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Population Growth Models: Geometric Growth

Population Growth Models: Geometric GrowthBrook MilliganDepartment of BiologyNew Mexico State UniversityLas Cruces, New Mexico 2009 Brook MilliganPopulation Growth Models: Geometric GrowthPopulation Models in GeneralPurpose of Population modelsProject into the future the current demography ( ,survivorship and reproduction)Guage the potential (or lack) for a Population to increaseDetermine the consequences of changes in the currentdemographyBrook MilliganPopulation Growth Models: Geometric GrowthPopulation Models in GeneralObservables:NorN(age) orN(stage)Project Population sizeNas a function of timetProjection in terms of fundamental parametersdescribing demographic events in an individual s , Pr(birth), Pr(death)enable understanding of how demographic vital rates affectthe whole populationBrook MilliganPopulation Growth Models: Geometric GrowthProjection versus PredictionNo Population experiences unlimited resourcesYet, all populations havepotentialfor exponential growthProjections describepotential, not what is actually predictedto occuranalogy: a speedometer projects potent

Geometric Growth Models General motivation Sequence of population sizes through time N t,N t+1,N t+2,... Change from one time to next increases due to births during period decreases due to deaths during period increases due to immigrants during period decreases due to emigrants during period Brook Milligan Population Growth Models: Geometric Growth

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Transcription of Population Growth Models: Geometric Growth

1 Population Growth Models: Geometric GrowthBrook MilliganDepartment of BiologyNew Mexico State UniversityLas Cruces, New Mexico 2009 Brook MilliganPopulation Growth Models: Geometric GrowthPopulation Models in GeneralPurpose of Population modelsProject into the future the current demography ( ,survivorship and reproduction)Guage the potential (or lack) for a Population to increaseDetermine the consequences of changes in the currentdemographyBrook MilliganPopulation Growth Models: Geometric GrowthPopulation Models in GeneralObservables:NorN(age) orN(stage)Project Population sizeNas a function of timetProjection in terms of fundamental parametersdescribing demographic events in an individual s , Pr(birth), Pr(death)enable understanding of how demographic vital rates affectthe whole populationBrook MilliganPopulation Growth Models: Geometric GrowthProjection versus PredictionNo Population experiences unlimited resourcesYet, all populations havepotentialfor exponential growthProjections describepotential, not what is actually predictedto occuranalogy: a speedometer projects potential travel onlyBrook MilliganPopulation Growth Models: Geometric GrowthGeometric Growth ModelsGeneral motivationSequence of Population sizes through timeNt,Nt+1,Nt+2.

2 Change from one time to nextincreases due to births during perioddecreases due to deaths during periodincreases due to immigrants during perioddecreases due to emigrants during periodBrook MilliganPopulation Growth Models: Geometric GrowthMathematical FormulationPopulation size after an interval of timeNt+1=Nt+B D+I E(1)B,D: birth, deathI,E: immigration, emigrationChange in Population size N=Nt+1 Nt(2)=B D+I E(3)Closed versus open populationsBrook MilliganPopulation Growth Models: Geometric GrowthGeometric Growth Model: AssumptionsClosed Population :I=E= 0 Constant per captita birth (b) and death (d) ratesB=bND=dNUnlimited resourcesNo genetic structurebanddidentical for all individuals regardless of genotypeNo age- or size-structurebanddidentical for all individuals regardless of size, age.

3 No time lagsbirth and death depend on current Population onlyBrook MilliganPopulation Growth Models: Geometric GrowthGeometric Growth Model: AssumptionsClosed Population :I=E= 0 Constant per captita birth (b) and death (d) ratesB=bND=dNUnlimited resourcesNo genetic structurebanddidentical for all individuals regardless of genotypeNo age- or size-structurebanddidentical for all individuals regardless of size, age, ..No time lagsbirth and death depend on current Population onlyBrook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Growth :Projection of Population SizeNt+1=Nt+B D(4)=Nt+(BNt DNt) Nt(5)=Nt+ (b d) Nt(6)= (1 + (b d)) Nt(7)= (1 +Rt) Nt(8)= t Nt(9) t=Nt+1Nt(10)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Growth :Change in Population Size N=Nt+1 Nt(11)= (1 +Rt)Nt Nt(12)=Nt+RtNt Nt(13)=RtNt(14)Rt= NNt(15)Brook MilliganPopulation Growth Models: Geometric GrowthFinite Rate of Increase: Nt+1= tNt(16) t=Nt+1Nt(17) Population increase: >1population stable: = 1population decrease: <1 Brook MilliganPopulation Growth Models.

4 Geometric GrowthProjection of Population SizeAssume a constant value of : , t= N1= N0(18)N2= N1(19)= ( N0)(20)= 2N0(21)Nt= Nt 1(22)= ( Nt 2)(23)= ( ( Nt 3))(24)= tN0(25)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Doubling TimeHow long does it take the Population to double in size?That is, how long does it take the Population to change fromN0to 2N0?Nt= tN0(26)2N0= tN0(27)2= t(28)ln(2)= ln( t)(29)ln(2) =t ln( )(30)t=ln(2)ln( )(31)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Doubling TimeHow long does it take the Population to double in size?That is, how long does it take the Population to change fromN0to 2N0?Nt= tN0(26)2N0= tN0(27)2= t(28)ln(2)= ln( t)(29)ln(2) =t ln( )(30)t=ln(2)ln( )(31)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Doubling TimeHow long does it take the Population to double in size?

5 That is, how long does it take the Population to change fromN0to 2N0?Nt= tN0(26)2N0= tN0(27)2= t(28)ln(2)= ln( t)(29)ln(2) =t ln( )(30)t=ln(2)ln( )(31)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Half LifeHow long does it take the Population to become half as large insize?That is, how long does it take the Population to change fromN0to12N0?Nt= tN0(32)12N0= tN0(33)12= t(34)ln(12)= ln( t)(35) ln(2)=t ln( )(36)t= ln(2)ln( )(37)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Half LifeHow long does it take the Population to become half as large insize?That is, how long does it take the Population to change fromN0to12N0?Nt= tN0(32)12N0= tN0(33)12= t(34)ln(12)= ln( t)(35) ln(2)=t ln( )(36)t= ln(2)ln( )(37)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Half LifeHow long does it take the Population to become half as large insize?

6 That is, how long does it take the Population to change fromN0to12N0?Nt= tN0(32)12N0= tN0(33)12= t(34)ln(12)= ln( t)(35) ln(2)=t ln( )(36)t= ln(2)ln( )(37)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population Model: Half LifeHow long does it take the Population to become half as large insize?That is, how long does it take the Population to change fromN0to12N0?Nt= tN0(32)12N0= tN0(33)12= t(34)ln(12)= ln( t)(35) ln(2)=t ln( )(36)t= ln(2)ln( )(37)Brook MilliganPopulation Growth Models: Geometric GrowthGeometric Population ModelQuantitative description of how a Population changes size astime progressesDepends directly on the finite rate of increase, in turn depends on the per capita rates of birth and death(through their difference only) measures therate of increase measures thepotentialfor a Population to growQuestions that can be answered:Is the Population increasing, decreasing, or stable?

7 What is the potential for the Population to increase?How long does it take for the Population to change by acertain amount?How will the answers change if the vital rates (bandd)change?Brook MilliganPopulation Growth Models: Geometric Growth

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