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1 Real Population Growth Ex1 The average annual growth rate for world population since 16 50 has been about 7 However the annual rate has varied signi64257cantly It peaked at about 1 during the 1960s and is currently about 2 Find the approximate dou ID: 36778

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Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionC SectionC.1:RealPopulationGrowth Ex.1Theaverageannualgrowthrateforworldpopulationsince1650hasbeenabout0:7%.However,theannualratehasvariedsignicantly.Itpeakedatabout2:1%duringthe1960sandiscurrentlyabout1:2%.Findtheapproximatedoublingtimeforeachofthesegrowthrates.Useeachtopredictworldpopulationin2050,basedona2000populationof6:0billion.1 Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionC SectionC.2:WhatDeterminestheGrowthRate? Denitionofoverallgrowthrate Theworldpopulationgrowthrateisthedifferencebetweenthebirthrateandthedeathrate:growthratebirthratedeathrate Ex.2Supposethatonaveragethereare8:5birthsper100peopleand6:5deathsper100peopleperyear.Whatisthepopulationgrowthrate? Ex.3In1950,theworldbirthratewas3:7birthsper100peopleandtheworlddeathratewas2:0deathsper100people.By1975,thebirthratehadfallento2:8birthsper100peopleandthedeathrateto1:1deathsper100people.Contrastthegrowthratesin1950and1975.2 Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionC SectionC.3:CarryingCapacityandRealGrowthModels Denitionofcarryingcapacity Foranyparticularspeciesinagivenenvironment,thecarryingcapacityisthemaximumpopulationthattheenvironmentcansupport. LogisticGrowthandOvershootandCollapse Logisticgrowthandovershootandcollapse Exponentialgrowthcannotcontinueindenitely.Indeed,humanpopulationcannotcontinuetogrowmuchlongeratthiscurrentrate,becausewewouldbeelbowtoelbowovertheentireEarthinjustafewcenturies.Theoreticalmodelsofpopulationgrowthassumethathumanpopulationislimitedbythecarryingcapacity.Twoimportantmodelsforpopulationsapproachingthecarryingcapacityare(1)agraduallevelingoff(logisticgrowth);(2)arapidincreasefollowedbyarapiddecrease(overshootandcollapse). Denitionoflogisticgrowth Alogisticgrowthmodelassumesthatthepopulationgrowthgraduallyslowsasthepopulationapproachesthecarryingcapacity.Whenthepopulationissmallrelativelytothecarryingcapacity,thelogisticgrowthisexponentialwithagrowthrateclosetothebasegrowthrater.Asthepopulationapproachescarryingcapacity,thelogisticgrowthrateapproacheszero.Thelogisticgrowthrateatanyparticulartimedependsonthepopulationatthattime,thecarryingcapacity,andthebasegrowthrater:logisticgrowthrater1population carryingcapacity 3 Chapter8:ExponentialAstonishmentLecturenotesMath1030SectionC Ex.4AssumethattheEarth'scarryingcapacityis12billionpeople.Giventhatthepopulationgrowthratepeakedinthe1960satabout2:1%whenthepopulationwasabout3billion,isitreasonabletoassumethathumanpopulationhasbeenfollowingalogisticgrowthpatternsincethe1960s?Isitreasonabletoassumethatpopulationhasbeengrowinglogisticallythroughoutthatpastcentury?Explain.Weneedtocomparethe2006growthrateofabout1:2%(seeExample1)tothegrowthratepredictedbyalogisticmodel.Since1:3%isclosetothecurrentgrowthrate(1:2%),itisreasonabletosaythathumanpopulationhasbeengrowinglogisticallysincethe1960s.However,humanpopulationhasnotbeenfollowinglogisticgrowthoverlongerperiods.Logisticgrowthrequiresacontinuallydecreasinggrowthrate,whichisconsistentwiththegrowthratepeakinginthe1960s.Inconclusion,itisstilltooearlytoknowwhetherthegrowthrateislogisticornot. Denitionofovershootandcollapse Alogisticmodelassumesthatthegrowthrateautomaticallyadjustsasthepopulationapproachesthecarry-ingcapacity.However,becauseoftheastonishingrateofexponentialgrowth,realpopulationoftenincreasebeyondthecarryingcapacityinarelativelyshortperiodoftime(overshoot).Whenapopulationovershootsthecarryingcapacityofitsenvironment,adecreaseinthepopulationisinevitable.Iftheovershootissubstantial,thedecreasecanberapidandsevere(collapse). 4