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疾病在食饵中传播的具有时滞的捕食被捕食模型的分析.docx

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疾病在食饵中传播的具有时滞的捕食被捕食模型的分析 Introduction: Foodborneillnessesareacommonprobleminmanypartsoftheworld.Infectiousagentscancontaminatefoodatanypointduringtheproduction,handling,andpreparationprocess,leadingtothespreadofdiseases.Inthispaper,weaimtoanalyzethetransmissionofdiseasesinafoodchain,wherepredatorsandpreyarebothexposedtoinfectiousagentsinthefoodtheyconsume.Wefocusonthetimedelayinthetransmissionofdiseasesanditseffectsonthedynamicsofthesystem. TheModel: Westartbyconsideringabasicfoodchainmadeupoftwospecies:apreyspeciesthatisthesourceoftheinfectiousagentandapredatorspeciesthatfeedsontheprey.Weassumethattheinfectiousagenthasatimedelayinitsspreadfromthepreytothepredatorspecies.Thisdelaycanrangefromafewhourstoseveraldays,dependingonthetypeofpathogenandthespecificfoodchainunderconsideration. Thedynamicsofthissystemcanbedescribedbyasetofdifferentialequations.LetN(t)andP(t)bethepopulationsizesofthepreyandpredatorspecies,respectively,attimet.LetI(t)andJ(t)bethedensitiesoftheinfectiousagentinthepreyandpredatorspecies,respectively,attimet.Finally,letDbethetimedelayinthetransmissionofthedisease.Thenthemodelcanbewrittenasfollows: dN/dt=rN-aNI dP/dt=baPJ-dP dI/dt=gN-cDI-h(t-D)I dJ/dt=cDI-eJ whereristheintrinsicgrowthrateofthepreyspecies,aisthepredationrateofthepredatorspecies,bistheconversionefficiencyofthepredatorspecies,disthenaturalmortalityrateofthepredatorspecies,gistherateofinfectionofthepreyspecies,cistherateofinfectionofthepredatorspecies,histherateatwhichtheinfectiousagentspreadsfromthepreytothepredatorspecies,andeisthenaturalmortalityrateoftheinfectiouspredatorspecies. Analysis: Ourmodelexhibitscomplexdynamics,includingbothstableandunstableequilibria,aswellaslimitcycles.Oneofthekeyfactorsaffectingthedynamicsofthesystemisthetimedelayinthetransmissionofthedisease.ForsmallvaluesofD,thepredatorpopulationcanbecomeinfectedquickly,leadingtoarapiddeclineinitspopulationsize.However,asthetimedelayincreases,thepredatorpopulationhastimetorecoverbeforeitbecomesinfected,leadingtomorest