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Fall2016Math1132Q(Section100)-Calculus2
MWF11:15am-12:05pm
Instructor:Dr.AngelynnAlvarezE-mail:[email protected]
Office:MONT305OfficeHours:MWF12:15-1:15pm,orbyappt
Section9.4–ModelsforPopulationGrowth
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Inthissection,weinvestigateseparabledifferentialequationsthatareusedtomodelpopulationgrowth:1. TheLawofNaturalGrowth2. TheLogisticEquation
Propertiesof𝒇 𝒙 = 𝒍𝒏(𝒙)toremember:• Thedomainofln (𝑥)is{𝑥 ∶ 𝑥 > 0}
• ln 𝑎 + ln 𝑏 = ln (𝑎𝑏)
• ln 𝑎 − ln 𝑏 = ln !!
• ln 𝑎! = 𝑏 ∙ 𝑙𝑛 𝑎
• 𝑒!andln (𝑥)areinversesofeachother.
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1. TheLawofNaturalGrowth• Let𝑃(𝑡)bethequantityofapopulationattime𝑡.• Therateofchangeof𝑃withrespectto𝑡 isproportionaltoitssize𝑃 𝑡 atanytime𝑡.
• Inequationform:𝑑𝑃𝑑𝑡
= 𝑘𝑃Note:• When𝑘 > 0,thepopulationisincreasing.• When𝑘 < 0,thepopulationisdecreasing.
Question:Howdoesthesolutionofaninitial-valueprobleminvolvingthelawofnaturalgrowthlooklike?
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Answer:Needtosolve𝑑𝑃𝑑𝑡
= 𝑘𝑃, 𝑃 0 = 𝑃!
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Thus:Thesolutiontotheinitial-valueproblem
𝑑𝑃𝑑𝑡
= 𝑘𝑃, 𝑃 0 = 𝑃!is*Wordproblemsinthissectionrequireustosolvefor𝑘(whengivenadditionalinformation)toanswerthegivenquestion.
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Example:Acertaintypeofbacteria,givenafavorablegrowthmedium,doublesinpopulationevery6.5hours.Giventhattherewereapproximately100bacteriatostartwith,howmanybacteriawilltherebeinadayandahalf?
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2. TheLogisticModel• Weneedtoconsiderthefactthatresourcesforlifeinanenvironmentcanbelimited.
• Weneedtoconsiderthecarryingcapacityofanenvironment.Ø Definition:Thecarryingcapacityofaparticularenvironmentisthemaximumpopulationsizethattheenvironmentcansupportinthelongrun.
• Let𝑃(𝑡)bethequantityofapopulationattime𝑡.• Let𝑀bethecarryingcapacityoftheenvironment.• Then,thelogisticdifferentialequationis:
𝑑𝑃𝑑𝑡
= 𝑘𝑃 1 −𝑃𝑀
Note:(Undertheassumptionthat𝑘 > 0)• If0 < 𝑃 < 𝑀,then!"
!"> 0and𝑃isincreasing.
• If𝑃 > 𝑀,then!"!"< 0and𝑃isdecreasing.
Question:Howdoesthesolutionofaninitial-valueprobleminvolvingthelogisticdifferentialequationlooklike?
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Answer:Thesolutiontotheinitial-valueproblem
𝑑𝑃𝑑𝑡
= 𝑘𝑃 1 −𝑃𝑀
, 𝑃 0 = 𝑃!is*Wordproblemsinthissectionrequireustosolvefor𝑘(whengivenadditionalinformation)toanswerthegivenquestion.
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Example:Supposeapopulationgrowsaccordingtoalogisticmodelwithinitialpopulation1,000andcarryingcapacity10,000.Ifthepopulationgrowsto2,500afteroneyear,whatwillthepopulationbeafteranother3 years?
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Example:Fishermenfilledahugefishtankwith400salmonandestimatedthecarryingcapacitytobe10,000.Thenumberofsalmontripledinthefirstyear.Usethelogisticequationtofindanexpressionforthesizeofthepopulationafter𝑡years,andthenfindouthowlongitwouldtakeforthepopulationtoreach5,000.
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Example:Acertainpopulationhasbeenmodeledbythedifferentialequation𝑑𝐵𝑑𝑡
= 𝑘𝐵(1 −𝐵𝑀)
where𝐵(𝑡)isthebiomassattime𝑡,and𝐵(𝑡)isinkilogramsand𝑡isinyears.Thecarryingcapacityisestimatedtobe𝑀 = 8kg,and𝑘 = 0.71peryear.If𝐵 0 = 2kg,howmanyyearswouldittakeforthebiomasstoreach4kg?
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Example:Solvethelogarithmicequationalgebraically:2 ln 𝑥 + 1 − ln( 𝑥 − 1 !) = ln 𝑥!