BIOL 4120: Principles of EcologyBIOL 4120: Principles of Ecology

Lecture 9: Population Lecture 9: Population Growth and RegulationGrowth and Regulation

Dafeng HuiDafeng Hui

Office: Harned Hall 320Office: Harned Hall 320

Phone: 963-5777Phone: 963-5777

Email: dhui@tnstate.eduEmail: dhui@tnstate.edu

World populationWorld population

OutlineOutline

9.1 Population grow by multiplication 9.1 Population grow by multiplication rather than additionrather than addition

9.2 Age structure influences population 9.2 Age structure influences population raterate

9.3 A life table summaries age-specific 9.3 A life table summaries age-specific schedules of survival and fecundityschedules of survival and fecundity

9.4 The intrinsic rate of increase can be 9.4 The intrinsic rate of increase can be estimated from the life tableestimated from the life table

9.5 Population size is regulated by density-9.5 Population size is regulated by density-dependent factorsdependent factors

9.1 Populations grow by multiplication rather 9.1 Populations grow by multiplication rather than additionthan addition

Definition: Population growth, how the number of Definition: Population growth, how the number of individuals in a population increases or decreases with timeindividuals in a population increases or decreases with time

Growth is controlled by rates of birth, immigration and Growth is controlled by rates of birth, immigration and death and emigration.death and emigration.

Open or closed population: no immigration and emigration, Open or closed population: no immigration and emigration, or immigration rate=emigration rate.or immigration rate=emigration rate.

In closed population, growth is determined by birth rate and In closed population, growth is determined by birth rate and death rate. Rate of growth is on a per capita basis. death rate. Rate of growth is on a per capita basis.

Demography: study of population.Demography: study of population.

Calculating population growth ratesCalculating population growth rates

Geometric growth

Assume a population, at t=0, population size is N(0), ratio of population size change in one year to the size of the proceeding year is .

If at time t, population size is N(t), then at time t+1,N(t+1)=N(t)

So N(1)=N(0) , N(2)=N(1) =N(0) 2 thus

N(t) = N(0) t

geometric growth model.

Geometric growth curve: N(0)=100, =1.5

(a): y on a arithmetic scale; (b) y on a logarithmic scale.

Exponential growthExponential growth

NNtt=N=N00exp(rt)exp(rt) Where r is the exponential growth rateWhere r is the exponential growth rate

Compare to N(t)=N(0) Compare to N(t)=N(0) tt

=exp(r) or r=ln(=exp(r) or r=ln())

These models are used to describe dynamics of These models are used to describe dynamics of populations. Geometric growth is used for population populations. Geometric growth is used for population generations not overlap (discrete time interval), generations not overlap (discrete time interval), exponential growth model is for continuous exponential growth model is for continuous population.population.

Population size change and model Population size change and model parameterparameter

Increasing population, Increasing population, >1, >1, r>0r>0

Constant pop. Size, Constant pop. Size, =1, =1, r=0r=0

Decreasing population, 0<Decreasing population, 0<<1, <1, r<0r<0

In a closed population, population size In a closed population, population size change is related to birth rate (b) and death change is related to birth rate (b) and death rate (d)rate (d)

The difference between birth rate and death The difference between birth rate and death rate is the intrinsic growth rate (r) rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth).(instantaneous per capita rate of growth).

r=b-dr=b-d Population growth is related to this intrinsic Population growth is related to this intrinsic

growth rate (r).growth rate (r).

dN/dt=(b-d)N=rN dN/dt=(b-d)N=rN

Calculating population growth rates from birth and death rates

Exponential population growthExponential population growth

Equations:Equations:• 1) dN/dt=rN (differential equation form)1) dN/dt=rN (differential equation form)

• 2) N2) Ntt=N=N0 0 exp(rt) (exponential growth exp(rt) (exponential growth model)model)

Conditions:Conditions:• Initial population is smallInitial population is small• No food or resource limitationNo food or resource limitation

An exampleAn example

Reindeer, St. Paul, Alaska.

Started in 1910 with only 4 males and 22 females

In 1940, there were nearly 2000

Whooping crane, an Whooping crane, an endangered speciesendangered species

recovered from near recovered from near extinction in 1941extinction in 1941

How to calculate r?

Software, Excel (trendline)

Arkansas National Wildlife Refuge

Nt=N0Exp(rt)

Give a time t, we can predict the population size.

An Example:

Deer population: N0=300, r=0.5, after 5 years, what’s the population size?

N5=N0 Exp(rt)=300*exp(0.5*5)=3655

(495, 815, 1344, 2216, 3655) t=10, ?

Prediction of population growthPrediction of population growth

44,524

9.2 Age structure influences 9.2 Age structure influences population growth ratepopulation growth rate

Age structure: the proportions of each Age structure: the proportions of each individual in each age class.individual in each age class.

Age structure influences population Age structure influences population growth rate, as only mature adults can growth rate, as only mature adults can reproduce and increase population size.reproduce and increase population size.

A A life tablelife table is an age-specific account of is an age-specific account of mortalitymortality

Age structure Age structure influences influences population growth population growth raterate

(t)=N(t+1)/N(t)

Age distributionAge distribution

Stable age distribution: by year 7, the proportion of each age group remain the same year after year.

Population is still growing at a constant rate.

In a stable age distribution, each age class grows at the same rate

Age structure of human population (2008 data) Different countries

(a) Stable for many years; (b) high birth (growth) rate

Population structure change over time in German

Birth rates are declining in many human populations

9.3 Life table summaries age-9.3 Life table summaries age-specific schedules of survival specific schedules of survival

and fecundityand fecundity Life table is an age-specific account of Life table is an age-specific account of

mortality.mortality.

Purpose of life table: to provide a clear and Purpose of life table: to provide a clear and systematic picture of mortality and survival systematic picture of mortality and survival within a population. Can also be used to within a population. Can also be used to simulate population size change.simulate population size change.

RecapRecap

Population growthPopulation growth

• Growth models: Geometric growth model vs Exponential Growth models: Geometric growth model vs Exponential growth modelgrowth model

• Life table and use life table to predict population growthLife table and use life table to predict population growth

Age structureAge structure

l(x)

Two types of life tablesTwo types of life tables Two typesTwo types

• Cohort or dynamic life tableCohort or dynamic life table as the above flycatcher, (plants, sessile animals)as the above flycatcher, (plants, sessile animals)

• Time-specific or static life table Time-specific or static life table

9.4 Life tables provide data for 9.4 Life tables provide data for mortality and survivorship curvesmortality and survivorship curves

Table is better than words, but a Table is better than words, but a graph is worth one thousand words.graph is worth one thousand words.

Mortality curve and survivorship Mortality curve and survivorship curve.curve.

Survivorship curvesSurvivorship curves

Log scale for Y axis I. deer, sheep, human, convex II: squire and adult birds, linear, not change with ageIII. Plants, fish, young bird, concave

9.5 Intrinsic rate of increase can 9.5 Intrinsic rate of increase can be estimated from the life tablebe estimated from the life table

Crude birthrate (demographers): # of birth over a period of time divided by population size at the beginning of the period*1000

Age-specific birthrates, bx

Mean # of females birth to a female in each age group.

(Only females give birth; birth rates vary with ages)

Gross reproduction rate: sum of the bx values across all age classes, provides an estimate of average offspring born to a female over her lifetime.

Birth rate and survivorship Birth rate and survivorship determine net reproductive ratedetermine net reproductive rate

Fecundity table: take survivorship column, Fecundity table: take survivorship column, llxx, from life table and add age-specific , from life table and add age-specific birthrate, bbirthrate, bxx..

Birth rate and survivorship Birth rate and survivorship determine net reproductive ratedetermine net reproductive rate

Net reproduction rate, RNet reproduction rate, R00: : number of female number of female

offspring a female at birth can produce (or average # of females offspring a female at birth can produce (or average # of females that will be produced (left) during a lifetime by a newborn that will be produced (left) during a lifetime by a newborn females.)females.)

RR00: depends on survivorship and fecundity: depends on survivorship and fecundity R0 =1, R0 =1, >1 >1 or <1or <1

Generation time: average age at which an Generation time: average age at which an individual gives birth to its offspringindividual gives birth to its offspring

T=Sum(xlT=Sum(xlxxbbxx)/sum(l)/sum(lxxbbxx) ) T=4.2/2.1=1.95T=4.2/2.1=1.95

N(t)/N(0)=N(t)/N(0)= t (T=1.95)

If t=T, then RIf t=T, then R00

== N(T)/N(0), then R00=T

=R01/T=2.1=2.11/1.95 =1.46; =1.46; r r = ln ln()=0.38

Generation time and growth rateGeneration time and growth rate

The growth potential of populationsThe growth potential of populations

Capacity of population growth:Capacity of population growth:

Pheasants: 2 males +6 females in 1937Pheasants: 2 males +6 females in 1937 in 5 years, in 5 years, 1325 (r=1.02, 1325 (r=1.02, =2.78)=2.78)

Elephant seal: in 1900, N0=100Elephant seal: in 1900, N0=100

2000: 150,000 (r=0.073, 2000: 150,000 (r=0.073, =1.076)=1.076)

2100: 225 millions2100: 225 millions

Field vole: Field vole: =24=24

Water flee: Water flee: =10=103030

The growth potential of populationsThe growth potential of populations

Doubling time: Doubling time:

t2 =ln(2)/ln(t2 =ln(2)/ln())

or or t2 =ln(2)/r t2 =ln(2)/r

Field vole: t2=0.22 yr, 79 days (Field vole: t2=0.22 yr, 79 days (=24)=24) Pheasants: t2=246 days (r=1.020)Pheasants: t2=246 days (r=1.020) Water flea, t2=3.6 days (r=69.07)Water flea, t2=3.6 days (r=69.07)

[Ln(2)=0.693, if r=0.01, t2=69.3][Ln(2)=0.693, if r=0.01, t2=69.3]

9.6 Limitation on Population Growth9.6 Limitation on Population Growth

Exponential growth model can not explain the population growth here

Need other equation or model

Barnacle

Birth rate and death rate change with population size N

dN/dt=rN

r=b-d

b=b0-aN

d=d0+cN

dN/dt=[(b0-d0)-(a+c)N]N

dN/dt=(b0-d0)[1-(a+c)/(b0-d0)*N]N

dN/dt=r N(1-N/K)

K=(b0-d0)/(a+c)K: carrying capacity: maximum sustainable population size under prevailing environment.

From exponential growth to logistic From exponential growth to logistic growthgrowth

Logistic population growth

Logistic Population GrowthLogistic Population Growth As resources are depleted, population As resources are depleted, population

growth rate slows and eventually stops: growth rate slows and eventually stops: logistic population growthlogistic population growth..• Sigmoid (S-shaped) Sigmoid (S-shaped) population growth population growth

curvecurve..• Carrying capacityCarrying capacity ( (KK) is the number of ) is the number of

individuals of a population the environment individuals of a population the environment can support.can support.

• (r) is per capita increase rate. (r) is per capita increase rate.

A few examples

US population (1790-1910): k=197 million, r=0.03134.

9.7 Population size is regulated by 9.7 Population size is regulated by density dependent factorsdensity dependent factors

Environment limits population growth by Environment limits population growth by altering birth and death rates.altering birth and death rates.• Density-dependent factorsDensity-dependent factors

E.g., Disease, Resource competitionE.g., Disease, Resource competition They influence a population in proportion to its sizeThey influence a population in proportion to its size Resulted in slowing the rate of increaseResulted in slowing the rate of increase

• Density-independent factorsDensity-independent factors E.g., Natural disastersE.g., Natural disasters Influence population without regard to the # of Influence population without regard to the # of

individuals or if the proportion of individuals affected individuals or if the proportion of individuals affected is the same at any densityis the same at any density

Density dependence in animals

Fecundity and life span of fruit fly decrease as population density increaseLab study

A long-term field study of common terns. Population started with 200, and leveled off after suitable nest sites are occupied on the Bird Island. Birds started to colonize Ram Island.

Why level off?

Density-dependent factors can control the size of natural population.

Song sparrow

(a) As density increases, more males can’t find territories; (b) number of fledglings produced per female and survival of those offspring (c) decrease.

Range quality influence percentage of white-tailed deer female pregnant and average number of embryos

Selective hunting may improve population caused by overgrazing.

Positive density dependence

Negative density dependence: common (right panel)Positive (or inverse) density dependence: left panel

Allee effect: with increase in density, easy to find mates, defend predators, increase diversity etc.

A: herring near icelandB: herring near North Sea

Allee effects also occur in plant population. Reproduction of Primula veris (a small herbaceous plant grow in nutrient poor grasslands) show positive density dependence. Seed mass is the average mass of individual seed. (pollination?)

Regulation of population size by density-dependent factors

Regulation of population size by density-dependent factors

Density dependence in plants

Size of flax (Linum) plants grown at different densities

Horseweed experiment

Kyoji Yoda, plant ecologist

Horseweed was planted at a very high density (100,000 seeds m-2)

Density declined in several months

Weight per seedling increased.Self-thinning: progressive decline in

density and increase in biomass of remaining individuals in a population.

The curve of the average weight and density is called self-thinning curve.-3/2 power law: the Regularity of this relationship is called -3/2 power law, as the slope is about -3/2.

ENDEND

Geometric GrowthGeometric Growth When generations do not overlap, growth When generations do not overlap, growth

can be modeled geometrically.can be modeled geometrically.

NNt t = N= Noott

• NNtt = Number of individuals at time t. = Number of individuals at time t.

• NNoo = Initial number of individuals. = Initial number of individuals. = Geometric rate of increase.= Geometric rate of increase.• t = Number of time intervals or generations.t = Number of time intervals or generations.

Another example of squirrelAnother example of squirrel

Predicted population size change Predicted population size change

Population size (N) increases every year.

Lambda (finite multiplication rate): =N(t+1)/N(t).

A 100-year-old ex-secretary who lived in a tiny cottage leaves behind a stunning surprise (Yahoo News, March 5, 2010)

3 shares of a stock, $60 each, today, $7,000,000 total. All donated to a college.

r determines the shape of the growth.

r=0, no change in population size

r<0, decrease in population size; r>0, increase in population size.

Properties of exponential growthProperties of exponential growth

Widely used in biology

Logistic equation incorporates the influences of population size and per capita growth rate.

Mortality curvesMortality curves

Logistic Population GrowthLogistic Population Growth

RecapRecap

Population growthPopulation growth

• Growth models: Geometric growth model vs Exponential Growth models: Geometric growth model vs Exponential growth modelgrowth model

• Life table and use life table to predict population growthLife table and use life table to predict population growth

RecapRecap

A life table summaries age-specific schedules of A life table summaries age-specific schedules of survival and fecunditysurvival and fecundity

The intrinsic rate of increase can be estimated The intrinsic rate of increase can be estimated from the life tablefrom the life table

Growth models: exponential growth and logistic Growth models: exponential growth and logistic growth modelgrowth model