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Life Table Analysis for Black-striped wallabies.
575 Words Assignment
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More details attached.
ECOL203/403Assignment 1: Age Structure of a
Population Using Life Tables
Introduction to Life Tables
Before you begin this exercise (or read any further) you should:
1. Read Chapter 13 of Attiwill and Wilson (2006), particularly the section
on life tables on page 220 – 223.
2. Make sure you have the life_table.xls file from the Assignment 1
folder)
3. Do the Molar Index and Skull aging Tutorial (Assignment 1 folder)
4. Download the Box of Skulls (Assignment 1 folder)
5. It is also advisable to read through this exercise completely before
starting on the spreadsheet in excel.
Background to the Data
The Black-striped wallaby, Macropus dorsalis
The black-striped wallaby is a medium-sized
macropod (females 7kg; males 16kg) that occurs
from northern Queensland to northern NSW. The
species is listed as ‘Endangered’ in NSW, but can
become overabundant in some parts of Queensland
– so wildlife ecologists need to manage their
numbers in some regions so that they do not cause
over-grazing of livestock pastures, while in other
place, the population needs to be stimulated to
increase in numbers to prevent them from becoming
locally extinct. The wallabies shelter in dense scrub
thickets (e.g. Brigalow) by day and graze adjacent
pasture or natural grasslands by night.
Debra White did her UNE Master of Natural Resource Science on black-
striped wallabies at the Brigalow Research Station near Theodore in central
Queensland (White 2004). She found that there was a high density of
wallabies sheltering in the patches of brigalow by day, and that at night, these
animals moved onto pasture, which they grazed heavily. White (2004) also
looked at age structure of wallabies at the site by aging skulls she collected,
and using these in a life table analysis. We will do a similar exercise in this
assignment using skulls collected at the same site used by White (2004), and
we will compare our results from the results from Debra White’s much larger
dataset.
Molar Progression in Macropods
Molar progression occurs only in the marsupial genera Macropus, Petrogale
and Peradorcas (Jackson 2003). These marsupials are among only a
relatively few mammals worldwide whose teeth erupt at the posterior end of
the jaw, and migrate forward along the jaw during life (the others are the
elephants). As the teeth wear down and become less useful for grazing, they
have moved sufficiently anterior in the jaw that they can fall out, ‘pushed’ from
behind by newly erupted teeth. In this way, macropods can maintain good
functioning teeth with high cusps for grazing on tough fibrous grasses
throughout life. This ‘molar progression’ is a handy way to age kangaroo and
wallaby skulls, and was used to generate the dataset you will use in this
assignment to examine the life history parameters of a ‘population’ of black-
striped wallabies (Macropus dorsalis) from the Brigalow Research Station in
southern Queensland.
Aging of Macropods Using Molar Index (MI)
Molar Index (MI) is calculated by measuring the position of molariform teeth
on the upper tooth row relative to a reference line drawn across the skull in
line with the anterior limits of the orbits (see Figure 1 below). For
convenience, ten stages of molar progression are recognized per tooth, and
given decimal notation in tenths. In the example below (Fig 1a), molar M1
has progressed beyond the orbit, while molar M2 is given a score of about
0.7, according to Figure 1b. Therefore, the skull has a molar index (MI) of
1.7. Consulting Table 1 below of published estimates from red-necked
wallabies (Macropus rufogriseus), a similar-sized macropod to the black-
striped wallaby, we see that this animal was around age 2 (in years) when it
died.
Figure 1. Skull showing reference line for age determination, and one-tenth
division in length for molar teeth of kangaroos and wallabies. Figures from
Kirkpatrick (1964), and Jackson (2003).
In this assignment, you will generate age data from the skulls of black-striped
wallabies collected in the field, and use them to generate a life table for the
population. You will measure these skulls and calculated their molar index,
and then convert these into ages for each animal (using Table 1). You can
then use those data to generate a life table for the population. Incidentally –
the skulls come from animals that died ‘naturally’ – that is, they weren’t culled
or harvested. Some might have died of old age; others may have been
caught up on a barbwire fence, while a dingo perhaps ate others. In any
event, from the age structure of the population, we can determine a profile for
the population in terms of average life expectancy, mortality rates, probability
of surviving to the next age bracket and so forth. Such tools are useful for the
ecologists, because they provide rich demographic data that allows one to
determine which individuals should be culled in order to contain or reduce a
population that is overabundant, or conversely, whether certain cohorts (age
groups) are particularly susceptible to some kind of mortality agent (such as
predation) and need to be protected so that the population has the best
chance or recovering to more sustainable numbers.
Black-striped wallaby painting by John Gould
Data for the life table analysis
Download and open the excel spreadsheet exercise called ‘life_table.xls’ from
the Assignment 1 folder on the unit website. Spend a minute looking around
this spreadsheet. There are three ‘worksheets’ (clickable tabs at the bottom
left of the opened workbook) within the spreadsheet:
1. Molar Index
2. Life Table
3. Data from White (2004)
MI Age (years)
<0.8 0
0.9 – 1.6 1
1.7 – 2.1 2
2.2 – 2.5 3
2.6 – 2.7 4
2.8 – 2.9 5
3.0 – 3.1 6
3.2 – 3.3 7
3.4 8
3.5 9
3.6 10
3.7 11
3.8 12
3.9 13
4.0 14
4.1 15
4.2 16
4.3 17
4.4 18+
Table 1: Relationship between molar
index (MI) and age (in years) of red-
necked wallabies. Equation modified from
Kirkpatrick (1965).
The first, Molar Index’, is where you will enter the age data of the population.
The second, ‘Life Table’ is where you will calculate the life history parameters
of the population. The third worksheet is data from Debra White’s thesis.
White examined the demography of wallabies at the Brigalow Research
Station in the exact way that we are doing – but because she worked with 667
skulls, where you only have 49 – we might expect her results to be a robust
estimate against which you can compare your own.
Graphs appearing on the ‘Life Table’ worksheet template already show
White’s results for some key life table parameters (take a look at these now –
Whites results are in light grey on Graphs A and B). As you complete this
exercise, your results will be graphed in red, so you can compare your results
to White’s results.
STEP 1 – Click the tab to bring up the worksheet labelled ‘Molar Index’. Enter
the molar index data you have gathered in the columns labelled ‘MI’ against
each of the 49 skulls. While in this worksheet, use Table 1 (above) to
determine the age at death for each wallaby, and put that number alongside
the molar index value (i.e. Column C) labelled ‘Age’ (Given age as the nearest
whole number in years). [HINT: use the sort function to sort Cells 2 to 50 in
Column A and B (as a block, sorted by MI) before you begin assigning ages –
this will make it much easier to generate ages, because wallabies will be
arranged from youngest to oldest – don’t worry that the skull numbers will be
jumbled – we won’t be using these in the analysis].
STEP 2 – Once all the MI and age data are entered alongside the
corresponding skull ID, click on the worksheet ‘Life table’. You will see here
that there is a template for a life table that has the following columns:
Column A – Actual numbers of skulls at each age
Column B – Age in years (x)
Column C – Age interval (yrs)
Column D – number surviving (nx)
Column E – Proportion surviving (lx)
Column F – Deaths at each age interval (dx)
Column G – mortality rate (qx)
Column H – Number surviving at agex at last birthday (bx)
Column I – Expectation of further life (ex) in years
If you had a look at this spreadsheet before you entered the age data in the
‘Molar Index’, you will now see that the sheet has changed – the ‘Actual
Numbers of skulls at each age’ column (Column A) is now filled with the data
you entered, neatly compiled by age class – you can click on a cell to see the
underlying formula for doing this – but don’t change the formula!
Definitions for columns requiring calculation of life table parameters are as
follows:
nx – The number of animals from the original cohort, that are still alive at each
age interval
lx – Proportion of animals surviving from birth to age x. Because all
individuals born are alive, l0 is proportional to the total number of animals
sampled (i.e., is ‘1’), and successive values of lx get smaller, as fewer of those
l0 animals live to older and older ages.
dx – Deaths at each age interval (dx) is the number of individuals alive at age
x that will die before age x+1.
qx – the mortality rate (qx) is the per capita mortality rate during an age
interval.
bx – Number surviving at agex at last birthday – this is a prediction of the
average number of individuals alive at the midpoint of age interval x, based
upon number alive at one interval (x), and the next interval (x+1).
ex – the expectation of further life (in years) for an individual that makes it to
age x. This is calculated by summing all the values of bx from that age
interval to the bottom of the table, divided by lx
So, to recap, you should now have a spreadsheet that has the first three
columns complete – is this correct? If so, is now time to calculate the
parameters that we can use to describe the population. We will do these
calculations, and compile the life table, using some simple Microsoft Excel
equations.
STEP 3 – Number Surviving (nx). Recall that nx is the number of wallabies
surviving from birth to age x. So, we need to write a formula to go in Column
D that describes survivorship at each age class. Starting at Cell D2, write the
following formula:
=SUM(A5:A$23)
Then hit the return key. What does this formula do? Well, simply put, it sums
all of the individuals in Column A, to generate a value that describes the
number of individuals surviving to Age 0. Because all individuals survive to
this age (i.e., if a wallaby was born, it must have survived to Age 0), this value
should be equal to the total animals in our sample – i.e., 49.
OK – next step is to copy this cell down to fill the column as far as Cell D23.
How do we do this? Hold the mouse over the bottom right corner of the cell –
the cursor should change to a square with arrows at top left and bottom
right… click and hold the mouse button down while dragging the cursor to Cell
D22, then release the mouse.
What happened? Hopefully, you generated a series of numbers that describe
survivorship at each age class. When you highlight Cell D6, you should see
the following formula:
=SUM(A6:A$23)
Note that the formula now sums from Cell A6 to A23 (rather than from A5),
similarly, in Cell D7, the formula will sum from A7, and so on (the ‘$’ preceding
the ‘23’ keeps Cell A23 constant in each subsequent equation, as the formula
changes). When you give this some thought, this makes sense – because we
only want survivorship from any particular age class, to the last (oldest) age
class. If you have completed this step correctly, you should see fewer and
fewer surviving wallabies with increasing age, until at age 18, only 1 wallaby
survives (and for our purposes, we will assume that this animal will die before
reaching the next age class).
STEP 4 – Proportion Surviving
If this is to be a representation of the whole population, we need to start
converting out numbers to something more universal – the previous column
simply told us how many of the 49 animals survived, but by converting this to
a proportion, we have a number at each interval that we could apply to a
population of any size, to predict how it might behave.
So, the next step involves a simple calculation to determine the proportion of
wallabies in the original cohort that survive to each age class. Since Column
D represents the number surviving to the next age class, Column E is simply
the number surviving, divided by our entire sample (N = 49). So, in Cell E2,
type the following formula:
=D5/49
– and hit return. Then, as before, click the mouse in the bottom right hand
corner of Cell E5, hold the mouse button down, drag the cursor down to Cell
E23, and then release the mouse. You should now have filled Column E,
which is a calculation of the proportion of animals born into the population that
survive to each age class. You will also see that “Figure A, the Proportion of
M. dorsalis Surviving (lx)” also was created, so you can, for the first time, see
graphically how you data compares to that of White’s (2004) data.
STEP 5 – Deaths at each age interval (dx), and mortality rate (qx)
Survivorship and mortality are clearly inter-related; mortality at any one age
interval is simply the difference between the numbers surviving from that age
interval to the next. So, the calculation of dx is simply nx – nx+1, so for our
spreadsheet, type the following in Cell F5:
=D5-D6
Copy this formulae down to Cell F23, as previously described at STEP 4.
STEP 6 – Mortality Rate
Now we are in a position to calculate an important life history parameter for
black-striped wallabies that is important in understanding any population – the
age-specific mortality rate, or put another way, the rate at which animals in
any particular age class would be expected to die before reaching the next
age class. Incidentally, this is one of the calculations insurance companies
use when determining what to charge you on a life insurance policy (see
Table 13.6 on page 221 of Attiwill and Wilson).
Look at Column D of the life table in the spreadsheet. For age interval 0 – 1
we should have 49 animals, but at other age intervals (at least, after about
age 3) we see less that this. What happened to those other animals? For
whatever reason, they didn’t make it to that particular age class – they died
before the next census period. So, the mortality rate at each age class is
clearly equal to dx/nx, and in the spreadsheet, we can calculate this by typing
the following equation in Cell G5:
=F5/D5
and then copying this down all the way to Cell G23 in the usual manner. You
now have an estimate of the rate at which animals at different age classes in
the population are dying – or put another way, “what is the likelihood of death
for a wallaby at age x before the next census date”. And, as for proportion
surviving in STEP 4, you now also have a graphical representation of this in
“Figure B: Mortality rate (qx) for M. dorsalis”, and can compare your data for
mortality directly with White’s (2004) data.
STEP 7 – Number surviving at agex at last birthday (bx) and expectation
of further life (ex).
Simply put, bx is the average number of individuals alive at the midpoint of
age interval x. So, to calculate this for each age class, you need to calculate
the average of one age class (x) and the next age class (x+1). So, at Cell H5,
enter the formula that describes this parameter:
=(D5+D6)/2
then, copy this down to H23. You now have a calculation of the ‘age
structure’ of the population. You may be asking at this point, so what? Well,
this is an intermediate calculation that will allow us to calculate a very useful
parameter for the population – expectation of further life. For any age class,
our calculations so far allow us to answer questions like: “What is the mortality
or survival rate of animals in that age class?” But importantly, as an individual
animal survives each age class, its expectation of further life should increase,
because it managed to survive (through finding food, evading predation and
disease etc) where others died. Life tables allow us to ask the question “For
animals at a given age, how much longer should we expect those animals to
live?” To calculate this, we sum the age structure of the remainder of the
population older than any particular age class, and divide this value by the
number of animals surviving at that age class. So, type the following formula
into Cell I5:
=SUM(H5:H$23)/D5
And copy this down to Cell I23, as previously described.
Have a look at the values you generated in this column. Starting at Age 0, we
can see that all animals, at birth, have an expectation of further life of about
7.4 years (or something similar). But, if we look at Age 8, we see that animals
that survived to 8 years of age have an expectation of further life of about 5
years. At age 11, these animals can be expected to live another 3.5 years,
and so on. Once again, insurance companies use calculations of this nature
on human census data to determine life insurance premiums.
STEP 8 – Generating a Survivorship Curve
The final step involves generating a survivorship curve for black-striped
wallabies that we can compare to theoretical survivorship curves commonly
used to summarise a population’s demography. These curves (e.g. Figure
13.12 on page 222 in Attiwill and Wilson) have a logarithmic scale on the y-
axis, so to compare to our survivorship for wallabies to these theoretical
curves, we must convert our data for nx to log values. To do this, type:
=LOG(D5)
in Cell D26, and copy this down to Cell D44. Presto! – you have a
survivorship curve for the population.
References
Attiwill, P. and Wilson, B. (2006). Ecology: An Australian Perspective. 2nd Edition. Oxford
University Press, Melbourne.
Jackson, S. (2003). Australian Mammals: Biology and Captive Management. CSIRO
Publishing, Melbourne.
Kirkpatrick, T.H. (1964). Molar progression and macropod age. Qld J. Ag. Anim. Sci. 21:
163–165.
Kirkpatrick, T.H. (1965). Studies of Macropodidae in Queensland. 2. Age estimation in the
grey kangaroo, the eastern wallaroo, and the red-necked wallaby, with notes on dental
abnormalities. Qld J. Ag. Anim. Sci. 22: 301–317.
White, D. (2004). Utilisation of remnant Brigalow communities and adjacent pasture by the
black-striped wallaby (Macropus dorsalis). Master of Resource Science, University of
New England, Armidale NSW.
Assessment Questions
As part of your assessment (the other part being the completed spreadsheet),
you are required to write a paragraph on each of the following questions (use
Microsoft Word for this), and submit your responses as part of your
assignment. The total length of this section should be no more than 600
words.
1. How do the graphs you made compare to White’s data? Was your
sample a reasonable representation of the black-striped wallaby
population?
2. Given your results, how would you respond if someone asked the
question: “To what age does a black-striped wallaby live at the
Brigalow Research Station?”
3. In reference to hypothetical survivorship curves (see pages 221-222 in
Attiwill and Wilson), what type of survivorship do black-striped
wallabies most likely exhibit? Does this fit the typical curve for
mammals?
4. Does the mortality rate fit the prediction for mammals? In very brief
terms, explain the pattern of mortality in the black-striped wallaby
population.
TO COMPLETE THIS ASSIGNMENT, DON’T FORGET
TO UPLOAD BOTH:
1. THE COMPLETED SPREADSHEET (results page
only, as a pdf file)
*AND*
2. THE ANSWERS TO ASSESSMENT QUESTIONS
(as a pdf file)
Weighting: Assignment is worth 20% of your final marks for the unit
Due Date: Monday 1 April, 2013 (by midnight)
Penalties for late submission: 5% per day
Instructions: Submit two files for the assignment (1) a spreadsheet file (the results sheet of your excel spreadsheet saved as a pdf) and (2) a written response to the questions listed below (also as a pdf). You should write a paragraph on each of the following questions. The total length of this section should be no more than 600 words.
In order to undertake this assignment, you need to follow the instructions given in the section ‘
Assignment 1 Resources
‘
Answer each of the following questions for this assignment:
1. How do the graphs you made compare to White’s data? Was your sample a reasonable representation of the black-striped wallaby population?
2. Given your results, how would you respond if someone asked the question: “To what age does a black-striped wallaby live at the Brigalow Research Station?”
3. In reference to hypothetical survivorship curves (see pages 221-222 in Attiwill and Wilson), what type of survivorship do black-striped wallabies most likely exhibit? Does this fit the typical curve for mammals?
4. Does the mortality rate fit the prediction for mammals? In very brief terms, explain the pattern of mortality in the black-striped wallaby population.
Black-striped wallaby Skulls
Please open the link bellow to measure the skulls size
http://moodle.une.edu.au/pluginfile.php/523661/mod_resource/content/5/boxofskulls/index.htm
Pre-Assignment 1 Practical: Aging of Kangaroos and Wallabies Using Molar Progression
Aging of Macropods Using Molar Progression
Molar progression occcurs only in the marsupial genera Macropus, Petrogale and Peradorcas (Jackson 2003). These marsupials are among only a relatively few worldwide whose teeth errupt ar the posterior end of the jaw, and migrate foward along the jaw during life (the others are the elephants). As the teeth wear down and become less useful for grazing, they have moved sufficiently anterior in the jaw that they can fall out, ‘pushed’ from behind by newly errupted teeth. In this way, macropods can maintain good functioning teeth with high cusps for grazing on tough fibrous grasses throughout life. This ‘molar progression’ is a handy way to age kangaroo and wallaby skulls, and will be demonstrated in this practical using a range of different-aged skulls from the eastern grey kangaroo, Macropus giganteus.
The position of teeth are estimated relative to a reference line drawn across the skull in line with the anterior limits of the orbits. Ten stages of molar progression are recognized and given decimal notation in tenths (see figure below, from Jackson 2003). In the example below (fig 1a), molar M1 has progressed beyond the orbit, while molar M2 is given a score of about 0.7, according to figure (b). Therefore, the skull has a molar index (MI) of 1.7. Consulting Table
4.1
below of published estimates from macropods, we see that this animal was somewhere between age 2 and 3 (in years) when it died.
Figure 1. Skull showing reference line for age determination, and one-tenth division in length for molar teeth of kangaroos and wallabies. Figure from Jackson (2003). Click image for a larger view.
Use the following rules when calculating molar index (MI):
1. If there is a premolar tooth, then the next one along in the tooth row is the M1. If there are not 4 molars, some of the teeth (i.e., M4, M3) have not yet erupted.
1. Some anterior teeth can be missing. If there is no sign of a premolar, then treat the last tooth that you can see in the skull as the M4, and count backwards from that to the M1.
As an example, I have anotated the images (Fig 2) below to indicate the two conditions that you might come across.
In the left-hand panel, the premolar remains in place and the skull has three erupted molars (M1-M3), plus (presumably) an additonal one that cannot be seen (M4), becuase it has not yet erupted. We therefore begin calculating MI at the boundary between the premolar (P2) and the first molar (M1) from 0.0, to yield a MI of 2.3. This corresponds to an age of about 3 years for this animal according to Table 1 (below).
In the second, much older animal (right-hand panel) the premolars have been shed along with the first molar (M1), and the remaining molars (M2-M4) have fully erupted. Therefore, we begin calculating the MI at 1.0, and count to the end of M4, to give a MI of
4.0
. This corresponds to an age of about 12 years for this animal according to Table 1.
Table 1. Molar indices (MI) of progression for age (in years) in Macropus giganteus. Table adapted from Jackson (2003) – for more species,
click here
.
|
Age |
MI |
|
|
1 |
0.4 |
|
|
2 |
1.4 |
|
|
3 |
2.0-2.3 |
|
|
4 |
2.4 |
|
|
5 |
2.7 |
|
|
6 |
3.0 |
|
|
7 |
3.2 |
|
|
8 |
3.4 |
|
|
9 |
3.6 |
|
|
10 |
3.8 |
|
|
11 |
3.9 |
|
|
12 |
4.0 | |
|
13 |
4.1 | |
|
14 |
4.2 |
|
|
15 |
4.3 |
|
|
16 |
4.5 |
|
|
17 |
4.6 |
|
|
18 |
||
|
19 |
4.7 |
|
|
20 |
4.8 |
Now its your turn. Examine the following photographs of eastern grey kangaroo (Macropus giganteus) skulls. For each skull, determine the age of the individual using the rules we have established for molar progression, and Table 1 above. For each skull, we have drawn in the reference line that you will need to use for aging acording to the molar progression method outlined above. The label given to each of the skulls corresponds with the museum specimen label in our collection in Ecosystem Management.
|
Skull 1 [NR0782] |
Skull 2 [NR1321] |
Skull 3 [NR1607] |
Skull 4 [NR2096] |
Skull 5 [NR2159] |
It is probably best that you make notes of molar Index (MI) and age of each skull (using the NR reference number) on a scrap of paper. Once you have done this, exit and click on ‘Take the skull quiz!’ to see whether you have got the hang of skull aging. Remember, this quiz is NOT ASSESSED – it is purely for your benefit.
>Molar Index
(bx)
Life Table 3213213
2102102
13 381
91 15916
5105105
29
4294294
84
3843844
942942943
2252252
861861862
666667
96096
0660661
027027
024024
009009
Proportion surviving (lx) Age (years) Age 0558376
235294
3333333
1111111
1 Mortality rate (qx) Age (years) 00
38 0.1 81 0.1 588 0.821 74 0.1 510 0.710 48 0.1 31 0.1 54 0.1 54 0.2 0.429 30 0.1 0.384 60 0.2 46 0.2 173 0.225 26 0.2 137 13 0.1 38 0.3 92 29 0.4 59 3 1.0 2 0.5 Can you please just use this as resource? http://www.oup.com.au/titles/higher_ed/science/biological_sciences/9780195550429 No more than this one Thanks
2
Skull ID
MI
Age
bsw
0
1
bsw02
bsw0
3
bsw0
4
bsw0
5
bsw0
6
bsw0
7
bsw0
8
bsw0
9
bsw
10
bsw
11
bsw
12
bsw
13
bsw
14
bsw
15
bsw
16
bsw
17
bsw
18
bsw19
bsw21
bsw22
bsw23
bsw24
bsw
26
bsw27
bsw28
bsw
29
bsw
30
bsw
31
bsw32
bsw33
bsw34
bsw35
bsw36
bsw37
bsw39
bsw40
bsw41
bsw42
bsw43
bsw
44
bsw45
bsw
46
bsw47
bsw
48
bsw49
bsw50
bsw51
bsw52
Life Table
Life Table for black-striped wallabies (Macropus dorsalis) based on skulls collected at the Brigalow Research Station, Queensland
Actual Numbers of skulls at each age
Age in Years (x)
Age interval (yrs)
Number surviving (nx)
Proportion surviving (lx)
Deaths at each age interval (dx)
Mortality rate (qx)
Number surviving at agex at last birthday
Expectation of Further Life (ex) in yrs
0 0
0 – 1
0 1
1 – 2
0 2
2 – 3
0 3
3 – 4
0 4
4 – 5
0 5
5 – 6
0 6
6 – 7
0 7
7 – 8
0 8
8 – 9
0 9
9 – 10
0 10
10 – 11
0 11
11 – 12
0 12
12 – 13
0 13
13 – 14
0 14
14 – 15
0 15
15 – 16
0 16
16 – 17
0 17
17 – 18
0 18
18 +
Log nx
1
0.9429429429
0.821
0.710
0.6
38
81
0.5
59
0.
510
0.4
0.3
0.2
0.225
0.1
0.1
666
0.109
60
0.066
0.027
0.024
0.009
0.0045045045
Proportion surviving (lx) (White (2004)
Proportion surviving (lx)
Figure A: Proportion of M. dorsalisSurviving (lx)Data from White (2004)
Logarithm of survival
Figure C: Survivorship Curve for M. dorsalis
0.0570570571
0.1289808917
0.1352833638
0.10147991
54
0.0729411765
0.
137
0.1
588
0.1048951049
0.234375
0.2346938776
0.
1
73
0.1048387097
0.3423423423
0.3972602
74
0.5909090909
0.
111
0.625
0.5
Mortality rate (qx) from White (2004)
Mortality rate (qx)
Figure B: Mortality rate (qx) for M. dorsalis
Actual Numbers of skulls at each age Age in Years (x) Age interval (yrs) Number surviving (nx) Proportion surviving (lx) (White (2004) Deaths at each age interval (dx) Mortality rate (qx) from White (2004) Number surviving at agex at last birthday Expectation of Further Life (ex) in yrs
38 0 0 – 1 666
1.0
647
6.6
81 1 1 – 2
628
0.943
6.0
74 2 2 – 3
547
5.8
48 3 3 – 4
473
449
5.7
31 4 4 – 5
425
0.638
410
5.2
54 5 5 – 6
394
0.5
92
367
4.6
54 6 6 – 7
340
0.511
313
4.3
30 7 7 – 8
286
271
4.0
60 8 8 – 9
256
226
3.4
46 9 9 – 10
196
0.294
3.3
26 10 10 – 11
150
3.1
13 11 11 – 12
124
0.186
118
2.7
38 12 12 – 13 111
0.167
1.9
29 13 13 – 14 73
0.110
1.7
26 14 14 – 15 44 0.066 26 0.6 31
1.5
2 15 15 – 16 18 0.027 2 0.1 17 1.9
10 16 16 – 17 16 0.024 10 0.6 11
1.1
3 17 17 – 18 6 0.009 3 0.5 5 1.0
3 18 18 + 3
0.005
