Chapter 11Chapter 11Population Growth Population Growth

(Part 1)(Part 1)

HomeworkHomeworkChapter 10 (Part 2)Chapter 10 (Part 2)

Question EQuestion EEffect of increased prey density (food) on life Effect of increased prey density (food) on life history traits of predator?history traits of predator?

LLxx::

mmxx::

RR00::

T:T:

r:r:

Increase ( ↑ vigor, better defenses)

Increase ( ↑ eggs, ↓ abortion)

Increase (given Lx and mx increase)

Decrease (rapid growth accelerates maturation)

Increase ( given ↓T & ↑R0 )

Homework Problem FHomework Problem F

AgeAge ddxx nnxx LLxx mmxx LLxx m mxx X LX Lxx m mxx

00 800800 850850 1.0001.000 00 00 00

11 2525 5050 0.0590.059 200200 11.811.8 11.811.8

22 1515 2525 0.0290.029 250250 7.257.25 14.514.5

33 55 1010 0.0120.012 300300 3.63.6 10.810.8

44 55 55 0.0060.006 350350 2.12.1 8.48.4

TotalTotal 850850 RR00 = = 24.7524.75 45.545.5

Problem F (continued)Problem F (continued)

Generation Time ( T )Generation Time ( T )T = Sum (X LT = Sum (X Lxx m mxx) / R) / R00

T = 45.5 / 24.75T = 45.5 / 24.75

T = 1.84T = 1.84

Per Capita Rate of Increase ( r )Per Capita Rate of Increase ( r )

r = Ln (24.75) / 1.85r = Ln (24.75) / 1.85

r = 1.74r = 1.74

Problem F (continued)Problem F (continued)

Type of Survivorship Curve ?Type of Survivorship Curve ?Type IIIType III Most individuals die at a very young age. Most individuals die at a very young age. Those that get past juvenile period have lower Those that get past juvenile period have lower mortality rate.mortality rate.

Type of Life History Pattern ?Type of Life History Pattern ?rr−selected:−selected: Life table indicates short life Life table indicates short life span, low juvenile survivorship, and high span, low juvenile survivorship, and high birth rates.birth rates.

Homework Problem GHomework Problem G

AgeAge nnxx LLxx mmxx LLxx m mxx X LX Lxx m mxx

00 850850 1.0001.000 00 00 00

11 800800 0.9410.941 00 00 00

22 750750 0.8820.882 00 00 00

33 700700 0.8240.824 11 0.8240.824 2.472.47

44 650650 0.7650.765 33 2.2952.295 9.189.18

55 600600 0.7060.706 33 2.1182.118 10.5910.59

66 500500 0.5880.588 33 1.7641.764 10.5810.58

77 200200 0.2350.235 33 0.7050.705 4.944.94

88 5050 0.0590.059 33 0.1770.177 1.421.42

R0 = 7.88 39.17

Problem G (continued)Problem G (continued)

Generation Time ( T )Generation Time ( T )T = Sum (X LT = Sum (X Lxx m mxx) / R) / R00

T = 39.17 / 7.88T = 39.17 / 7.88

T = 4.97T = 4.97

Per Capita Rate of Increase ( r )Per Capita Rate of Increase ( r )

r = Ln (7.88) / 4.97r = Ln (7.88) / 4.97

r = 0.415r = 0.415

Problem G (continued)Problem G (continued)

Type of Survivorship Curve ?Type of Survivorship Curve ?Type IType I

Most individuals survive juvenile age. Most Most individuals survive juvenile age. Most mortality is in oldest age classes.mortality is in oldest age classes.

Type of Life History Pattern ?Type of Life History Pattern ?

KK−selected:−selected: Life table indicates longer Life table indicates longer life life span, high juvenile survivorship, and low span, high juvenile survivorship, and low

birth rates.birth rates.

Chapter 11Chapter 11Population Growth Population Growth

(Part 1)(Part 1)

Estimating Population Estimating Population Growth Rate and Projecting Growth Rate and Projecting

Future Population SizeFuture Population Size

Estimating Population Growth Estimating Population Growth Rate ( r ) From Life Table DataRate ( r ) From Life Table Data

Ln (RLn (R00))rr == __________ TT

r < 0r < 0 Population size N decreasingPopulation size N decreasing

r = 0r = 0 Population size constantPopulation size constant

r > 0r > 0 Population size increasingPopulation size increasing

Estimating Population Growth Estimating Population Growth Rate by Direct Observation: Rate by Direct Observation:

Geometric Rate of Increase (Geometric Rate of Increase (λλ))

NNttλλ = = ___ ___ NNt-1t-1

λλ < 1 < 1 Population size N decreasingPopulation size N decreasing

λλ = 1 = 1 Population size N constantPopulation size N constant

λλ > 1 > 1 Population size N increasingPopulation size N increasing

““Lambda”Lambda” The number of individuals The number of individuals in population at time tin population at time t

Per individual in population Per individual in population at time tat time t−1−1

Geometric Rate of Increase

Different way to say Different way to say the same thing.the same thing.

Example Calculations for Example Calculations for λλ

Population of a plant species is 250 this Population of a plant species is 250 this year, compared to 200 last year.year, compared to 200 last year.

λλ = N = Ntt / N / Nt−1t−1 = 250 / 200 = 1.250 = 250 / 200 = 1.250

– For each individual alive last year, there For each individual alive last year, there are 1.25 individuals in the population this are 1.25 individuals in the population this year. year.

– This population is increasing in sizeThis population is increasing in size..

Example Calculations for Example Calculations for λλ

Population of an animal species is 125 Population of an animal species is 125 this year, compared to 140 last year.this year, compared to 140 last year.

λλ = N = Ntt / N / Nt-1t-1 = 125 / 140 = 0.893 = 125 / 140 = 0.893

– For each individual alive last year, there For each individual alive last year, there are 0.89 individuals in the population this are 0.89 individuals in the population this year.year.

– This population is decreasing in size.This population is decreasing in size.

Computing Computing λλ for any Time Interval for any Time Interval

λλ = (N = (Ntt / N / N00))

Where:Where:

NN00 == Population size at some earlier timePopulation size at some earlier time

NNtt == Population size at some later time tPopulation size at some later time t

tt == Number of time intervals (years, Number of time intervals (years, days, hours)days, hours)

(1/t)(1/t)

Example Calculation for Example Calculation for λλ

Suppose there are currently 500 goats Suppose there are currently 500 goats on Santa Catalina island. A census on Santa Catalina island. A census done 5 years ago counted 350.done 5 years ago counted 350.

λλ == (500 / 350)(500 / 350)(1/5)(1/5)

== (1.429)(1.429)0.20.2

== 1.0741.074

Example Calculation for Example Calculation for λλ

Suppose a census of a population of a Suppose a census of a population of a currently endangered butterfly species currently endangered butterfly species counted 2873 individuals in 1925. A re-counted 2873 individuals in 1925. A re-census in 2001 found only 95 census in 2001 found only 95 individuals.individuals.

λλ == (95 / 2873)(95 / 2873)(1 / 76)(1 / 76)

= (0.0331)0.0132

= 0.956

Relationship Between r and Relationship Between r and λλ

r = Ln (r = Ln (λλ))λλ = = e e rr

Examples:Examples:If If λλ = 1.250 = 1.250 r = Ln(1.250) =r = Ln(1.250) = 0.2230.223If If λλ = 0.893 = 0.893 r = Ln(0.893) =r = Ln(0.893) = - 0.113- 0.113

If r = 0.075If r = 0.075 λλ = = ee 0.075 0.075 = = 1.0781.078If r = −0.10If r = −0.10 λλ = = ee −0.10 −0.10 = 0.905 = 0.905

Ln is the natural Ln is the natural log function.log function.

ee is the exponent- is the exponent-iation function.iation function.

Expressing Population Growth Expressing Population Growth as a Percent (as a Percent (PP))

PP = 100 ( = 100 (λλ – 1) – 1)Examples:Examples:

If If λλ = 1.25 = 1.25 PP = 100 (1.25 − 1) = = 100 (1.25 − 1) = +25%+25%

Population increasing at 25% per yearPopulation increasing at 25% per year

If If λλ = 0.893 = 0.893 PP = 100 (0.893 − 1) = = 100 (0.893 − 1) = −10.7%−10.7%

Population decreasing at 10.7% per year.Population decreasing at 10.7% per year.

Expressing Population Growth Expressing Population Growth as Doubling Time (Tas Doubling Time (T22))

TT22 = Ln (2) / Ln ( = Ln (2) / Ln (λλ))Examples:Examples:

If If λλ = 1.250 = 1.250 TT22 = Ln(2) / Ln(1.250) = = Ln(2) / Ln(1.250) = 3.113.11

This population doubles in 3.11 yrsThis population doubles in 3.11 yrs

If If λλ = 0.893 = 0.893 TT22 = Ln(2) / Ln (0.893) = = Ln(2) / Ln (0.893) = −6.12−6.12

This population decreases by half in 6.12 yrsThis population decreases by half in 6.12 yrs

Predicting Future Population Size:Predicting Future Population Size:Calculation Based on Calculation Based on λλ

NNtt = N = N00 λλtt

Where:Where: NNtt = = Population size at some futurePopulation size at some future

time (after t time intervals)time (after t time intervals)

NN00 = = Current (original) population Current (original) population

sizesize

t =t = Number of time intervalsNumber of time intervals

Note:Note: Exponentiation precedes multiplication Exponentiation precedes multiplication

Example CalculationsExample Calculations

Suppose that NSuppose that N00 = 5000 and = 5000 and λλ = 0.893 = 0.893What will the population size be in 15 What will the population size be in 15 years?years?

NN15 15 = 5000 (0.893) = 5000 (0.893)1515 = 916 individuals = 916 individuals

Suppose that NSuppose that N00 = 250 and = 250 and λλ = 1.25 = 1.25What will the population size be in 10 What will the population size be in 10 years?years?

NN1010 = 250 (1.25) = 250 (1.25)1010 = 2328 = 2328 individualsindividuals

Predicting Future Population Size:Predicting Future Population Size:Calculation Based on rCalculation Based on r

NNtt = N = N00 e er tr t

Where:Where: NNtt = = Population size at some future Population size at some future time (after t time intervals)time (after t time intervals)

NN00 = = Current (original) population sizeCurrent (original) population size

t =t = Number of time intervalsNumber of time intervals

Note:Note: Multiply r by t, press =, press Multiply r by t, press =, press e e x , multiply by N , multiply by N00

Example CalculationsExample Calculations

Suppose that NSuppose that N00 = 3300 and r = -0.025 = 3300 and r = -0.025What will the population size be in 25 What will the population size be in 25 years?years?

NN25 25 = 3300 = 3300 ee -0.025 x 25-0.025 x 25 = 1766 = 1766

Suppose that NSuppose that N00 = 50,000 and r = 0.003 = 50,000 and r = 0.003What will the population size be in 20 What will the population size be in 20 years?years?

NN2020 = 50,000 = 50,000 e e 0.003 x 200.003 x 20 = 53,092 = 53,092

ExponentialExponentialPopulation GrowthPopulation Growth

Historical and Projected Global Population

Exponential Population Growth (N0 = 2, λ = 2)

0

500000

1000000

1500000

2000000

2500000

0 5 10 15 20

Year

Po

pu

lati

on

Siz

e

0

500

1000

1500

2000

2500

0 2 4 6 8 10

Year

Po

pu

lati

on

Siz

e

Population grows from 2 to 2048Population grows from 2 to 2048in the first 10 yrs.in the first 10 yrs.

Population grows from Population grows from 2048 to 2,097,152 in 2048 to 2,097,152 in second 10 yrs.second 10 yrs.

560 Trillion in 50 years

Exponential Growth of Scots Pine PopulationExponential Growth of Scots Pine Population

Body Size and Intrinsic Rate of IncreaseBody Size and Intrinsic Rate of Increase

Intrinsic Rate of Increase (rIntrinsic Rate of Increase (rmm ) is the maximum rate of ) is the maximum rate of

population growth that occurs when environmental population growth that occurs when environmental conditions and resource availability are optimum.conditions and resource availability are optimum.

Population Growth and Body Size:

Thalia democratica Vs. the Gray Whale

r = 0.91/dayP = 148%/dayT2 = 0.76 day

r = 0.041 / yrP = 4.2% / yrT2 = 17 yr

Exponential Growth of the Collared Exponential Growth of the Collared Dove PopulationDove Population

Why ?

Factors That Limit Population GrowthFactors That Limit Population Growth

Competition for Limited ResourcesCompetition for Limited Resources– Food / LightFood / Light– WaterWater– Space Space – Nest sitesNest sites

CatastrophesCatastrophes– Fire, Flood, Severe Storms, Freeze Events, Fire, Flood, Severe Storms, Freeze Events,

Extended DroughtsExtended Droughts

Density-Dependent

Density-Independent

Density-Dependent Population Density-Dependent Population RegulationRegulation

Increasing number of individuals depletes Increasing number of individuals depletes available resourcesavailable resources

Physiological ConsequencesPhysiological Consequences of Competition of Competition for Limited Resources:for Limited Resources:– Reduced growth rateReduced growth rate– Reduced fat / starch storageReduced fat / starch storage– Increased stress from competitive interactionsIncreased stress from competitive interactions– Reduced immune function (animals) or chemical Reduced immune function (animals) or chemical

defense (plants)defense (plants)

Density-Dependent Population Density-Dependent Population RegulationRegulation

Population ConsequencesPopulation Consequences of Competition: of Competition:

– Reduced growth rate Reduced growth rate

– Reduced storageReduced storage

– Increased stressIncreased stress

– Reduced defensesReduced defenses

Increased TIncreased T

Decreased Decreased mmxx

Population size stabilizes or declinesPopulation size stabilizes or declines

Decreased LDecreased Lxx

Decreased RDecreased R00

Decreased rDecreased r

The End The End Part 1Part 1

LogisticLogisticPopulation GrowthPopulation Growth

Logistic Population GrowthLogistic Population Growth

What Is Carrying Capacity (K) ?What Is Carrying Capacity (K) ?

Definition:Definition: The maximum number of The maximum number of individuals for a given species that the individuals for a given species that the resources of an environment can resources of an environment can sustainsustain over the long term.over the long term.

Interaction between amount of resources in Interaction between amount of resources in the environment and the the environment and the per capitaper capita rate of rate of resource consumption.resource consumption.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Logistic Growth by a Yeast PopulationLogistic Growth by a Yeast Population

Bir

ths

> D

eath

s

Bir

ths

> D

eath

s

Births = DeathsBirths = Deaths

Population Growth of Population Growth of Balanus balanoidesBalanus balanoides

Logistic Population Growth by African BuffaloLogistic Population Growth by African Buffalo

Logistic Equation for Population Logistic Equation for Population GrowthGrowth

dNdN K K−−NN____ = N r= N rmm ____ ____dtdt KK

Rate ofRate ofPopulationPopulationGrowthGrowth

Number ofNumber ofReproductiveReproductiveIndividualsIndividuals

Limits rate of Limits rate of growth when growth when population is population is smallsmall

IntrinsicIntrinsic(maximum)(maximum)Rate ofRate ofIncreaseIncrease

Availability ofAvailability ofResourcesResources

Limits rate of Limits rate of growth when growth when population is population is largelarge

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Logistic Growth by a Logistic Growth by a ParameciumParamecium Population Population

Low NLow N

Low (KLow (K−−N)/KN)/K

Population Size

and Per Capita Rate of

Increase

r = rr = rmm (K (K−N)/K−N)/K

Density versusDensity versusPer Capita Rate Per Capita Rate of Increase in aof Increase in aDaphnia cultureDaphnia culture

▲ ▲ DensityDensity

↓↓▼ ▼ rr

Density-Density-DependentDependentPopulationPopulation

GrowthGrowth

Density-Independent Influences Density-Independent Influences on Population Growthon Population Growth

Disturbances (storms, fires, freeze Disturbances (storms, fires, freeze events, floods) directly cause mortalityevents, floods) directly cause mortality

Fluctuations in resource availability Fluctuations in resource availability due to environmental variation (Ex. due to environmental variation (Ex. droughts) influence birth and death droughts) influence birth and death ratesrates

Rainfall and Galapagos Medium Rainfall and Galapagos Medium Ground Finch PopulationGround Finch Population

Availability of Availability of CaterpillarsCaterpillars

and Number of and Number of Chicks FledgedChicks Fledged

Dry yearDry year Wet yearWet year

Annual Annual Rainfall Rainfall

vs.vs.NumberNumberOf Egg Of Egg

ClutchesClutches

Interaction ofEnvironmentalAnd BiologicalInfluences on

Population Growth:

El Nino, Finches, and Cactus

Flowers

Wet conditionsWet conditionsduring the 1983during the 1983

El Nino decimatedEl Nino decimatedthe cactus the cactus populationpopulation

When finch population greater When finch population greater than its carrying capacity, than its carrying capacity, resource degradation occursresource degradation occurs

Growth of Human Growth of Human PopulationsPopulations

Distribution of the Human PopulationDistribution of the Human Population

Variation in Human Population DensityVariation in Human Population Density

Age Distributions for Human Populations:Age Distributions for Human Populations:

Predictors of Future Population GrowthPredictors of Future Population Growth

Population SIZE Population SIZE will be Stablewill be Stable

Growth Rate = 0Growth Rate = 0

Ag

e C

las

s

% of Population

Juvenile

Reproductive

Post-Reproductive

Proportion of population in reproductive age classes ≈ juvenile age classes.

Age Distributions for Human Populations:Age Distributions for Human Populations:

Predictors of Future Population GrowthPredictors of Future Population Growth

Population SIZEPopulation SIZEWill DeclineWill Decline

Growth Rate < 0Growth Rate < 0

Ag

e C

las

s

% of PopulationJuvenile

Reproductive

Post-Reproductive

Proportion of population in reproductive age classes >> juvenile age classes.

Many adults produce few children (low r).

Age Distributions for Human Populations:Age Distributions for Human Populations:

Predictors of Future Population GrowthPredictors of Future Population Growth

Population SIZEPopulation SIZEWill IncreaseWill Increase

Growth Rate >> 0Growth Rate >> 0

Ag

e C

las

s

% of PopulationJuvenile

Reproductive

Post-Reproductive

Proportion of population in reproductive age classes << juvenile age classes.

Few adults produce many children (high r).

Historical and Projected Human Populations

Genocide Event and War Increased Death Rates

Collapse of Communist System and Reduced Birth Rates

??

Rwandan genocide due to inadequate crop land w/ the population at 7.5 million

Can the current Can the current growth rate of the growth rate of the

global human global human population be population be

sustained ?sustained ?

If not, what will If not, what will slow or reverse slow or reverse

human population human population growth ?growth ?

What Will the Future Bring ?What Will the Future Bring ?

Conference of Rome, 1970Conference of Rome, 1970

ChineseChineseFamineFamine

Is the Global Human Population Is the Global Human Population Already at Carrying Capacity ?Already at Carrying Capacity ?

Trend of decreasing per capita availability of Trend of decreasing per capita availability of farmland and freshwaterfarmland and freshwater

Trend of decreasing total crop land, range Trend of decreasing total crop land, range land, and forestland, and forest

13 of 15 major marine fisheries near or total 13 of 15 major marine fisheries near or total collapsecollapse

Humans consume 40% of global primary Humans consume 40% of global primary productivityproductivity

Possible FuturesPossible Futures

Death Rate Solution:Death Rate Solution: Decrease LxDecrease Lx– MalnutritionMalnutrition– DiseaseDisease– Warfare (Terrorism)Warfare (Terrorism)– PollutionPollution

ChemicalChemical

RadiationRadiation

Birth Rate Solution:Birth Rate Solution: Decrease RDecrease R00

– Increase age of Increase age of reproduction (T)reproduction (T)

Education / employment Education / employment for girls / womenfor girls / women

– Decrease mDecrease mxx

Universal availability of Universal availability of contraceptivescontraceptives

Decrease infant mortalityDecrease infant mortality

Increase standard of Increase standard of livingliving

Reduces “bet-hedging” birthsReduces “bet-hedging” births

Desire for children competes Desire for children competes with desire for “stuff”with desire for “stuff”

Making the Birth Rate Solution Making the Birth Rate Solution HappenHappen

Micro-bank programs make fair credit (loans) Micro-bank programs make fair credit (loans) available to poor women.available to poor women.

Programs that produce “soap opera” Programs that produce “soap opera” entertainment that give women role models entertainment that give women role models for small families.for small families.

International family planning NGO’s provide International family planning NGO’s provide contraception and safe sterilization.contraception and safe sterilization.

Building schools for Building schools for GIRLSGIRLS in Afghanistan. in Afghanistan.

Economic development aid to improve health Economic development aid to improve health care systems and employment.care systems and employment.

Can Technology Save Us From Can Technology Save Us From the Death Rate Solution (Again) ?the Death Rate Solution (Again) ?

Green Revolution (Part 2) ???Green Revolution (Part 2) ???

Renewable and clean energy sources Renewable and clean energy sources (solar, wind, hydro-electric)(solar, wind, hydro-electric)

Medical research to combat new and Medical research to combat new and resistant diseasesresistant diseases

Warp Drive to Other Planets ???Warp Drive to Other Planets ???

I guess we will just have to I guess we will just have to wait and see….wait and see….

Or maybe we should be doing Or maybe we should be doing something NOW ?something NOW ?

HomeworkHomeworkChapter 11 (Part 1)Chapter 11 (Part 1)

Chapter 11 Problem AChapter 11 Problem A

1.1. NNtt = N = N00 eertrt

t = 3t = 3 NNtt = 50 = 50 e e (0.4)(3)(0.4)(3) = 166

t = 10 Nt = 50 50 e e (0.4)(10)(0.4)(10) = 2730

t = 25 Nt = 50 50 e e (0.4)(25)(0.4)(25) = 1,101,323

2. λ = e r = e 0.4 = 1.49 / yr

3. P = 100 (λ – 1) = 100 (1.49 – 1) = 49% / yr

4. T2 = Ln (2) / Ln (1.49) = 1.74 yrs

5. 1.74 yrs to double in BOTH cases.

Chapter 11 Problem BChapter 11 Problem B

1.1. λλ = (N = (Ntt / N / N00))(1/t) = (700/300)(1/3) = 1.33 / day

2. P = 100(λ-1) = 100(1.33-1) = 33% / day

3. r = Ln(λ) = Ln (1.33) = 0.285

4. T2 = Ln(2) / Ln(1.33) = 2.43 days

5. N20 = 300 (1.33)20 = 89,982

Chapter 11 Problem CChapter 11 Problem C

YearYear t (yrs)t (yrs) NNt (billion)t (billion) λλ PP (%) (%) TT2 (yrs)2 (yrs)

18041804 -- 11

19271927 123123 22 1.00571.0057 0.570.57 123123

19601960 3333 33 1.01241.0124 1.241.24 5656

19741974 1414 44 1.02081.0208 2.082.08 3434

19871987 1313 55 1.01731.0173 1.731.73 4040

19991999 1212 66 1.01531.0153 1.531.53 4646

Population will double to 12 billion by year 1999 + 46 = Population will double to 12 billion by year 1999 + 46 = 20452045

Assumption ? Assumption ? Population growth rate Population growth rate λλ remains constant remains constant

The EndThe End