somesuitablesufficientconditionsfortheglobalattractivityoftheinteriorequilibriumisobtained.Ourstudyshowsthattheconstantproportionofpreyrefugecouldleadtomorecomplicatedynamicbehaviors.Numericalsimulationsarealsopresentedtoillustratethefeasibilityofthemainresults.