Section III Population Ecology

Section III Population Ecology

鄭先祐生態主張者 Ayo

[email protected]

2003 chap.6 Population Growth生態學 2

Section Three Population Ecology

• Chap.6 Population growth ( 族群成長 )

• Chap.7 Physical environment ( 物理環境 )

• Chap.8 Competition and coexistence ( 競爭與共存 )

• Chap.9 Mutualism ( 共生 )

• Chap.10 Predation ( 掠食 )

• Chap.11 Herbivory ( 素食 )

• Chap.12 Parasitism ( 寄生 )

• Chap.13 Evaluating the controls on population size


Chap. 6 Population Growth1. Tabulating changes in population age structure

through time– Time-specific life tables

– Age-specific life tables

2. Fecundity schedules and female fecundity, and estimating future population growth

3. Population growth models– Deterministic models

– Geometric models

– Logistic models

– Stochastic models

Road Map


6.1 Life tables• The construction of life tables is termed demograph

y.

– Construct life tables

– Demonstrate the age structure of a population

• Time-specific life table

– Snapshot – age structure at a single point in time (time-specific life table)

– Useful in examining long-lived animals

• Ex. Dall Mountain Sheep (Figure 6.1 and Table 6.1)


Time-specific life table

• Snapshot – age structure at a single point in time (time-specific life table)

• Useful in examining long-lived animals

– Ex. Dall Mountain Sheep (Figure 6.1 and Table 6.1)


Life Tables

• Useful parameters in the life tables – x = age class or interval

– nx = number of survivors at beginning of age interval x.

– dx = number of organisms dying between age intervals = nx – nx+1

– lx = proportion of organisms surviving to the beginning of age interval x = ns / n0


Life Tables

• Useful parameters in the life tables

– qx = rate of mortality between age intervals = dx / ns

– ex = the mean expectation of life for organisms alive at the beginning of age x

• Lx = average number alive during an age class = (nx+ nx+1) / 2

• Tx = intermediate step in determining life expectancy = Lx

• ex = Tx / nx



0

0.5

1

1.5

2

2.5

3

3.5

Age (years)

n

(log

sca

le)

x1

0

1 142 3 4 5 6 7 8 9 10 11 12 13

Fig. 6.2 Time-specific survivorship curve


Assumptions that limit the accuracy of time-specific life tables

• Equal number of offspring are born each year– Favorable climate for breeding?

• A need for an independent method for estimating birth rates of each age class

• As a result, age-specific life tables are typically reported

– Of 31 life tables examined, 26 were age specific and only 5 were time specific.


Age-specific life tables

• Needed for short-lived organisms– Time-specific life tables biased toward the stage

common at the moment

• Follows one cohort or generation

• Population censuses must be frequent and conducted over a limited time– Ex. Table 6.2 and Figure 6.3

• Comparison in the accuracy of life tables (Figure 6.5)



0

0.5

1

1.5

2

2.5

3

3.5

n (l

og s

cale

) x

Age (years)

1 2 763 4 5

Fig. 6.3 Age-specific survivorship curve for the American robin.


Fig. 6.5 Hypothetical comparison of cohort survivorship of humans born in 1930.

Comparison in the accuracy of life tables


General types of survivorship curves (Figure 6.4)• Type I

– Most individuals are lost when they are older

– Vertebrates or organisms that exhibit parental care and protect their young

– Small dip at young age due to predators

• Type II– Almost linear rate of loss

– Many birds and some invertebrates

• Type III– Large fraction are lost in the juvenile stages

– Invertebrates, many plants, and marine invertebrates that do not exhibit parental care

– Large losses due to predators


Type I

Type II

Type III

Many mammals

Many birds,small mammals,lizards, turtles

Many invertebrates

Age

Num

ber

of

surv

ivors

(n )

(log s

cale

) x

1000

100

10

1

0.1

Fig. 6.4


6.2 Reproductive rate

• Fecundity– Age-specific birth rates

– Number of female offspring produced by each breeding female

• Fecundity schedules– Fecundity information in life table

– Describe reproductive output and survivorship of breeding individuals.

– Ex. Table 6.3



Fecundity schedules

• Table components

– lx = survivorship (number of females surviving in each age class

– mx = age-specific fecundity

– Ro = population’s net reproductive rate = lx mx

• Ro = 1; population is stationary

• Ro > 1; population is increasing

• Ro < 1; population is decreasing

• Table 6.3


Fecundity schedules

• Variation in formula for plants

– Age-specific fecundity (mx ) is calculated differently

– Fx = total number of seeds, or young deposited

– nx = total number of reproducing individuals

– mx = Fx / nx

– Table 6.4



6.3 Deterministic Models: Geometric Growth• Predicting population growth ( 預測族群的成

長 ) ，需要知道：– Ro

– Initial population size

– Population size at time t

• Population size of females at next generation = Nt+1= RoNt

– Ro = net reproductive rate

– Nt = population size of females at this generation


Geometric Growth

• Dependency of Ro

– Ro < 1; population becomes extinct

– Ro = 1; population remains constant• Population is at equilibrium

• No change in density

– Ro > 1; population increases• Even a fraction above one, population will increase

rapidly

• Characteristic “J ” shaped curve

• Geometric growth

• Figure 6.7


100

200

300

400

500Pop

ula

tion in

siz

e (

N)

0

Generations30

R =1.20 0

R =1.15 0

R =1.10 0

R =1.05 0

10 20

N +1 = R N t 0 t

Fig. 6.7

R 值愈大，族群的成長愈快


Geometric Growth

– Ro > 1; population increases (cont.).

• Something (e.g., resources) will eventually limit growth

• Population crash

• Figure 6.8a

• Figure 6.8b

• Figure 6.8c


1910 1920 1930 1940 1950

Num

ber

of

rein

deer

2000

1500

1000

500

0

YearFig. 6.8 a

2003 chap.6 Population Growth生態學 27Fig6.8b和c


Geometric Growth：Human population growth

• Prior to agriculture and domestication of animals (~10,000 B.C.)– Average annual rate of growth: ~0.0001%

• After the establishment of agriculture– 300 million people by 1 A.D.

– 800 million by 1750

– Average annual rate of growth: ~0.1%


Geometric Growth：Human population growth• Period of rapid population growth

– Began 1750

– From 1750 to 1900• Average annual rate of growth: ~0.5%



• Reasons for rapid growth– Advances in medicine

– Advances in nutrition

– Trends in growth (Figure 6.9)


1830

1930

1960

1975

1987

1998

2009

2020

2033

20462100

0

1

2

3

4

5

6

7

8

9

10

11

13

12

14

Bill

ions

of

people

2-5 millionYears ago

7,000BC

6,000BC

5,000BC

4,000BC

3,000BC

2,000BC

1,000BC

1AD

1,000AD

2,000AD

3,000AD

Year

4,000AD

Fig. 6.9 The world population explosion.


Human population statistics

– Population is increasing at a rate of 3 people every second

– Current population: over 6 billion

– UN predicts population will stabilize at 11.5 billion by 2150

• Developed countries– Average annual rate of growth from 1960-1965: 1.19%

– Average annual rate of growth from 1990-1995: 0.48%

• Developing countries– Average annual rate of growth from 1960-1965: 2.35%

– Average annual rate of growth from 1990-1995: 2.38%



•Fertility rates • Theoretic replacement rate: 2.0– but Actual replacement rate: 2.1


Overlapping generations

• Many species in warm climates reproduce continually and generations overlap.

• Rate of increase is described by a differential equation– dN / dt = rN = (b – d)N

– N = population size

– t = time

– r = per capita rate of population growth

– b = instantaneous birth rate

– d = instantaneous death rate

– dN = the rate of change in numbers

– dN / dt = the rate of population increase


0

1

2

3

4

5

20 40 60 80 100

r = 0.02

r =0.01

r = 0(equilibrium)

Time (t)

In (

N)

Fig. 6.10

•The starting population is N=10

–r is analogous to Ro

» In a stable population» r = (ln Ro) / Tc

• Tc generation time


族群加倍的時間

• Nt =N0ert

• Nt / N0 = ert

• If Nt / N0 = 2, ert = 2

• ln(2) = rt

• 0.69315 = rt

• t = 0.69315 / r

• r = 0.01 t = 69.3

• r = 0.02 t = 34.7

• r = 0.03 t = 23.1

• r = 0.04 t = 17.3

• r = 0.05 t = 13.9

• r = 0.06 t = 11.6


Logistic growth equations

• dN / dt= rN[(K-N)/K]; or

• dN / dt = =rN[1-(N/K)]– dN / dt = Rate of population change

– r = per capita rate of population growth

– N = population size

– K = carrying capacity

• S-Shaped Curve: Figure 6.11


Pop

ula

tion s

ize

K

Time

Logistic “S” shaped curve

Geometric “J” shaped curve


Logistic growth assumptions

1. Relation between density and rate of increase is linear

2. Effect of density on rate of increase is instantaneous

3. Environment (and thus K) is constant

4. All individuals reproduce equally

5. No immigration and emigration


Logistic growth assumptions

• Testing assumptions – Early laboratory cultures Pearl 1927

• Figure 6.12

– Complex studies and temporal effects• Figure 6.13


150

300

450

600

750A

mount

of

yeast

K = 665

0 2 4 6 8 10 12 14 16 18 20

Time (hrs)Fig. 6.12 yeast


200

400

600

800Time

N

Num

ber

per

12

gra

ms

of

wheat

Logistic curve predicted by theory

Time (weeks)

50 100 180

Callandra oryzaeRhizopertha dominica

Fig. 6.13 grain beetles


Difficulty in meeting assumptions in nature

1. Each individual added to the population probably does not cause an incremental decrease to r

2. Time lags, especially with species with complex life cycles

3. K may vary seasonally and/or with climate

4. Often a few individuals command many matings

5. Few barriers to prevent dispersal


Effect of time lags– Robert May (1976)

– Incorporated time lags into logistic equation

– dN / dt = rN[1-(Nt- /K)]

• dN / dt = Rate of population change

• r = per capita rate of population growth

• N = population size

• K = carrying capacity.

• Nt-= time lag between the change in population size and its effect on population growth, then the population growth at time t is controlled by its size at some time in the past, t -

• Nt-= population size in the past


Effect of time lags

• Ex. r = 1.1, K = 1000 and N = 900– No time lag, new population size

• dN / dt = 1.1 x 900 (1 – 900/1000) = 99• New population size = 900 + 99 = 999• Still below K

– With time lag, where a population is 900, although the effects of crowding are being felt as though the population was 800

• dN / dt = 1.1 x 900 (1 – 800/1000) = 198• New population size = 900 + 198 = 1098• Possible for a population to exceed K


Effect of response time• Ratio of time lag () to response time (1/r) or r

controls population growth (Figure 6.14)– r is small (<0.368)

• Population increases smoothly to carrying capacity– rt is large (>1.57)

• Population enters into a stable oscillation called a limit cycle

• Rising and falling around K• Never reaching equilibrium

– rt is intermediate (>0.368 and <1.57)• Populations undergo oscillations that dampen with

time until K is reached


K

Time (t)

Smooth response

K

Time (t)

Num

ber

of

indiv

iduals

(N

)

K

Time (t)

Damped oscillations

period

am

plit

ude

Stable limit cycle

r small (<0.368)

r medium (>0.368,<1.57)

r large (>1.75)

Num

ber

of

indiv

iduals

(N

)N

um

ber

of

indiv

iduals

(N

)

Fig. 6.14


Species with discrete generations

• Nt+1 = Nt + rNt [1 – (Nt / K)]– In discrete generations, the time lag is 1.0

• r is small (2.0)– Population generally reaches K smoothly

• r is between 2.0 and 2.449– Population enters a stable two-point limit cycle with sharp

peaks and valleys• r is between 2.449 and 2.570

– More complex limit cycles• r is larger than 2.57

– Limit cycles breakdown– Population grows in a complex, non-repeating patterns,

know as ‘chaos’• Figure 6.15


N

N

N

t

t

t

r small (2.000–2.499)

r medium (2.499–2.570)

r large (>2.570)

Fig. 6.15


6.4 Stochastic Models

• Models are based on probability theory

– Figure 6.16

• dN / dt = rN = (b – d) N– If b = 0.5, d = 0, and N0 = 10,

– integral form of equation Nt = N0ert

– So for the above example, Nt= 10 x 1.649 = 16.49

• Path of population growth (Figure 6.17)


0

0.10

0.20

0.30

Pro

port

ion o

f obse

rvati

ons

6 8 10 12 14

Population size

Fig. 6.16 stochastic frequency distribution


Pop

ula

tion d

ensi

ty

Time

Extinction

Possible stochastic

path

Fig. 6.17


Stochastic Models

• Probability of extinction = (d/b)N0

– The larger the initial population size

– The greater the value of b – d

– The more resistant a population is to extinction

• Introduce biological variation into calculations of population growth– More representative of nature

– More complicated mathematics


Applied Ecology

Human Population growth and the use of contraceptives

• 1992 Johns Hopkins study– Developed countries

• 70% of couples use contraceptives

– Developing countries• ~45% of couples use contraceptives

• Africa, 14%

• Asia, 50%

• Latin America, 57%


Human Population growth

• China– 1950s and 1960s

• Fertility was six children per woman

– 1970s• Government planning and incentives to reduce

population growth

– 1990• 75% use birth control

• Fertility rate dropped to 2.2


Human Population growth

Other governments– 1976, only 97 governments supported family planning

– 1988, 125 governments supported family planning

– As of 1989, in 31 countries, couples have no access to family planning

• Women– Women in developing countries want fewer children

– In virtually every country outside of Saharan Africa, the desireds number of children is below 3


Low growth rates

• Countries concerned about low growth rates– Some Western European countries and other

developed countries

– Total fertility has dropped below the replacement level of 2.1

• Reduced populations concerns – Affect political strength

– Economic structure


問題與討論！

[email protected]

Ayo 台南站： http://mail.nutn.edu.tw/~hycheng/

Documents

Section III Population Ecology