Chapter 6 Population Growth. 生態學的分科：以生物組織水準來分個體生態學 Autecology 種群（族群）生態學 Population ecology 群體（群落）生態學

Chapter 6Population

Growth

生態學的分科：以生物組織水準來分

• 個體生態學 Autecology• 種群（族群）生態學 Population ecology• 群體（群落）生態學 Synecology:

community ecology• 生態系統生態學 Ecosystem ecology

Outline• Tabulating changes in population

age structure through time– Time-specific life tables– Age-specific life tables

Outline• Fecundity schedules and female

fecundity, and estimating future population growth

• Population growth models– Deterministic models– Geometric models

Outline• Population growth models (cont.).

– Logistic models– Stochastic models

Demography and Population Growth

Demography 族群統計學、人口統計學The quantitative description of a population

Population 族群A group of conspecifics inhabits a specific place at a specific time.

. The demographic and genetic populations are not necessarily the same.. Demographic distinction and genetic difference are not necessarily corresponding.

Demography and Population Growth

. Population characteristics

. Quantitative parameters

. Population difference: density, age distribution. Population growth and quantitative methods. Population determination factors. Offspring production and energy invest. Population genetics variation. Intrapopulation behavior

Population density 族群密度

. abundant 豐富的

. common 普遍

. rare 稀有

. endangered 瀕危 ( 絕 ) 的

. extinct 絕滅的eg. Zacco pachycephalus 粗首獵 ( 溪哥 ) endemic to Taiwan, but common in west coasts

Sampling methods:. Traps 陷阱. Fecal pellets 糞便. Vacuolization frequencies 鳴叫頻度. pelt records 生皮記錄. Catch per unit effort (CPUE) 單位努力漁獲量. Percentage ground cover 遮蔽度. Frequency of abundance along transects or in quadrants of known area. 穿越線或四分區豐度。. Feeding damage 食害程度。. Roadside spotting in a standard distance 單位長度內視察的記錄。

Estimates of absolute density-1

. Proportion of population marked = (no. of animals marked at t1) / (total no, of animals in population). Proportion of sample marked = (no. of marked animals captured at t2) / (total no. of animals captured at t2)

Estimates of absolute density-2

If random sampling, Proportion of population marked = proportion of sample marked

t1 標示個體 / 族群數量 ﹦ t2 再捕獲標示個體數 /t2 捕獲數（已知數）（未知數）（已知數）（已知數）

Assumptions:1. 標記不會對於個體有增加死亡率的危險2. 標記不會影響再捕捉的機率（記憶與學習）3. 實驗期間族群內個體沒有移入或移出的問題4. 實驗期間沒有死亡或新生的變化

櫻花鉤吻鮭

#13Chapt. 06

Salmon Watch—Skin Diving

#14Chapt. 06

Salmon Watch — Skin Diving and Counting

1800

1090

660

1136

606

941

616

959

253

943

278

565

1227

1718

346408

3428

494

782

728

796

1854

788

1155

857

637

2495

1870

638679

679

0

500

1000

1500

2000

2500

3000

3500

4000

普查時間（Ｓ：春季；Ｗ：冬季）

數量（尾）

瑞

伯

颱

風

、

芭

比

絲

颱

風

碧

莉

絲

颱

風

、

象

神

颱

風

溫

泥

颱

風

賀

伯

颱

風

枯

水

期

八

月

大

雨

六

月

大

雨

九

月

洪

水

九

月

乾

旱

琳

恩

颱

風

韋

恩

颱

風

桃

芝

颱

風

、

納

莉

颱

風

七家灣溪流域櫻花鉤吻鮭歷年族群變化圖

重大天災以紅色圖說標於圖中

歷年最高數量3428尾

2000秋繁殖季節遭逢象神颱風

1994年以來最低族群數量

346尾

頭前溪毛蟹

Dispersion 分散情形. random distribution: 闊葉林中的樹. aggregated: 草地上的韓國薊. hyperdispersed, regular in dispersion: 繁殖區的海鳥

雪山冷杉林

Statistical methods:-1

. Poisson distributionPx=axe-a/x!

P = Poisson probability x = No. of occurrencesa = The mean number of

occurrencese = The base of the natural log

蘭嶼熱帶森林

Statistical methods:-2

方差 / 平均數比率指標

If s2/x (mean / variance) > 1 hyperdispersionIf s2/x (mean / variance) < 1 aggregation Significant test (n-1)s2/x x : meanS2 : variance

Life Tables 壽命表是一種記載和描述出生後的一定數量的個體，隨著生命過程而損失，減少的規律的一種方法，是描述種群的死亡與存活過程的一種有效手段。

Life Tables 壽命表1. age specific (cohort analysis) 群對分析 , 年齡群特性。Short-lived animal ： 2. time-specific (static) 靜態分析 , 時間特性。long-lived animals.

age-specific life table 特定年齡壽命表

=dynamic life table=horizontal life table以一特定世代群體的命運為根據，跟蹤該世的整個生活史，每隔一定時間統計存活數與死亡數。 . 優點

. 可反映出與密度相關的死亡率與繁殖率變化。

. 分析種群數量變動與調節的機制。缺點

. 不易掌握全世代的生物資料。

Time specific life table 特定時間壽命表

=Stationary or Vertical life table由不同年齡組組成的群體及估算的各年齡組的死亡率為根據。優點

. 可在短時間內獲得所需資料缺點

. 不同世代會有不同歷史背景

. 推理不易來自隨機的取樣

. 種群結構不易穩定

Composite life table 綜合壽命表

取不同來源的資料平均值作壽命表 (B) birth 種群大小 death (D) dn=B-D

Net reproductive rate: RoRo = lx mxln = surviving proportionmx = age-specific fertility

四種族群年齡結構的組成圖

已開發和開發中國家在 1995～ 2025年之間的年齡結構

三種不同的存活曲線模型

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

Deterministic Models 宿命論模式

Discrete Generations 不連續世代 Geometric Carrying Capacity logistick-n 尚未利用於種群增長的潛力

Stochastic Models 推計學模式

Discrete Generation Overlapping Generations

細菌的對數生長

海豹族群的成長模式

族群成長率與族群數量增加之間的關係

邏輯模型所預測的族群成長情形

指數成長和對數成長的比較

Life Tables• Date on numbers of individuals at

each age– Construct life tables– Demonstrate the age structure of a

population

• Life table construction – demography

Life Tables• Life table construction –

demography (cont.).– How a population will grow– Construction of data

•Follow a cohort from births to deaths (age specific life table)

Life Tables– 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)

0

0.5

1

1.5

2

2.5

3

3.5

Age (years)

n (l

og s

cale

) x

10

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

Life Tables•Useful parameters in the life tables

(Deevey 1947)– 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

(Deevey 1947) (cont.).– 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 = (ns+ ns+1) / 2

» Tx = intermediate step in determining life expectancy = Lx

Life Tables– ex = the mean expectation of life for

organisms alive (cont.).» Ex = Tx / nx

•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

Life Tables•Assumptions that limit the accuracy of

time-specific life tables (cont.). – As a result, age-specific life tables are

typically reported

– Age-specific life tables •Needed for short-lived organisms

– Time-specific life tables biased toward the stage common at the moment

Life Tables– Follows one cohort or generation– Population censuses must be

frequent and conducted over a limited time

– Ex. Table 6.2 and Figure 6.3

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

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

Life Tables•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

Life Tables•Type III (cont.).

– Large losses due to predators

– Comparison in the accuracy of life tables•Figure 6.5

Reproductive Rate• Fecundity

– Age-specific birth rates– Number of female offspring produced

by each breeding female

• Fecundity schedules– Fecundity information in life table

Reproductive Rate• Describe reproductive output and

survivorship of breeding individuals

• Fecundity schedules (cont.).– Ex. Table 6.3

Reproductive Rate– Table components

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

•mx = age-specific fecundity•Ro = population’s net reproductive rate

= lx mx

Reproductive Rate•Rx = population’s net reproductive rate

= lxmx

– Ro = 1; population is stationary– Ro > 1; population is increasing– Ro < 1; population is decreasing– Table 6.3

Reproductive Rate•Variation in formula for plants

– Age-specific fecundity (m ) is calculated differently

– Fx = total number of eggs, seeds, or young deposited

– nx = total number of reproducing individuals– mx = Fx / nx

– Figure 6.5

x

Reproductive Rate•Variation in formula for plants (cont.).

– Table 6.4

Reproductive Rate•Variation in formula for plants (cont.).

– Figure 6.6

Deterministic Models: Geometric Growth

• Predicting population growth• Need to know;

– Ro

– Initial population size– Population size at time t

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


• Population size (cont.).– Ro = net reproductive rate– Nt = population size of females at this

generation


• Dependency of Ro

– Ro < 1; population becomes extinct– Ro = 1; population remains constant

•Population is at equilibrium•No change in density


• Dependency of R– Ro > 1; population increases

•Even a fraction above one, population will increase rapidly

o


– Ro > 1; population increases (cont.).•Characteristic “J” shaped curve•Geometric growth•Figure 6.7

100

200

300

400

500Popula

tion in s

ize (

N)

0Generations

30

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


– Ro > 1; population increases (cont.).•Something (e.g., resources) will

eventually limit growth•Population crash•Figure 6.8a

1910 1920 1930 1940 1950

Num

ber

of

rein

deer

2000

1500

1000

500

0

Year


– Ro > 1; population increases (cont.).•Figure 6.8b


– Ro > 1; population increases (cont.).•Figure 6.8c


• 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%


– 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%


– Period of rapid population growth•Reasons for rapid growth

– Advances in medicine– Advances in nutrition

•Trends in growth (Figure 6.9)

1830

1930

1960

1975

19871998

20092020

2033

20462100

01

23

4

5

67

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


– 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%


– Human population statistics (cont.).– Average annual rate of growth from 1990-

1995: 0.48%

•Developing countries– Average annual rate of growth from 1960-

1965: 2.35%


– Human population statistics (cont.).•Developing countries

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


• Fertility rates– Average number of live births typically borne

by a woman during her lifetime (Table 6.6)


• Fertility rates (cont.).– Theoretic replacement rate: 2.0– Actual replacement rate: 2.1– Decline in fertility rate

» 1960-1965: ~5.0» 1990: 3.3


– Overlapping generations•Many species in warm climates

reproduce continually and generations overlap.


– Overlapping generations (cont.).•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


– Overlapping generations (cont.).•Rate of increase is described by a

differential equation– dN / dt = rN = (b – d)N– b = instantaneous birth rate– d = instantaneous death rate– dN = the rate of change in numbers


•Rate of increase is described by a differential equation (cont.).

– dN / dt = the rate of population increase– Produces a “J” shaped curve– Plot with the natural logarithm, produces a

straight line (Figure 6.8)

0

1

2

3

4

5

20 40 60 80 100

r = 0.02

r =0.01

r = 0(equilibrium)

Time (t)

In (

N)


– r is analogous to Ro

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

– Logistic Growth•Occurs in populations where resources

are or can be limiting•Logistic growth equations

•Tc generation time


•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


» (K - N) / K = unused resources remaining

» S-Shaped Curve: Figure 6.11

•As N gets larger, the amount of resources remaining gets smaller•When N = K, zero growth will occur

Popul a

t ion s

ize

K

Time


•Logistic growth assumptions– 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


– Testing assumptions – Early laboratory cultures Pearl 1927

» Figure 6.12

150

300

450

600

750A

mount

of

yeast

K = 665

0 2 4 6 8 10 12 14 16 18 20

Time (hrs)


– Complex studies and temporal effects (cont.).» Figure 6.13

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


•Difficulty in meeting assumptions in nature

– 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


•Difficulty in meeting assumptions in nature (cont.).

– Often a few individuals command many matings

– Few barriers to prevent dispersal

•Effect of time lags – Robert May (1976)


•Effect of time lags (cont.).– 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


– dN / dt = rN[1-(N t- /K)] (cont.).» 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– Ex. r = 1.1, K = 1000 and N = 900

» No time lag, new population size


» No time lag, new population size (cont.).

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


» No time lag, new population size(cont.).

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

• Still below K

•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» i. Inversely proportional to r = (1/r)

– Ratio of time lag () to response time (1/r) or r controls population growth

•r is small (<0.368)– Population increases smoothly to carrying capacity– (Figure 6.14)

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

)


• r is large (>1.57)– Population enters into a

stable oscillation called a limit cycle

– Rising and falling around K– Never reaching equilibrium

• r is intermediate (>0.368 and <1.57)

– Populations undergo oscillations that dampen with time until K is reached


– Species with discrete generations» Logistic equation

• 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


– Species with discrete generations» Logistic equation

• r is between 2.449 and 2.570– More complex limit cycles– 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)


– Species with discrete generations» Logistic equation (cont.).

• r is larger than 2.57– Limit cycles breakdown– Population grows in a complex,

non-repeating patterns, know as ‘chaos’

Stochastic Models• Models are based on probability

theory– Figure 6.16

0

0.10

0.20

0.30

Pro

port

ion o

f obse

rvati

ons

6 8 10 12 14

Population size

Stochastic Models• Stochastic models of geometric

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

integral form of equation

Stochastic Models– Stochastic models of integral form of

equation•Nt = N0ert

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

Stochastic Models• Stochastic models of geometric

growth (cont.).– Path of population growth

•Figure 6.17

Popula

tion d

ensi

ty

Time

Extinction

Possible stochastic

path

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

Stochastic Models• Introduce biological variation into

calculations of population growth– More representative of nature– More complicated mathematics

Applied Ecology• 1992 Johns Hopkins study

– Developed countries•70% of couples use contraceptives

– Developing countries•~45% of couples use contraceptives•Africa, 14%

Applied Ecology• Developing countries (cont.).

•Asia, 50%•Latin America, 57%

Applied Ecology• 1992 Johns Hopkins study (cont.).

– China•1950s and 1960s

– Fertility was six children per woman

•1970s– Government planning and incentives to

reduce population growth

Applied Ecology– China (cont.).

•1990– 75% use birth control– Fertility rate dropped to 2.2

– Other governments•1976, only 97 governments supported

family planning

Applied Ecology– Other governments (cont.).

•1988, 125 governments supported family planning

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

Applied Ecology– Women

•Women in developing countries want fewer children

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

Applied Ecology• 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

Applied Ecology• Countries concerned about low

growth rates (cont.).– Reduced populations concerns

•Affect political strength•Economic structure

Summary

• Life tables provide information on how populations are structured. For longer-lived organisms, time specific tables provide a snapshot of the population’s age structure.

Summary

• For shorter-lived organisms, age specific life tables are used.

• Three types of survivorship– Type I: Most individuals die after

they have reached their sexual maturity

Summary

• Three types of survivorship– Type II: An almost linear rate of loss– Type III: Most individuals are lost as

juveniles

Summary

• With information on survivorship and age-specific fecundity, net reproductive rate, R0, can be calculated and population growth determined

Summary

• Population growth can be examined using deterministic or stochastic models

Summary

• When environments are favorable, and resources are not limiting– Populations undergo geometric or

exponential growth– Results in J-shaped population growth

curves

Summary

• Limits on population growth– Exhaustion of space or resources– Waste product accumulation– Results in an S-shaped growth curve

Summary

• Limits on population growth– Upper level is the carrying capacity, K

• Population growth curves are not always smooth in trajectory

Summary

• Population growth curves (cont.).– Curves may oscillate or appear to be

random fluctuations– Result of time lags

Discussion Question #1• What limits the population growth of

organisms most commonly – weather, resources, pollutants, natural enemies, disease or social factors? Pick some local plants and animals and make some educated guesses.

Discussion Question #2• What do you think will limit human

population growth? Do you think the carrying capacity of the earth for humans has still to be reached or have we overshot it? Remember, time lags can cause carrying capacity to be exceeded. If fossil fuels are used up, could the human population crash?

Discussion Question #3 • Compare the reproduction output of a

family line, which has mothers always having twin girls at age 20 years old, versus a family line which has mothers having triplet girls at age 33. What conclusions can you draw? (Hint: compare reproductive output after 100 and 200 years.). Should all couples stop reproducing after two children?

Documents

Chapter 6 Population Growth. 生態學的分科： 以生物組織水準來分 個體生態學 Autecology 種群（族群）生態學 Population ecology 群體（群落）生態學

Chapter 6 Population Growth. 生態學的分科：以生物組織水準來分個體生態學 Autecology 種群（族群）生態學 Population ecology 群體（群落）生態學