Combining survivorship and fecundity
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Life tables.
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Table of lx and mx
Allow p
projections
j
of
population dynamics
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0.5
Fertiliity (mx)
Proportion Surviving (lx)
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0.6
0.3
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0.2
25
35
45
5
15
25
35
45
Age Class
R0 = Net Reproductive Rate
Average number of
offspring left by each
female
Can calculate from
survival and fecundity
for each age class
R0 = Σ(lxmx)
T = Generation time
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Average time from being
an egg to laying an egg
(being a baby to having
a baby)
Calculate from the timing
of births and deaths
T = Σ(xlxmx) / Ro
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r (per capita rate of increase)
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Overall birth rate – overall death rate
r = ln(Ro) / T
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Critical Values
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X is the midpoint of the
age class
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Counts are of females
Births are of female offspring
0.0
15
Age Class
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In many cases, ignore males
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0.4
R0 (net reproductive rate)
T ((Generation time))
r (per capita rate of increase)
0.1
0.100
5
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Important summaries of a life table are:
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1.000
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Combining survivorship and fecundity
Annuals: annual, x=1, and
T=1
Perennials: x varies with
age class
Life Table – see spreadsheet
ln is the natural logarithm
r > 0: grow
r < 0: shrink
In this case (US 1998), r= 0.02%/year
1
S.A.D.
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λ (Geometric Rate of Increase)
IF lx and mx stay
constant, a population
will eventually reach a
STABLE AGE
DISTRIBUTION.
Each bar will be a
constant proportion of
the bar above it
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Ratio of the population
sizes at two different times
λ = Nt+1 / Nt
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λ >1: grow
λ <1 shrink
Most appropriate when
growth occurs in ‘pulses’
E.g., Yearly breeding season
(deer, bears, annuals)
The parable of the
grain of rice
Conclusions
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then λ = 2
Population will double
Critical Values:
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If Nt = 1, and Nt+1 = 2,
A survivorship curve summarizes the pattern of
survival in a population
Patterns of births in a population can vary from
semelparous
p
to interoparous
p
Age distribution reflects the history of survival,
reproduction, and the potential for future growth
of a population
A life table (lx and mx) can be used to estimate
net reproductive R0 (net reproductive rate), λ
(Geometric rate of increase), T (Generation
time), and r (per capita rate of increase)
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Prince rewards the inventor of chess
The inventor asks for “one grain of rice”
on the first square of the chessboard
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Was this a reasonable reward?
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2 on the second
4 on the third,
And so forth for all 64 squares
A) Yes
B) No
(How much rice will the prince give out?)
www.sjgames.com
Doubles
each
square
1
http://ncowie.files.wordpress.com/2007/10/grains-falling-on-a-pile-of-uncooked-white-rice_1.jpg
2
4
8
16
32
64
128
www.sjgames.com
Doubles
each
square
1
2
4
8
16
32
64
128
256 512 1024 2048 4096 8192 10 4 10 4
http://ncowie.files.wordpress.com/2007/10/grains-falling-on-a-pile-of-uncooked-white-rice_1.jpg
www.sjgames.com
2
Doubles
each
square
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1
2
4
8
16
32
64
128
256 512 1024 2048 4096 8192 10 4 10 4
One grain =25 mg
Halfway (square
32): 100,000 kg
10 5
10 6
10 7
10 10
1014
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Total for 64 squares is 4 billion
tons
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13
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1016
1015
1017
At 25 mg/grain
/ i = 2 billi
billion ttons
Global rice production in 2008:
~ 30 million tons
1018
http://ncowie.files.wordpress.com/2007/10/grains-falling-on-a-pile-of-uncooked-white-rice_1.jpg
www.sjgames.com
Living things multiply
Population Growth
With additive growth, the inventor would have
1+1+1… = 2,080 grains = 5.2 grams
With multiplicative growth, 4 billion tons
1E+18
Grains of Rice
8E+18
6E+18
4E+18
2E+18
„
http://www.youtube.com/watch?v=ecj768NinEo
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What elements of population growth does
this show well?
What elements of population growth in this
do not show up well?
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1E+21
1E+19
Grains of Rice
On the 64th square:
9,223,372,036,854,780,000
grains (263)
10 11
10
10 12
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10 8
10 9
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Doubling 64 times
1E+15
1E+12
1E+09
1E+06
1000
0
1
0
20
40
Square
60
0
20
40
60
Square
Arithmetic vs. Geometric growth
3
4