Fall2016Math1132Q(Section100)-Calculus2

MWF11:15am-12:05pm

Instructor:Dr.AngelynnAlvarezE-mail:angelynn.alvarez@uconn.edu

Office:MONT305OfficeHours:MWF12:15-1:15pm,orbyappt

Section9.4–ModelsforPopulationGrowth

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.

1. TheLawofNaturalGrowth• Let𝑃(𝑡)bethequantityofapopulationattime𝑡.• Therateofchangeof𝑃withrespectto𝑡 isproportionaltoitssize𝑃 𝑡 atanytime𝑡.

• Inequationform:𝑑𝑃𝑑𝑡

= 𝑘𝑃Note:• When𝑘 > 0,thepopulationisincreasing.• When𝑘 < 0,thepopulationisdecreasing.

Question:Howdoesthesolutionofaninitial-valueprobleminvolvingthelawofnaturalgrowthlooklike?

Answer:Needtosolve𝑑𝑃𝑑𝑡

= 𝑘𝑃, 𝑃 0 = 𝑃!

Thus:Thesolutiontotheinitial-valueproblem

𝑑𝑃𝑑𝑡

= 𝑘𝑃, 𝑃 0 = 𝑃!is*Wordproblemsinthissectionrequireustosolvefor𝑘(whengivenadditionalinformation)toanswerthegivenquestion.

Example:Acertaintypeofbacteria,givenafavorablegrowthmedium,doublesinpopulationevery6.5hours.Giventhattherewereapproximately100bacteriatostartwith,howmanybacteriawilltherebeinadayandahalf?

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?

Answer:Thesolutiontotheinitial-valueproblem

𝑑𝑃𝑑𝑡

= 𝑘𝑃 1 −𝑃𝑀

, 𝑃 0 = 𝑃!is*Wordproblemsinthissectionrequireustosolvefor𝑘(whengivenadditionalinformation)toanswerthegivenquestion.

Example:Supposeapopulationgrowsaccordingtoalogisticmodelwithinitialpopulation1,000andcarryingcapacity10,000.Ifthepopulationgrowsto2,500afteroneyear,whatwillthepopulationbeafteranother3 years?

Example:Fishermenfilledahugefishtankwith400salmonandestimatedthecarryingcapacitytobe10,000.Thenumberofsalmontripledinthefirstyear.Usethelogisticequationtofindanexpressionforthesizeofthepopulationafter𝑡years,andthenfindouthowlongitwouldtakeforthepopulationtoreach5,000.

Example:Acertainpopulationhasbeenmodeledbythedifferentialequation𝑑𝐵𝑑𝑡

= 𝑘𝐵(1 −𝐵𝑀)

where𝐵(𝑡)isthebiomassattime𝑡,and𝐵(𝑡)isinkilogramsand𝑡isinyears.Thecarryingcapacityisestimatedtobe𝑀 = 8kg,and𝑘 = 0.71peryear.If𝐵 0 = 2kg,howmanyyearswouldittakeforthebiomasstoreach4kg?

Example:Solvethelogarithmicequationalgebraically:2 ln 𝑥 + 1 − ln( 𝑥 − 1 !) = ln 𝑥!