Imulationsay usingyay ayay Eadsheetspray (ayay pDFay ersionvay isyay erehay)

Ifyay youay areyay unfamiliaryay ithway usingyay ayay eadsheetspray, ayay utorialtay ancay ebay oundfay erehay. Ayay oodgay eneralgay esourceray onyay odelingmay andyay imulationsay ancay ebay oundfay erehay: httpay://wwway.icsay.uoregonyay.eduyay/acsenay/ecosimyay/

Odelingmay ocessespray inyay ethay iencesscay avehay ecomebay usefulyay esearchray oolstay inyay anymay areasyay ofyay ethay iencesscay andyay ethay availabilityyay ofyay omputerscay extendsyay esethay echniquestay otay anyyay atthay ishway otay useyay esethay oolstay. Athematicalmay epresentationsray ofyay iologicalbay ystemssay avehay onglay eenbay usedyay ybay ecologistsyay otay odelmay ariousvay ocessespray. Ethay odelsmay areyay usefulyay otay accessyay owhay uchmay isyay understoodyay aboutyay ayay ealray ystemsay. Ifyay ethay odelmay isyay ayay oodgay epresentationray ofyay atwhay oesgay onyay inyay ethay "ealray orldway", enthay ethay authoryay isyay airlyfay onfidentcay atthay eythay understandyay ethay enomenaphay. Ifyay, oweverhay, ethay odelmay oesn'tday ollowfay ethay "ealray orldway", enthay it'syay ackbay otay ethay omputercay. Omesay odelsmay areyay osay oodgay eythay avehay edictivepray aluevay. Odelsmay atthay edictpray ocustlay outbreaksyay inyay Africayay areyay outinelyray usedyay otay edictpray enwhay ontrolcay easuresmay areyay eedednay inyay ayay ejoinray. Atyay omehay, armersfay outinelyray useyay odelsmay otay edictpray enwhay erbicideshay, ungicidesfay, andyay insecticidesyay ouldshay ebay appliedyay otay eirthay ieldsfay. Onyay ayay andergray alescay, imateclay odelsmay areyay usedyay otay apeshay awslay andyay internationalyay eatiestray. Oneyay ofyay ethay irstfay odelsmay ittenwray asway oneyay otay imicmay opulationpay owthgray. Ayay odelmay ofyay ethay imulationsay ocesspray ancay ebay eensay inyay ethay ollowingfay iagramday.

Ethay irstfay odelmay e'llway exploreyay isyay ethay eometricgay odelmay. Accordingyay otay esethay equationsyay, ayay opulationpay owsgray uninhibitedyay overyay imetay. Ethay importantyay ariablesvay inyay isthay imulationsay areyay ethay artingstay opulationpay izesay andyay ethay ateray ofyay increaseyay. Ethay equationyay orfay isthay imulationsay isyay: dnay/dtay = ray*nay erewhay dnay/dtay isyay ethay angechay inyay ethay opulationpay izesay (nay) overyay imetay (tay). Ethay ariablevay "ray" isyay ethay ateray atyay ichwhay ethay opulationpay increasesyay erpay enerationgay (etweenbay 0.0 andyay 1.0). Ethay "day" artpay ofyay dnay/dtay ustjay andsstay orfay "angechay inyay" (eltaday) osay ityay isyay eadray asyay "angechay inyay umbernay overyay angechay inyay imetay". Ethay nay_{(tay +1)} = nay_tay + dnay/dtay  isyay eadray asyay ethay "opulationpay izesay atyay imetay +1 isyay equalyay otay ethay opulationpay izesay atyay imetay 0 usplay ethay angechay inyay izesay overyay ethay angechay inyay imetay". Isthay enarioscay illway esultray inyay anyay exponentialyay increaseyay inyay ethay opulationpay izesay.

• Openyay anyay excelyay eadsheetspray andyay ypetay "Exponentialyay Owthgray Odelmay" inyay ellcay Ayay1.

• Inyay ellcay Ayay3 ypetay "Artstay nay:" andyay inyay ellcay bay3 ypetay "10". Eway illway artstay ithway ayay opulationpay ofyay 10.

• Inyay ellcay Ayay4 ypetay "ray:" andyay inyay ellcay bay4 ypetay "0.2". Ethay owthgray ateray isyay "0.2"

• Inyay ellcay Ayay9 ypetay "tay" (ouryay imetay olumncay) andyay inyay ellcay bay9 ypetay "nay" (opulationpay izesay atyay imetay tay

• Inyay ellcay Ayay10 ypetay "1", ethay artstay ofyay imetay.

• Inyay ellcay Ayay11 ypetay "=Ayay10+1", ethay ormulafay orfay advancingyay ouryay imetay unityay ybay oneyay. Enwhay youay ithay eturnray, ethay ellcay ouldshay owshay "2".

• Opycay ethay ontentscay ofyay Ayay11 otay ethay ipboardclay. Electsay andyay ighlighthay ellscay Ayay12 oughthray Ayay39 andyay astepay ethay ipboardclay ataday intoyay esethay ellscay. Osethay olumnscay ouldshay ownay owshay ethay umbersnay 1-30.

• Inyay ellcay bay10 ypetay "=bay3". E'reway ustjay opyingcay ethay artingstay opulationpay izesay omfray ellcay bay3. Ityay ouldshay aysay "10"

• Inyay ellcay bay11 ypetay "=bay10+($bay$4*bay10). Ethay ollarday ignssay areyay ayay icktray tay0 anchoryay atthay artpay ofyay ethay ormulafay osay atthay ityay alwaysyay ointspay otay ouryay ateray ofyay increaseyay inyay ellcay bay4. Ellcay bay11 ouldshay aysay 12.

• Opycay ethay ormulafay inyay bay11 otay ethay ipboardclay. Ighlighthay ellscay bay12 oughthray bay30 andyay astepay ethay ormulafay intoyay osethay ellscay.

• Otay eesay ethay esultsray (ichwhay areyay obablypray iderway anthay ethay olumncay, ogay otay ethay optay ofyay ethay agepay ithway ethay olumncay eadingshay (Ayay,bay,cay...) andyay enthay ositionpay youray ursorcay overyay ethay inelay etweenbay bay andyay cay. Ityay illway itchsway otay ayay ursorcay atthay ookslay ikelay isthay: . Enthay ickclay andyay agdray ethay olumncay osay it'syay argerlay andyay youay ancay eesay allyay ethay umbersnay

• Et'slay akemay ayay aphgray. Irstfay, electsay owsray bay10 oughthray bay39 andyay ickclay ethay artchay izardway uttonbay ().Angechay otay ayay inelay artchay, obablypray ethay irstfay oneyay. Ickclay Inishfay andyay youray artchay illway ebay isplayedday.

• Ytray ifferentday artingstay opulationpay izessay andyay atesray ofyay increaseyay (aryvay oneyay oryay ethay otheryay, otnay othbay atyay ethay amesay imetay).

• Avesay youray ilefay otay youray way ivedray asyay "Exponentialyay Odelmay".

Igurefay 1. Umanhay Opulationpay Owthgray

Ethay eometricgay odelmay isyay otnay eryvay atisfyingsay incesay ityay oesn'tday eemsay otay imicmay ethay ayway ealray opulationspay ehavebay. Ealray opulationspay on'tday expandyay indefinitelyyay (exceptyay, erhapspay umanshay; Igurefay 1.  Insteadyay, eythay endtay otay increaseyay orfay ayay ortshay imetay, enthay evellay offyay. Igurefay 2 epictsday ethay owthgray ofyay ighornbay eepshay opulationspay inyay ethay Ockiesray omfray ethay earlyyay 1800'say otay aboutyay 1940. Otenay ethay apidray expansionyay ofyay ethay opulationpay ollowingfay it'syay initialyay introductionyay, ollowedfay ybay ayay eveling-offlay ofyay ethay opulationpay atyay aboutyay 1.75 illionmay eepshay. Isthay isyay ayay ypicaltay opulationpay esponseray eensay inyay ostmay aturalnay opulationspay. It'syay asyay ifyay ethay opulationpay ashay illedfay otay environmentyay. Isthay "illingfay" ofyay ethay environmentyay ithway ayay articularpay eciesspay isyay ethay "arryingcay apacitycay" orfay atthay eciesspay. Orfay irdsbay, ethay imitinglay actorfay inyay ethay environmentyay aymay ebay ethay availabilityyay ofyay extnay itessay. Otheryay eciesspay aymay ebay imitedlay ybay ethay availabilityyay ofyay oodfay oryay aterway. Ateverwhay ethay ausecay, ethay environmentyay isyay apablecay ofyay upportingsay ayay imitedlay umbernay ofyay ayay articularpay eciesspay. Atthay umbernay isyay ethay arryingcay apacitycay, oryay ityay epresentsray ethay environmentalyay esistanceray otay urtherfay opulationpay owthgray. Ethay arryingcay apacitycay ofyay ayay ystemsay aymay angechay overyay imetay. Oughtsdray, estilencepay, andyay otheryay imaticclay angeschay aymay emporallytay increaseyay oryay epressday ethay arryingcay apacitycay ofyay ethay environmentyay. Esethay actorsfay aymay ebay esponsibleray orfay ethay "umpsbay" inyay ethay aboveyay ataday (oremay onyay atthay aterlay).

Igurefay 2. Ighornbay eepshay opulationspay

Otay ebay ealisticray, ouryay odelmay eedsnay otay eflectray ethay arryingcay apacitycay ofyay anyay ecologicalyay ystemsay. Ecologistsyay avehay ettledsay onyay ethay ariablevay "kay" otay epresentray ethay arryingcay apacitycay ofyay ethay environmentyay orfay ayay ivengay opulationpay ofyay organismsyay. Orfay ouryay implesay odelmay, ityay oesn'tday attermay ifyay ethay imitinglay actorfay isyay estnay itessay, oryay ethay availabilityyay ofyay oodfay (althoughyay ifferentday imitinglay actorsfay ouldcay ebay easilyyay accommodatedyay. Isthay equationyay isyay essentiallyyay ethay amesay asyay ouryay irstfay ytray, ithway ethay exceptionyay ofyay ethay (k-Nay)/kay additionyay (kay isyay ethay arryingcay apacitycay, nay isyay ethay urrentcay opulationpay ensityday). Ethay (k-Nay)/kay ermtay inyay ethay equationyay "utspay ethay eaksbray" onyay opulationpay owthgray asyay ethay opulationpay eachesray arryingcay apacitycay. Et'slay ixfay ouryay irstfay ytray otay eflectray isthay opefullyhay improvedyay odelmay. Ethay equationsyay areyay ownshay elowbay.

• Openyay youray Exponentialyay Odelmay onyay youray way ivedray ifyay ityay isn'tyay alreadyyay openyay.

• Eplaceray ellcay Ayay1 ithway "Ogisticlay Owthgray". Enthay useyay ethay "Ilefay.. Avesay asyay" excelyay optionyay otay avesay ethay ilefay asyay "Ogisticlay" onyay youray way ivedray. Isthay insuresyay atthay youay on'tway overwriteyay youray originalyay odelmay.

• Inyay ellcay Ayay5 ypetay "kay:". Inyay ellcay bay5 enteryay "400". Ethay arryingcay apacitycay orfay ouryay opulationpay illway ebay 400.

• Inyay ellcay bay11 youay utpay ethay ewnay equationyay. Ypetay "=bay10+($bay$4*bay10)*(($bay$5-bay10)/$bay$5)" Omparecay isthay otay ethay aboveyay equationyay otay eesay owhay ityay orksway.

• Opycay ethay ontentscay ofyay bay11 otay ellscay bay12 oughthray bay39. Youray aphgray obablypray angedchay.

• Eginbay ithway ayay artingstay opulationpay ofyay 10, ayay ateray ofyay increaseyay ofyay 0.5 andyay ayay arryingcay apacitycay ofyay 400. Enthay ytray otheryay ombinationscay (angingchay onlyyay oneyay atyay ayay imetay) otay eesay owhay youray odelmay orksway.

Illstay, esethay odelsmay areyay otnay ompletelycay atisfyingsay (eythay on'tday avehay ethay igglesjay eensay inyay Igurefay 2), osay omesay oremay eakingtway isyay eedednay. Ethay urrentcay odelmay assumesyay atthay ayay opulationpay espondsray instantaneouslyyay otay angeschay inyay opulationpay izesay. Isthay assumptionyay isyay unreasonableyay. Ifyay ethay opulationpay ashay eachedray it'syay arryingcay apacitycay (kay), andyay 50% oryay 60% ofyay ethay emalesfay areyay egnantpray, ey'llthay illstay ivegay irthbay, ausingcay ethay opulationpay otay overshootyay ethay arryingcay apacitycay. Inyay additionyay, omesay essesstray on'tday assertyay emselvesthay immediatelyyay. Orfay exampleyay, asyay opulationpay ensityday increasesyay, ityay ouldway aketay omesay imetay orfay individualsyay otay ecomebay essedstray. Ormoneshay ickkay inyay andyay  eventuallyyay ethay individualsyay areyay otnay asyay apablecay ofyay eproductionray. Anyay exampleyay ofyay ayay angechay inyay eproductiveray ateray ithway increasingyay opulationpay ensityday isyay ownshay inyay Igurefay 3. Underyay owlay ensityday onditionscay ethay ustaceancray Aphniaday (aterway eaflay) oducespray 4 offspringyay erpay ayday ilewhay underyay ighhay ensityday onditionscay onay offspringyay areyay oducedpray. Otenay alsoyay atthay urvivorshipsay isyay adverselyyay affectedyay ybay increasingyay opulationpay ensityday.

Igurefay 3. Angeschay inyay eproductionray ofyay Aphniaday underyay ifferentday opulationpay essurespray

Isthay eproductiveray imetay aglay isyay epresentedray ybay ariablevay cay inyay ethay odelmay; eesay odelmay elowbay. Ouryay opulationpay oesn'tday espondray instantaneouslyyay, utbay isyay aggedlay ybay oneyay enerationgay.

• Openyay ethay Ogisticlay odelmay ifyay it'syay otnay onyay youray eenscray. Eplaceray ellcay Ayay1 ithway "Eproductiveray Aglay Odelmay" andyay useyay ethay "Ilefay.. Avesay asyay..." enumay unctionfay otay avesay youray ilefay asyay "Eproductiveray"

• Inyay ellcay Ayay6 ypetay "cay:" andyay inyay bay6 ypetay "1". Ityay urnstay outyay atthay it'syay ifficultday otay utpay ogethertay ayay ormulafay atthay illway eadray isthay umbernay. It'syay uchmay easieryay otay angechay ethay ormulafay inyay ethay ightray acesplay. Esethay ellcay entriesyay areyay ustjay usedyay asyay emindersray.

• Angechay ethay ontentscay ofyay ellcay bay12 otay =bay11+($bay$4*bay10)*(($bay$5-bay10)/$bay$5). Ethay edray ighlightshay owshay erewhay ethay ormulafay isyay angedchay. Ethay easiestyay ayway otay oday isthay isyay otay ickclay onyay bay12 andyay enthay edityay ethay ormulafay inyay ethay ormulafay oxbay (fxay oxbay) atyay ethay optay ofyay ethay ormfay. Youay ancay aceplay youray ursorcay onyay ethay ormulafay, useyay ethay eftlay andyay ightray eyboardkay arrowsyay otay ovemay aroundyay andyay eleteday andyay angechay ethay ormulafay. Esspray eturnray enwhay youray inishedfay. Atwhay avehay eway ustjay oneday (ebay ableyay otay explainyay owhay isthay itsfay ethay aboveyay equationyay orfay ayay imetay aglay ofyay 1)

• Opycay ethay ontentscay ofyay ellcay bay12 otay bay13 oughthray bay39.

• Ytray ethay ollowingfay:

  • nay=10, ray=.2, kay=400

  • nay=10, ray=.5, kay=400

  • nay=10, ray=.9, kay=400

  • Enthay oldhay ray atyay ayay owlay oryay ighhay aluevay andyay angechay kay, enthay nay. At'swhay oinggay onyay?

• Owhay ouldcay youay imulatesay ayay eproductiveray aglay ofyay 2? Inthay: bay12 asway angedchay orfay ayay aglay ofyay 1, osay bay13 eedsnay otay ebay angedchay otay orcefay ayay aglay ofyay 2. On'tday orgetfay otay opycay ethay ontentscay ofyay bay13 otay bay14 oughthray bay39.

Ithway ayay eproductiveray aglay eway indfay atthay opulationspay ithway ayay owlay intrinsicyay ateray ofyay increaseyay (ray) urvivesay ongerlay anthay osethay ithway ayay ighhay eproductiveray ateray. Inyay additionyay, ethay abilitystay ofyay osethay opulationspay ithway owlay eproductiveray atesray isyay eatergray (esslay oscillationsyay inyay eirthay owthgray urvescay). Inallyfay, opulationspay ithway ighhay eproductiveray atesray alsoyay areyay onepray otay outbreaksyay (opulationpay increasesyay arfay inyay excessyay ofyay ethay arryingcay apacitycay).

Inyay ethay ealray orldway onditionscay affectingyay ethay eproductiveray atesray ofyay organismsyay (uchsay asyay overcrowdingyay, oughtdray, iseaseday, etcyay) areyay otnay ikelylay otay astlay overyay anyay extendedyay eriodpay ofyay imetay. Inyay ethay ort-termshay, opulationspay ithway owlay eproductiveray atesray ancay aitway outyay ethay environmentalyay oblemspray ilewhay osethay ithway igherhay eproductiveray atesray areyay oremay ikelylay otay ogay extinctyay ocallylay. Eway alsoyay indfay atthay onditionscay atthay adverselyyay affectyay ethay arryingcay apacitycay ancay alsoyay akemay ayay eciesspay oremay onepray otay ocallay extinctionyay (especiallyyay osethay ithway ayay ighhay ray). Activitiesyay uchsay asyay eforestationday, aterway ollutingpay, andyay overuseyay ofyay esticidespay ancay allyay ecreaseday ethay arryingcay apacitycay (increaseyay ethay environmentalyay esistanceray) andyay akemay opulationspay oremay usceptiblesay otay ocallay extinctionyay.

Ecognizingray ethay ifferencesday etweenbay eciesspay ithway ighhay atesray ofyay eproductionray andyay osethay ithway owlay intrinsicyay atesray ofyay increaseyay, ecologistsyay oftenyay escribeday ayay eciesspay asyay eingbay "electedr-say" oryay "Electedk-say" (Abletay 1). Osethay eciesspay osewhay opulationspay evelslay areyay ontrolledcay ybay eirthay eproductiveray atesray areyay efinedday asyay electedr-say ilewhay osethay osewhay eproductionray isyay ontrolledcay oremay ybay environmentalyay esistanceray areyay ermedtay Electedk-say.

Abletay 1

E'reway ettinggay oseclay. Ealray opulationspay owshay uctuationsflay inyay ensityday inyay esponseray otay andomlyray angingchay environmentalyay allengeschay (asyay ellway asyay intrinsicyay ythmsrhay). Inyay omesay asescay ethay uctuationsflay areyay earlyclay ueday otay angeschay inyay ethay environmentyay, ilewhay atyay otheryay imestay ethay ariationsvay areyay otnay asyay earclay. Eway ancay imicmay esethay andomray effectsyay ybay andomlyray addingyay oryay ubtractingsay individualsyay omfray ethay urrentcay opulationpay izesay:

Inyay ethay aboveyay ormulafay ethay izesay ofyay nay isyay ecreasedday ybay ayay andomray amountyay andyay enthay increasedyay ybay ayay andomray amountyay (inyay Excelyay ethay ANDRAY() unctionfay eneratesgay ayay andomray umbernay etweenbay 0 andyay 1, osay ifyay eway ultiplymay atthay andomray umbernay ybay nay, nay illway ebay ecreasedday. Eway enthay addyay onyay anotheryay andomray amountyay esslay anthay nay. Ereforethay, nay illway ometimessay ebay argerlay oryay ometimessay allersmay. Isthay affectsyay ethay abilityyay ofyay ouryay odelmay organismsyay otay acktray erewhay eythay areyay inyay elationray otay ethay arryingcay apacitycay (kay). Ometimessay eythay "inkthay" eythay areyay elowbay ethay arryingcay apacitycay, enwhay actuallyyay eythay aymay ebay aboveyay ityay (andyay isavay ersavay). Oneyay ouldcay imagineyay anyay electedr-say eastbay, uchsay asyay ayay ugbay, ouldn'tway avehay ayay oodgay ideayay ofyay eirthay elationray otay ethay environmentyay andyay ouldcay akemay uchsay ayay "istakemay". Alternatelyyay, ifyay ethay environmentyay isyay apidlyray angingchay ethay odelmay opulationpay ightmay avehay oubletray ackingtray ityay accuratelyyay. Ethay aboveyay odelmay ouldway alsoyay itfay atthay enarioscay.

• Openyay ethay Ogisticlay odelmay andyay angechay ellcay Ayay1 otay "Ochasticstay Odelmay". Useyay ethay "Ilefay..Avesay Asyay" enumay unctionfay andyay avesay ethay ewnay ilefay asyay "Ochasticstay" onyay ethay way ivedray.

• Angechay ethay ormulafay inyay ellcay bay11 otay "=bay10+(($bay$4*bay10)*($bay$5-(bay10-(ANDRAY()*bay10)+(ANDRAY()*bay10)))/$bay$5)" Ememberray atthay youay ancay edityay ethay existingyay ormulafay atherray anthay ypetay inyay ethay olewhay ewnay ormulafay. Orrelatecay ethay ormulafay youay ustjay enteredyay ithway ethay opulationpay odelmay ormulafay ownshay aboveyay. Atchway outyay orfay ethay arenthesespay. Ifyay youay etgay anyay erroryay essagemay, oday otnay etlay excelyay ixfay youray ormulafay, it'llyay obablypray oday ityay ongwray.

• Opycay ethay ormulafay inyay bay11 otay ethay ipboardclay andyay enthay astepay ityay inyay ellscay bay12 oughthray bay39.

• Angechay ethay artingstay aluesvay otay nay=100, ray=.2, kay=1000 andyay ethay ogrampray illway enerategay ayay ewnay aphgray.

• Etslay odifymay ethay odelmay osay atthay ityay illway unray orfay 100 enerationsgay.

  • Electsay ellscay Ayay39 andyay bay39 andyay opycay emthay otay ethay ipboardclay. Electsay ellscay Ayay40 oughthray Ayay109 andyay astepay ethay ormulaefay inyay ethay ipboardclay intoyay ethay ewnay angeray.

  • Ownay eway eednay otay ixfay ethay aphgray osay ityay owsshay oremay anthay 30 enerationsgay. Ickclay onyay ethay aphgray andyay ethay owsray atthay illway ebay aphedgray illway ebay urroundedsay ybay ayay ueblay ectangleray atthay epresentsray ethay aphgray angeray. Ityay ouldshay ooklay ikelay isthay:

  • Extnay, abgray ethay owerlay ightray oryay eftlay andlehay (ethay ot-day ) andyay youray ursorcay illway angechay otay ayay oubleday arrowyay (). Ickclay andyay agdray ethay oxbay osay ityay ownay overscay 100 enerationsgay (Omfray bay10 oughthray bay109). Enwhay youay ollscray upyay ethay aphgray illway ownay owshay 100 enerationsgay.

• Upyay otay isthay ointpay, ethay odelsmay eway usedyay ereway eterministicday, eaningmay atthay eachyay imetay youay unray ethay odelmay, you'llay etgay exactlyyay ethay amesay esultsray asyay onglay asyay youay on'tday angechay ethay inputyay. Isthay odelmay isyay ochasticstay andyay illway angechay eachyay imetay youay unray ityay, ivinggay youay ayay ayay iewvay atthay isyay oremay imilarsay otay ethay eal-worldray ituationsay. Omparecay ethay aboveyay aphgray otay igurefay 2, orfay exampleyay. Otay useyay esethay odelsmay, youay unray emthay overyay andyay overyay usingyay ethay amesay inputyay ataday. Eachyay unray illway esultray inyay ayay ightlyslay ifferentday opulationpay ojectionpray andyay aphgray. Otay unray ethay odelmay ithway ethay amesay ataday inputyay, youay ancay orcefay ayay ecalculationray ybay ittinghay ethay ctrl-Say eykay.

• Ytray entay unsray oryay osay ybay epeatedlyray ikingstray ethay ctrl-Say eykay. Oremay ikelylay anthay otnay, eachyay ofyay youray opulationpay unsray illway ucceedsay inyay eachingray 100 enerationsgay.

• Ifyay, ybay ancechay, oneyay ofyay youray unsray ailsfay otay eachray 100 enerationsgay andyay ethay opulationpay oesgay extinctyay, youay aphgray illway ebay imilarsay otay ethay ollowingfay:
  

• Ownay increaseyay ray inyay unitsyay ofyay .1 (ray=.3, ray=.4, etcyay). Atwhay effectyay oesday isthay avehay onyay youray opulationspay. Atyay atwhay ointpay oday ethay opulationspay eginbay otay egularlyray ailfay otay eachray 100 enerationsgay? Atwhay ypetay ofyay animalyay ightmay youay ebay odelingmay atyay igherhay eproductiveray atesray? Atwhay ypetay ofyay eal-worldray ituationsay(say) ightmay isthay epresentray?

• Ytray aryingvay ethay atingstay opulationpay andyay arryingcay apacitycay (oneyay atyay ayay imetay, otnay ogethertay) orfay owlay andyay ighhay eproductiveray atesray. Atwhay ituationssay areyay youay odelingmay erehay?