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Section III Population Ecology

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- 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

Road Map

- 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

- 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

- 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

- 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

- 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

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

- 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

- 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

2003 生態學 chap.6 Population Growth

3.5

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

2003 生態學 chap.6 Population Growth

- 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

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

- 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

2003 生態學 chap.6 Population Growth

- 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

- 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

2003 生態學 chap.6 Population 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

- 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

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

- 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

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和c

2003 生態學 chap.6 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

- 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

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

- 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

2003 生態學 chap.6 Population Growth

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

2003 生態學 chap.6 Population Growth

- 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

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

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

2003 生態學 chap.6 Population Growth

- 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

Geometric “J” shaped curve

Population size

Logistic “S” shaped curve

Time

2003 生態學 chap.6 Population Growth

- 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

- 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

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

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

- 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

- 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

- 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

- 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`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

- 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)

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

- 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

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

Population density

Possible

stochastic

path

Extinction

Time

2003 生態學 chap.6 Population Growth

- 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

- 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

- 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

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

- 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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2003 生態學 chap.6 Population Growth