Lecture 7: Life Tables

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Lecture 7 Life Tables

• EEES 3050

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Exam

• 1st Remember there is a test in a week
• Will consist of approx. 50 multiple choice,
  short answer, and 50 written questions.
• 2nd you can help write the exam
• Send me a question via e-mial.
• Can be multiple choice, T/F, short answer, essay,
  etc.

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Exam

• 3rd
• Sigma Xi Research Symposium
• When Saturday from 9-330. Ecology talks are
  primarily in the morning.
• Each student presentation is 15 minutes long.
• You can use 3 student talks for your research
  critique.
• Possible extra credit on exam for attending as
  well.
• Guaranteed brownie points.

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Demographics and vital statistics

• Populations can be quantified
• Determine mortality rates.
• Rates of reproduction
• Determine if population should increase or
  decrease.
• Comparisons between different time periods
• Comparisons between different populations
• Yes this mathematics and statistics

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Life Tables

• Summary of the mortality/survival of cohort (age
  group) of individuals.
• Life Tables contain information on
• x age
• Nx number alive at age x
• lx proportion surviving from initial stage to
  age x
• dx Number dying from age x to x 1
• qx per capita rate of mortality from age x to x
  1
• bx of offspring per female at age x.

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Life Tables background

• Originally developed by human demographers.
• Used by ecologists starting in 1921
• Raymond Pearl Three types of survivorship
  curves.

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Life tables

• How do gather data?
• 1) Cohort life table
• Following a cohort through time.
• 2) Static life table
• Based on a cross section of a population.
• Why would these tables differ?
• Differing birth/death rates.

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Basic Life Table
Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx)
0 115 1.0 90 0.78
1 25 0.217 6 0.24
2 19 0.165 7 0.37
3 12 0.104 10 0.83
4 2 0.017 1 0.50
5 1 0.009 1 1.0
6 0 0 ---- -----
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Basic Life Table
Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx)
0 115 1.0 90 0.78
1 25 0.217 6 0.24
2 19 0.165 7 0.37
3 12 0.104 10 0.83
4 2 0.017 1 0.50
5 1 0.009 1 1.0
6 0 0 ---- -----
lx nx/n0 or the number in age x divided the
initial number.
l4 n4/n0 2/115 0.017
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Basic Life Table
Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx)
0 115 1.0 90 0.78
1 25 0.217 6 0.24
2 19 0.165 7 0.37
3 12 0.104 10 0.83
4 2 0.017 1 0.50
5 1 0.009 1 1.0
6 0 0 ---- -----
dx nx nx 1 or the number in age x minus
number in x1.
d4 n4 n5 2 - 1 1
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Basic Life Table
Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx)
0 115 1.0 90 0.78
1 25 0.217 6 0.24
2 19 0.165 7 0.37
3 12 0.104 10 0.83
4 2 0.017 1 0.50
5 1 0.009 1 1.0
6 0 0 ---- -----
qx dx/nx or the dying at age x divided the
number alive at age x.
q4 d4/n4 1/2 0.50
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Survivorship curves.
1000
No. Alive
Time
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Survivorship curves.
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3 Basic survivorship curves.
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3 Basic survivorship curves.
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3 Basic survivorship curves.

• Type 1
• High young survival rates.
• Example Humans
• Type 2
• Constant survival rate
• Example birds
• Type 3
• High mortality of young
• Examples many fishes, invertebrates

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Not everything follows these patterns.
Mediterranean fruit flies
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Ecologists and life tables

• How do ecologists construct life tables.
• Survivorship directly observed.
• Possible for short-lived organisms
• Age at death observed.
• Not easy or common
• Need to identify remains of organisms that died
  naturally.
• Example African buffalo
• Age structure directly observed.
• Get a sample of fish, or core trees,
• Relies on many assumptions.

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Data gathered by collecting skulls of animals
that died naturally.
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So far looked at mortality now add reproduction

• Termedintrinsic capacity for increase.
• Or Malthusian parameter.
• Background on rates
• A numerical proportion between two sets of
  things.
• Example 27 of 350 fail a test 7.7 failure
  rate.
• Example 8 or 12 seedlings die 66.7 mortality
  rate.

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R0 net reproductive rate.
Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx) No. of offspring per female (bx) lx bx
0 115 1.0 90 0.78 0 0
1 25 0.217 6 0.24 4 0.868
2 19 0.165 7 0.37 1 0.165
3 12 0.104 10 0.83 0.5 0.052
4 2 0.017 1 0.50 0 0
5 1 0.009 1 1.0 0 0
6 0 0 ---- ----- ---- ----
R0 Sum of lxbx 0.868 0.165 0.052 1.085
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Population growth with R0 1.085
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Class Life Table
Age in years (x) of students alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx) No. of offspring per female (bx) lx bx
1) lt21
2) 21-25
3) 26-30
4) gt 30
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Population growth with R0 3
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Lotka population change

• When population reaches a stable age
  distribution, can use a differential equation.
• dN/dt rN
• Or Nt N0ert , where e 2.71828
• Important parameter is r

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To calculate r

• First need to know generation length defined as
  the mean period elapsing between the production
  of parents and the production of offspring.

Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx) No. of offspring per female (bx) lx bx
0 115 1.0 90 0.78 0 0
1 25 0.217 6 0.24 4 0.868
2 19 0.165 7 0.37 1 0.165
3 12 0.104 10 0.83 0.5 0.052
4 2 0.017 1 0.50 0 0
5 1 0.009 1 1.0 0 0
6 0 0 ---- ----- ---- ----
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To calculate r

• First need to know generation length defined as
  the mean period elapsing between the production
  of parents and the production of offspring.

Age in years (x) of birds alive (Nx) Proportion surviving (lx) dying within interval (dx) Rate of mortality (qx) No. of offspring per female (bx) lx bx
0 115 1.0 90 0.78 0 0
1 25 0.217 6 0.24 4 0.868
2 19 0.165 7 0.37 1 0.165
3 12 0.104 10 0.83 0.5 0.052
4 2 0.017 1 0.50 0 0
5 1 0.009 1 1.0 0 0
6 0 0 ---- ----- ---- ----
1.248
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To calculate r
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• N0ert , where e 2.71828
• r gt 0 ?
• r 0 ?
• r lt 0 ?

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Finite rate of increase

• Finite rate einstantaneous rate
• ? er
• For our example r 0.0653
• ? e0.0653
• ? 2.718280.0653 1.067
• This means that for each individual in time x,
  there will be 1.067 individuals in time x 1

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These rates can be affected by abiotic factors.
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(No Transcript)
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Reproductive Value

• Contribution a female will make to the future
  population
• Or present progeny expected future progeny

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Reproductive Value
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Age distributions

• Constant age structure in a population is
  attained only if the lx and bx distributions are
  unchanging.
• Stationary age distribution
• Occurs in a population with a constant size and
  where fertility equals mortality.
• Stable age distribution
• Constant age-specific mortality and fertility
  rates.

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Age distributions
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Age distributions
Note Dominant year classes.
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Evolution of demographic traits.

• Examples
• Big-bang reproduction
• Salmon spawn once and die
• Repeated reproduction
• Oak trees produce thousands of acorns for 100s
  of years.

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Evolution of demographic traits.
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Summary

• Life Tables are a summary of the
  mortality/survival of cohort (age group) of
  individuals.
• 3 Basic survivorship curves.
• Constructing life tables can be difficult
• R0 and r
• Age distributions can predict future of
  population.