Transcript Populations IV: Life Tables

Life Tables
Grizzly Bears (Ursus arctos horribilis)
Grizzly Bears (Ursus arctos horribilis)
– Yellowstone grizzly
population was declining
– Age-structured models
– survival of mothers was most
important to the population
– Legislation resulting from his
findings got tourists out of
areas with mothers and cubs
– Yellowstone grizzlies began
to recover
Population structure
• Sex (gender)
• Age
• Size (better for plants)
• Birth, death and movement rates vary in
different sex, age, or size categories.
Human age and sex structure
Baby boom and social security
Structured data
• Known (or marked)
individuals
• Carcasses
• Age structure (how many
of what age)
• Sex ratio (how many of
what sex)
Life tables
Cohort (dynamic): follow all
individuals born in one
time interval (e.g. year)
until they die
Cross-sectional (static): take
a snap-shot of the current
age-structure
Composite: data taken from
multiple years
Notation
x=
nx =
lx
age of the individual
number of individuals of age x
= number (or %) of individuals alive at age x
mx
= fecundity rate
average # of female offspring produced per female per time period
--------------------------------------------------------------------------------------------------
px = survival rate
probability of surviving from age x to x+1
qx = mortality rate
probability of dying between age x and x+1
Aphids Aphidoidea
Age, days
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Number of survivals Mean offspring per parent
1000
0
900
0
820
0
750
0
680
0
620
0
550
1
500
2
450
5
400
10
350
12
300
10
250
8
200
6
100
3
50
1
0
0
Excel data sheet – start with 1000
Cohort study
Survival – px
px – the probability of surviving from x to x+1
px = Nx+1/Nx
Nx
1000
900
820
750
680
620
550
500
450
400
350
300
250
200
100
50
0
px
1.00
0.90
0.91
0.91
0.91
0.91
0.89
0.91
0.90
0.89
0.88
0.86
0.83
0.80
0.50
0.50
0.00
1
0.8
0.6
Px
Age, days (x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.4
0.2
0
0
5
10
Age (days)
15
20
Survivorship, lx
lx – survival to age x - survivorship
lx = Nx/N0
Three types of survivorship curves – plotted on a semi-log plot
Survivorship of Aphids
log lx
1
0.1
0.01
0
5
10
15
Age (days)
Looks like Type I – lab conditions
20
Survivorship, lx
lx – survival to age x - survivorship
lx = Nx/N0
Nx
1000
900
820
750
680
620
550
500
450
400
350
300
250
200
100
50
0
lx
1
0.9
0.82
0.75
0.68
0.62
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.1
0.05
0
N0= 1000
Survivorship of Aphids
1
0.8
0.6
lx
Age, days (x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.4
0.2
0
0
2
4
6
8
Age (days)
10
12
14
16
Fecundity table (mx)
Usually recorded as # of females
produced per female of age x
Can be interpreted as the probability that
a female of age x will give birth to a
daughter during that time interval
14
Aphids
12
mx
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
Age (days)
10 11 12 13 14 15 16
Age, days (x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Nx
1000
900
820
750
680
620
550
500
450
400
350
300
250
200
100
50
0
lx
1
0.9
0.82
0.75
0.68
0.62
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.1
0.05
0
mx
0
0
0
0
0
0
1
2
5
10
12
10
8
6
3
1
0
Net reproductive number
R0: the mean number of female
offspring produced by a female
during her lifetime.
R0 < 1 population is
declining
R0 > 1 increasing population
R0 = 1 indicates a stationary
population
If lx is a proportion:
R0   lx mx
Age, days (x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Nx
1000
900
820
750
680
620
550
500
450
400
350
300
250
200
100
50
0
lx
1
0.9
0.82
0.75
0.68
0.62
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.1
0.05
0
mx
0
0
0
0
0
0
1
2
5
10
12
10
8
6
3
1
0
lxmx
0
0
0
0
0
0
0.55
1
2.25
4
4.2
3
2
1.2
0.3
0.05
0
R0 = 18.55
Generation Time, T
T – average generation time, is the average age a female gives
birth to one offspring
n
 xl m
x
T
x 0
x
R0
T=183.85/18.55 = 9.11
An average female Aphid
gives birth to one offspring
at 9.11 days
Age, days (x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Nx
1000
900
820
750
680
620
550
500
450
400
350
300
250
200
100
50
0
lx
1
0.9
0.82
0.75
0.68
0.62
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.1
0.05
0
mx
0
0
0
0
0
0
1
2
5
10
12
10
8
6
3
1
0
lxmx
0
0
0
0
0
0
0.55
1
2.25
4
4.2
3
2
1.2
0.3
0.05
0
xlxmx
0
0
0
0
0
0
3.3
7
18
36
42
33
24
15.6
4.2
0.75
0
R0=18.55 Σxlxmx=183.85
Net Reproductive Number or l?
Note that l does not equal R0
l is a rate per time step
l = N t+1/Nt
R0 is a rate per lifetime/generation.
R0   lx mx
Domestic Sheep (Ovis aries)
1.00
0.80
lx
0.60
0.40
0.20
0.00
1
2
3
4
5 6 7 8 9
Age Class (years)
10 11 12
1.00
mx
0.80
0.60
0.40
0.20
0.00
1
2
3
4
5
6
7
8
Age Class (years)
9 10 11 12
Age (x)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
lx
1.00
0.85
0.82
0.80
0.76
0.70
0.63
0.53
0.42
0.29
0.16
0.06
mx
0.00
0.05
0.39
0.47
0.48
0.55
0.54
0.50
0.47
0.46
0.43
0.42
Caughley, 1967
Age (x)
0
1
2
3
4
5
6
7
8
9
10
11
lx
1.00
0.85
0.82
0.80
0.76
0.70
0.63
0.53
0.42
0.29
0.16
0.06
mx
0.00
0.05
0.39
0.47
0.48
0.55
0.54
0.50
0.47
0.46
0.43
0.42
lxmx
0.00
0.04
0.32
0.38
0.37
0.38
0.34
0.27
0.20
0.13
0.07
0.03
xlxmx
0.00
0.04
0.64
1.13
1.46
1.91
2.04
1.87
1.56
1.19
0.70
0.28
R0=2.51
Σxlxmx=12.83
T= 5.11
Age (x)
0
1
2
3
4
5
6
7
8
9
10
11
lx
1.00
0.85
0.82
0.80
0.76
0.70
0.63
0.53
0.42
0.29
0.16
0.06
mx
0.00
0.05
0.39
0.47
0.48
0.55
0.54
0.50
0.47
0.46
0.43
0.42
lxmx
0.00
0.04
0.32
0.38
0.37
0.38
0.34
0.27
0.20
0.13
0.07
0.03
xlxmx
0.00
0.04
0.64
1.13
1.46
1.91
2.04
1.87
1.56
1.19
0.70
0.28
R0=2.51
Σxlxmx=12.83
T= 5.11
Age (x)
0
1
2
3
4
5
6
7
8
9
10
11
lx
1.00
0.85
0.82
0.80
0.76
0.70
0.63
0.53
0.42
0.29
0.16
0.06
mx
0.00
0.05
0.39
0.47
0.48
0.55
0.54
0.50
0.47
0.46
0.43
0.42
lxmx
0.00
0.04
0.32
0.38
0.37
0.38
0.34
0.27
0.20
0.13
0.07
0.03
xlxmx
0.00
0.04
0.64
1.13
1.46
1.91
2.04
1.87
1.56
1.19
0.70
0.28
R0=2.51
Σxlxmx=12.83
T= 5.11
Node-arc notation
3 ages or stages
Ages – small bird
reproduces in its second year
Xantus' Murrelet Synthliboramphus hypoleucus
0
1
2
Node-arc notation
3 ages or stages
Stages – plant which can hang out
in a vegetative state
or progress to a reproductive state
Prickly lettuce Lactuca serriola
0
1
2
Why is age-structure useful?
• Life-expectancy calculations
– Life insurance companies like it
– Planning for future funds (politics)
• Harvesting
– When are fish going to be big
enough to eat?
– What is the population turnover?
• Conservation/Control issues
– Which age is most susceptible to
mortality
– If females aren’t surviving to
reproduce, then no point in saving
the babies
Turtle Excluding Devices (TEDs)
• The loggerhead turtle (Caretta caretta)
– Very high egg loss due to beaches
being developed, eggs poached
– Also very high early juvenile loss
due to predation – as they disperse
into the ocean
• Turtle conservation in the 1980s
focused on protecting eggs and beaches
Turtle Excluding Devices (TEDs)
1987 – Crouse et al. - programs
focusing on preserving turtle eggs
may be least effective; late juvenile/
early adult survival is more
important
• Often caught in fish nets – huge source
of mortality
• Create TEDs to prevent turtles and
other large by-catch species drowning
• 1997 – Grand and Beissinger –
move the eggs and the picture
changes – we must protect eggs on
beaches AND use TEDs
TEDs used in Australia
The end