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Section III Population Ecology

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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

2003 生態學 chap.6 Population Growth

Chap. 6 Population Growth

- Tabulating changes in population age structure through time
- Time-specific life tables
- Age-specific life tables

- Fecundity schedules and female fecundity, and estimating future population growth
- Population growth models
- Deterministic models
- Geometric models
- Logistic models
- Stochastic models

2003 生態學 chap.6 Population Growth

6.1 Life tables

- The construction of life tables is termed demography.
- 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)

2003 生態學 chap.6 Population Growth

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)

2003 生態學 chap.6 Population Growth

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

2003 生態學 chap.6 Population Growth

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 = SLx
- ex = Tx / nx

2003 生態學 chap.6 Population Growth

Fig. 6.2 Time-specific survivorship curve

3.5

3

2.5

2

n (log scale)

10

1.5

x

1

0.5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Age (years)

2003 生態學 chap.6 Population Growth

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.

2003 生態學 chap.6 Population Growth

Age-specific life tables 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)

2003 生態學 chap.6 Population Growth

3.5 tables

3

2.5

2

n (log scale)

1.5

x

1

0.5

0

1

4

2

3

5

6

7

Age (years)

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

2003 生態學 chap.6 Population Growth

Comparison in the accuracy of life tables tables

Fig. 6.5 Hypothetical comparison of cohort survivorship of humans born in 1930.2003 生態學 chap.6 Population Growth

General types of survivorship curves (Figure 6.4) tables

- 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

2003 生態學 chap.6 Population Growth

Type I tables

1000

Many birds,

small mammals,

lizards, turtles

Many mammals

100

x

Number of survivors (n ) (log scale)

Type II

10

Many invertebrates

1

Type III

0.1

Fig. 6.4

Age

2003 生態學 chap.6 Population Growth

6.2 Reproductive rate tables

- 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

2003 生態學 chap.6 Population Growth

Fecundity schedules tables

- Table components
- lx = survivorship (number of females surviving in each age class
- mx = age-specific fecundity
- Ro = population’s net reproductive rate = Slx mx
- Ro = 1; population is stationary
- Ro > 1; population is increasing
- Ro < 1; population is decreasing
- Table 6.3

2003 生態學 chap.6 Population Growth

Fecundity schedules tables

- 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

2003 生態學 chap.6 Population Growth

6.3 Deterministic Models: tables 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

2003 生態學 chap.6 Population Growth

Geometric Growth tables

- 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

2003 生態學 chap.6 Population Growth

R =1.20 tables

0

500

R =1.15

0

R值愈大，族群的成長愈快

400

300

Population in size (N)

N +1 = R N

200

t 0 t

R =1.10

0

100

R =1.05

0

10

0

20

30

Fig. 6.7

Generations

2003 生態學 chap.6 Population Growth

Geometric Growth tables

- Ro > 1; population increases (cont.).
- Something (e.g., resources) will eventually limit growth
- Population crash
- Figure 6.8a
- Figure 6.8b
- Figure 6.8c

2003 生態學 chap.6 Population Growth

2000 tables

1500

Number of reindeer

1000

500

0

1910

1920

1930

1940

1950

Fig. 6.8 a

Year

2003 生態學 chap.6 Population Growth

Fig6.8b tables和c

2003 生態學 chap.6 Population Growth

Geometric Growth tables：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%

2003 生態學 chap.6 Population Growth

Geometric Growth tables：Human population growth

- Period of rapid population growth
- Began 1750
- From 1750 to 1900
- Average annual rate of growth: ~0.5%

- From 1900 to 1950
- Average annual rate of growth: ~0.8%

- From 1950 to 2000
- Average annual rate of growth: ~1.7%

- Reasons for rapid growth
- Advances in medicine
- Advances in nutrition
- Trends in growth (Figure 6.9)

2003 生態學 chap.6 Population Growth

14 tables

Year

13

12

11

2100

2046

10

2033

9

2020

8

Billions of people

2009

7

1998

6

1987

5

1975

4

1960

3

1930

2

1830

1

0

2-5 million

Years ago

7,000

BC

6,000

BC

5,000

BC

4,000

BC

3,000

BC

2,000

BC

1,000

BC

1

AD

1,000

AD

2,000

AD

3,000

AD

4,000

AD

Fig. 6.9 The world population explosion.

2003 生態學 chap.6 Population Growth

Human population statistics tables Developed countries Developing countries

- 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

- Average annual rate of growth from 1960-1965: 1.19%
- Average annual rate of growth from 1990-1995: 0.48%

- Average annual rate of growth from 1960-1965: 2.35%
- Average annual rate of growth from 1990-1995: 2.38%

2003 生態學 chap.6 Population Growth

- Fertility rates tables
- Theoretic replacement rate: 2.0
- but Actual replacement rate: 2.1

2003 生態學 chap.6 Population Growth

Overlapping generations tables

- 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

2003 生態學 chap.6 Population Growth

5 tables

r = 0.02

4

r =0.01

3

In (N)

r = 0

(equilibrium)

2

- r is analogous to Ro
- In a stable population
- r = (ln Ro) / Tc
- Tc generation time

1

The starting population is N=10

0

20

60

80

100

40

Fig. 6.10

Time (t)

2003 生態學 chap.6 Population Growth

N tablest =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

族群加倍的時間2003 生態學 chap.6 Population Growth

Logistic growth equations tables

- 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

2003 生態學 chap.6 Population Growth

K tables

Geometric “J” shaped curve

Population size

Logistic “S” shaped curve

Time

2003 生態學 chap.6 Population Growth

Logistic growth assumptions tables

- Relation between density and rate of increase is linear
- Effect of density on rate of increase is instantaneous
- Environment (and thus K) is constant
- All individuals reproduce equally
- No immigration and emigration

2003 生態學 chap.6 Population Growth

Logistic growth assumptions tables

- Testing assumptions
- Early laboratory cultures Pearl 1927
- Figure 6.12

- Complex studies and temporal effects
- Figure 6.13

- Early laboratory cultures Pearl 1927

2003 生態學 chap.6 Population Growth

750 tables

K = 665

600

450

Amount of yeast

300

150

18

0

2

4

6

8

10

12

14

16

20

Time (hrs)

Fig. 6.12 yeast

2003 生態學 chap.6 Population Growth

Logistic curve tables

predicted by theory

N

Time

800

600

Rhizopertha dominica

Number per 12 grams of wheat

Callandra oryzae

400

200

50

180

100

Time (weeks)

Fig. 6.13 grain beetles

2003 生態學 chap.6 Population Growth

Difficulty in meeting assumptions in nature tables

- Each individual added to the population probably does not cause an incremental decrease to r
- Time lags, especially with species with complex life cycles
- K may vary seasonally and/or with climate
- Often a few individuals command many matings
- Few barriers to prevent dispersal

2003 生態學 chap.6 Population Growth

Effect of time lags tables

- Robert May (1976)
- Incorporated time lags into logistic equation
- dN / dt = rN[1-(Nt-t /K)]
- dN / dt = Rate of population change
- r = per capita rate of population growth
- N = population size
- K = carrying capacity.
- Nt-t= 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 - t
- Nt-t= population size in the past

2003 生態學 chap.6 Population Growth

Effect of time lags tables

- 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

- No time lag, new population size

2003 生態學 chap.6 Population Growth

Effect of response time tables

- Ratio of time lag (t) to response time (1/r) or rt controls population growth (Figure 6.14)
- rt 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

- rt is small (<0.368)

2003 生態學 chap.6 Population Growth

r tables`small (<0.368)

Smooth response

Number of

individuals (N)

K

Time (t)

rtmedium (>0.368,<1.57)

Damped oscillations

Number of

individuals (N)

K

Time (t)

r t large (>1.75)

Stable limit cycle

period

Number of

individuals (N)

K

amplitude

Fig. 6.14

Time (t)

2003 生態學 chap.6 Population Growth

Species with discrete generations tables

- 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

2003 生態學 chap.6 Population Growth

r small (2.000–2.499) tables

N

t

r medium (2.499–2.570)

N

t

r large (>2.570)

N

Fig. 6.15

t

2003 生態學 chap.6 Population Growth

6.4 Stochastic Models tables

- 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)

2003 生態學 chap.6 Population Growth

0.30 tables

0.20

Proportion of observations

0.10

0

8

6

10

12

14

Population size

Fig. 6.16 stochastic frequency distribution

2003 生態學 chap.6 Population Growth

Fig. 6.17 tables

Population density

Possible

stochastic

path

Extinction

Time

2003 生態學 chap.6 Population Growth

Stochastic Models tables

- 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

2003 生態學 chap.6 Population Growth

Applied Ecology tablesHuman 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%

- Developed countries

2003 生態學 chap.6 Population Growth

Human Population growth tables

- 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

- 1950s and 1960s

2003 生態學 chap.6 Population Growth

Human Population growth tables Women

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 in developing countries want fewer children
- In virtually every country outside of Saharan Africa, the desireds number of children is below 3

2003 生態學 chap.6 Population Growth

Low growth rates tables

- 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

2003 生態學 chap.6 Population Growth

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