Title: Section III Population Ecology
1Section III Population Ecology
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- ????? Ayo
- Japalura_at_hotmail.com
2Section Three Population Ecology
- Chap.6 Population growth (????)
- Chap.7 Physical environment (????)
- Chap.8 Competition and coexistence (?????)
- Chap.9 Mutualism (??)
- Chap.10 Predation (??)
- Chap.11 Herbivory (??)
- Chap.12 Parasitism (??)
- Chap.13 Evaluating the controls on population size
3Chap. 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
46.1 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)
5Time-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)
6Life 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 nx1 - lx proportion of organisms surviving to the
beginning of age interval x ns / n0
7Life 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 nx1) / 2 - Tx intermediate step in determining life
expectancy SLx - ex Tx / nx
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9Fig. 6.2 Time-specific survivorship curve
3.5
3
2.5
2
n (log scale)
10
1.5
x
1
0.5
0
1
14
2
3
4
5
6
7
8
9
10
11
12
13
Age (years)
10Assumptions 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.
11Age-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)
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133.5
3
2.5
2
n (log scale)
1.5
x
1
0.5
0
1
2
7
6
3
4
5
Age (years)
Fig. 6.3 Age-specific survivorship curve for the
American robin.
14Fig. 6.5 Hypothetical comparison of cohort
survivorship of humans born in 1930.
Comparison in the accuracy of life tables
15General 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
16Type 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
176.2 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
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19Fecundity schedules
- Table components
- lx survivorship (number of females surviving in
each age class - mx age-specific fecundity
- Ro populations net reproductive rate Slx mx
- Ro 1 population is stationary
- Ro gt 1 population is increasing
- Ro lt 1 population is decreasing
- Table 6.3
20Fecundity 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
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226.3 Deterministic Models
Geometric Growth
- Predicting population growth (???????),????
- Ro
- Initial population size
- Population size at time t
- Population size of females at next generation
Nt1 RoNt - Ro net reproductive rate
- Nt population size of females at this generation
23Geometric Growth
- Dependency of Ro
- Ro lt 1 population becomes extinct
- Ro 1 population remains constant
- Population is at equilibrium
- No change in density
- Ro gt 1 population increases
- Even a fraction above one, population will
increase rapidly - Characteristic J shaped curve
- Geometric growth
- Figure 6.7
24R 1.20
0
500
R 1.15
0
R???,???????
400
300
Population in size (N)
N 1 R N
200
t 0 t
R 1.10
0
100
R 1.05
0
0
10
20
30
Fig. 6.7
Generations
25Geometric Growth
- Ro gt 1 population increases (cont.).
- Something (e.g., resources) will eventually limit
growth - Population crash
- Figure 6.8a
- Figure 6.8b
- Figure 6.8c
262000
1500
Number of reindeer
1000
500
0
1910
1920
1930
1940
1950
Fig. 6.8 a
Year
27Fig6.8b?c
28Geometric GrowthHuman 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
29Geometric GrowthHuman 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)
3014
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.
31Human 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
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33- Fertility rates
- Theoretic replacement rate 2.0
- but Actual replacement rate 2.1
34Overlapping 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
355
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 N10
0
20
60
80
100
40
Fig. 6.10
Time (t)
36???????
- 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
- Nt N0ert
- Nt / N0 ert
- If Nt / N0 2, ert 2
- ln(2) rt
- 0.69315 rt
- t 0.69315 / r
37Logistic growth equations
- dN / dt rN(K-N)/K or
- dN / dt rN1-(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
38K
Geometric J shaped curve
Population size
Logistic S shaped curve
Time
39Logistic 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
40Logistic growth assumptions
- Testing assumptions
- Early laboratory cultures Pearl 1927
- Figure 6.12
- Complex studies and temporal effects
- Figure 6.13
41750
K 665
600
450
Amount of yeast
300
150
0
2
4
6
8
10
12
14
16
18
20
Time (hrs)
Fig. 6.12 yeast
42Logistic 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
43Difficulty 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
- Often a few individuals command many matings
- Few barriers to prevent dispersal
44Effect of time lags
- Robert May (1976)
- Incorporated time lags into logistic equation
- dN / dt rN1-(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
45Effect 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
46Effect of response time
- Ratio of time lag (t) to response time (1/r) or
rt controls population growth (Figure 6.14) - rt is small (lt0.368)
- Population increases smoothly to carrying
capacity - rt is large (gt1.57)
- Population enters into a stable oscillation
called a limit cycle - Rising and falling around K
- Never reaching equilibrium
- rt is intermediate (gt0.368 and lt1.57)
- Populations undergo oscillations that dampen
with time until K is reached
47r small (lt0.368)
Smooth response
Number of individuals (N)
K
Time (t)
r t medium (gt0.368,lt1.57)
Damped oscillations
Number of individuals (N)
K
Time (t)
r t large (gt1.75)
Stable limit cycle
period
Number of individuals (N)
K
amplitude
Fig. 6.14
Time (t)
48Species with discrete generations
- Nt1 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
49r small (2.0002.499)
N
t
r medium (2.4992.570)
N
t
r large (gt2.570)
N
Fig. 6.15
t
506.4 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)
510.30
0.20
Proportion of observations
0.10
0
8
6
10
12
14
Population size
Fig. 6.16 stochastic frequency distribution
52Fig. 6.17
Population density
Possible stochastic path
Extinction
Time
53Stochastic 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
54Applied EcologyHuman 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
55Human 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
56Human 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
57Low 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
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