Transcript Demographic calculations continued

Demographic Calculations (cont.)
To quantify population dynamics, we clearly must take into
account both survivorship and natality. To start, the variable
we add to the life table is the product of lx and mx. In that way
we compute the expected reproduction traceable to an
individual of age x. mx gives us the number of offspring the
average female of age x has (if she is alive), and lx tells us the
probability that she will have survived to bear those young.
Summing what that proverbial average female does over her
lifespan, we determine a number called the Net Replacement
Rate, R0. It represents number of offspring left behind by an
average member of the initial cohort.
Since we're interested in what an average female achieves, the
lx has been re-computed from total numbers to fractions of the
original cohort alive at age x.
Here are those calculations for the sample life table we’ve
been using:
Age x
0
1
2
3
4
5
lx
1.000
.800
.600
.400
.200
0
R0 
mx
0
0
1
2
0
-
lm
x0
x
l0
lxmx
0
0
.6
.8
0
-
x
 14
.
Another value is sometimes reported; it is called the Gross
Replacement Rate (or GRR). It is the sum of mx values
across all ages in the table.
If an average female leaves more than 1 female offspring
behind over her lifespan, then R0 > 1; the population is
growing. If R0 < 1, it's declining.
There are 2 minor caveats to mention:
1) We have artificially divided the age structure into distinct
classes, when processes occur continuously. The proper
mathematical notation is in terms of integrals rather than
sums. If the age structure has been appropriately divided,
the effect is unimportant.
2) Reference to the 'average' female makes it apparent that
instantaneous measures of lx at the 'birthday' could be
inappropriate; we are assuming that lx is an adequate
representation of the 'pivotal' age within the interval. If
survivorship is a smooth function (no sudden, sharp
changes within an age class) we are relatively safe.
To this point, we have studiously avoided speaking of
generations. A typical insect life history, for example, is 1
year long, and generations do not overlap, i.e. offspring do
not begin reproducing during the lifespan of their parents.
In many populations of larger, longer-lived organisms there
are 'grandparents'; offspring begin reproduction while their
parents are still alive. In such cases R0 may represent the
contribution of an average individual, but will not
adequately represent the rate of population growth over
time.
To assess growth rate in those populations, we need to know
what the generation time is within the population to go
further. We can define the generation time in terms of
'averages' - T (alternatively  or G) is the time between the
birth of a female and the birth of her medianth offspring.
To be exact, this population should also be in a stable age
distribution, but even when it isn't exactly in a SAD this
approximation is usually within about 10% of the real
generation time.
We weight every offspring produced by the cohort by the age
at which the reproductive event occurs, and then calculate the
average age of reproducing parents.
Using our sample life table, you calculate xlxmx, sum these
‘age-weighted’ births, and divide by the total number of
births, R0. The formula:


 xl m
x 0
x
R0
x
Age x
0
1
2
3
4
5
lx
1.000
.800
.600
.400
.200
0
mx
0
0
1
2
0
-
lxmx
0
0
.6
.8
0
1.4
xlxmx
0
0
1.2
2.4
0
3.6
 = G = 3.6/1.4 = 2.57
Once we've calculated 'r', next on the list, there are methods to
correct our initial estimate, and the corrections are mostly
related to errors resulting from using summation
approximations rather than integrals (while also fixing any
error due to the approximation for generation time).
A useful aside: The calculation of growth rate, r, in a
population with overlapping generations is analgous to
calculating interest on your bank account with compoundingAmount = Principal(1+interest rate/periods per year)periods*years
or
Nt = N0(1 + r/p)pt
period, p, is the number of compounding intervals per unit of
time (e.g. monthly compounding equals 12 periods per year,
and banks normally tell you the annual interest rate), and t is
the number of time units (e.g. years for banks), so that pt is
the total time over which the account (or population) is
growing.
Daily compounding gets you more total interest (at the same
interest rate) because interest dollars start earning their own
interest earlier.
r is the interest rate when interest is compounded
instantaneously. Here is the derivation:
let r/p = 1/z (z is just a dummy variable)
then
Nt = N0 (1 + 1/z)rzt
= N0 [(1 + 1/z)z]rt
and in the limit as z approaches   (1 + 1/z)z = e
thus
Nt = N0 ert
This is our initial approximation of r. If the time t in this
equation is set equal to the generation time G, then the
population should have grown by a factor R0.
After one generation time, for each 'average' female we
started with, we should now have R0 females in the
population. Therefore:
R0 = erG
ln(R0) = rG
r = ln(R0)/G
For our example life table:
G = 2.57,
R0 = 1.4,
ln(R0) =.336,
and
r = .131
 = er = 1.14
Now remember that the generation time was only an
approximation. The mathematical method for correction (still
an approximation to using integrals, is called Euler’s
equation…
1 =  e-rxlxmx
For our sample life table, using the approximate ‘r’ = 0.131:
Age x
0
1
2
3
4
5
lx
1.000
.800
.600
.400
.200
0
mx
0
0
1
2
0
-
lxmx
0
0
.600
.800
0
e-rxlxmx
0
0
.462
.540
0
_____
1.002
If you increase the value for r, the e-rx multiplier gets smaller,
and will decrease the sum. So, try r = .132…
Now the sum is 0.999. So try a value between those – set
r = 0.1315…
Then the sum is 1.000. This is the corrected value for r.
One of the other things we'd like to know is the age structure
of our population when a stable age distribution has been
reached. Once the corrected r has been determined, a bit of
algebra using Fisher's (or Euler's) equation can give us a
formula for the proportions in each age class. That formula
is:
Cx = (e-rxlx)/[ e-rxlx]
for the sample life table…
Age x lx
e-rxlx
0 1.000
1
1
.800
.7014
2
.600
.4612
3
.400
.2696
4
.200
.1182
5
0
__0__
 = 2.5504
Cx
.392
.275
.181
.106
.046
These proportions can be turned into numbers corresponding
to any starting population you like, e.g. fractions of a cohort
of 1000.
Sharpe and Lotka (1911), and later Lotka (1921) were the
ones who proved that with a constant life table (i.e. lx and mx)
a population inevitably reaches a stable age distribution.
(An aside: when such a population reaches equilibrium total
size, i.e. K, the population is then said to be in a stationary
age distribution.)
To project populations forward in time, we could use r, but
there is a simple, ‘brute force’ method that is straightforward.
To use it, we need to calculate one more variable from the life
table, px, called proportional survivorship:
px = lx+1/lx = 1 - qx
the proportional survivorship for the sample life table:
Age x
0
1
2
3
4
5
lx
1.000
.800
.600
.400
.200
0
mx
px
0
0
1
2
0
-
.8
.75
.66
.50
0
-
Fx=pxmx Nt+1
0
0
.66
1
0
 ntFx
n0p0
n1p1
n2p2
n3p3
Explaining the population in the next time period:
The number of newborns is the product of the number of
organisms of each age times the fecundity. Fecundity is the
number of babies each age has discounted by the probability
that the female survives the age interval.
The number in each other age class is the number one time
unit younger one time unit ago times the proportion of them
that survive.
We can anticipate the matrix approach to calculations. In it we
first calculate a fertility for each age class, Fi, which is, for
age class i, the product of the mx and px. This product is the
number of offspring produced by a female which survived
throughout the period, discounted by the probability of
survival through the period.
Note that this refers to age class, rather than age.
The total number of newborns in the next period is the sum of
fertilities for all age classes:
n0,t+1 =  Fxnx,t
Now we are at a point where introduction of the Leslie matrix
can, if you are willing to work with matrices, simplify
calculations…
This matrix has the fertilities (Fx) for age classes as its first row
and the proportional survivorship (px) as subdiagonal elements.
If, alternatively, data was collected in the form of the stages of
organisms in the population, the probabilities of transition
between stages (or of remaining in the same stage into an
additional time period) are recorded in the matrix, which is
then properly a transition (or Lefkovich) matrix.
Before even beginning to set up and use the Leslie matrix, you
need to understand the basics of matrix algebra…
An Introduction to Matrix Algebra
Matrices are simply rectangular arrays of numbers or
variables. The size of a matrix is important. Size is indicated
by dimensions (m x n) which are the number of rows (m) and
the number of columns (n). Matrices can take the form of
column vectors (m x 1) or row vectors (1 x n), as well.
Matrices can be added, subtracted, multiplied by a constant (a
`scalar`), or by other matrices (as long as they have
appropriate numbers of rows and columns), they may have an
inverse, and they can be transposed. There are, additionally,
certain characteristics associated with matrices: determinants,
eigenvalues and eigenvectors.
Addition and subtraction of matrices is straightforward.
Corresponding elements (the value in a given row x column
position) are added or subtracted.
Addition and subtraction of matrices, like addition and
subtraction of numbers, is transitive, i.e. A + B = B + A.
|a11 a12| + |b11 b12| = |a11+b11 a12+b12|
|a21 a22|
|b21 b22|
|a21+b21 a22+b22 |
The matrix sum is the sum of the corresponding elements in
the individual matrices. Subtraction of matrices occurs the
same way.
To multiply a matrix by a scalar, each element in the matrix is
multiplied by that scalar value:
X x |a11 a12| = |xa11 xa12|
|a21 a22|
|xa21 xa22|
The transpose of a matrix A, designated A`, is constructed by
switching the rows and columns of it. The elements in the
matrices above have subscripts. By reversing the subscripts
for each element, and placing the modified elements in the
positions then appropriate, the transpose has been constructed.
|a11 a12|T = |a11 a21|
|a21 a22|
|a12 a22|
Multiplication of two matrices is slightly more complicated.
The product only exists if the number of columns in the first
matrix equals the number of rows in the second. The process is
also intransitive, i.e. A x B is not equal to B x A. Each element
in the matrix which results from multiplication is the sum of
the products of corresponding elements in the first row of
matrix A and the first column of matrix B, the second row of A
with the second column of B, etc…
In a standard notation:
ci,j =  k=1 to n [ai,kbk,j]
A sample multiplication:
|1 2| x |5 6| = |5+14 6+16 | = |19 22|
|3 4| |7 8|
|15+28 18+32|
|43 50|
The inverse of a matrix corresponds in a general way to the
inverse of a number or function in ordinary algebra, i.e. if
y = ax
then
a-1y = x
The parallel matrix equation, which would be equivalent to a
set of linear equations, would be:
y = Ax
and the 'solution' would involve the use of the inverse matrix,
i.e.:
A-1y = x
Actual calculation of the inverse of a matrix is best left to
computer programmes, and inverses only exist for square
matrices (those in which the numbers of rows and columns are
equal).
For a 2 x 2 matrix, the calculations are readily done by hand.
Here is the abstract ‘formula’ to calculate that inverse:
|a b| -1 = 1
|d -b|
|c d|
ad-bc |-c a|
The determinant of a matrix is another important
characteristic. It is straightforward to calculate for a 2 x 2
matrix, but best left to computer programmes for larger
matrices. For that 2 x 2:
det |a b| = ad - bc
|c d|
The determinant of a matrix has a geometrical meaning; it is
the volume of the n-dimensional parallelogram formed by the
vectors represented by each of the n rows of the matrix. (In a
simple expression for what that means, Oy vey!!)
The eigenvector(s) of a matrix are vectors which remain the
same when multiplied by the matrix, or are changed only by
scalar multiplication. That is, x is an eigenvector of the matrix
L, specifically a right eigenvector (note that it is written to the
right of the matrix) if:
Lx = x
In this equation  is an eigenvalue of the matrix. There are
also left eigenvectors for a matrix. They satisfy the equation:
y'L =  y'
It would be possible now to calculate eigenvalues and
eigenvectors of a matrix. They have meaning when they are
determined for a Leslie matrix:
 is the dominant eigenvalue of the matrix.
The left and right eigenvectors are the stable age structure
(Cx) and reproductive values (Vx) for the life table. We won’t
even try that for any matrix derived from a life table.
However, like other calculations, it can be done without too
much hassle for a 2 x 2 matrix…
Eigenvalues and eigenvectors are calculated by using the
characteristic equation. If
Lx = x,
then:
Lx -  x = 0
To reorganize this equation, we need to use the 'identity
matrix'. This special matrix has 1s along the diagonal, and 0s
elsewhere. It is a matrix version of the number 1, i.e. IL = LI
= L. Using the identity matrix:
(L -  I)x = 0
For this equation to hold and have a non-zero solution for x,
det(L - I) = 0
Making this equation hold involves solving an nth degree
equation in  for its roots, which are the eigenvalues of the
matrix.
For a simple 2x2 example, the eigenvalues of the matrix
|2 1|
|3 4|
det (L - I) = det |2-  1|
| 3 4- |
s are solutions of the parabolic equation:
2 - 6  + 5 = 0
( - 5)( - 1) = 0
and the eigenvalues are lambda1 = 1 and lambda2 = 5.
By substituting the eigenvalues into the characteristic
equation, the eigenvectors can be determined. Substitute  = 5
into the matrix multiplication below, and try x1 = 1. You will
then find that x2 = 3, so that one eigenvector is (1,3)'.
Substituting the other eigenvalue ( = 1), then if x1 = 1,
x2 = -1, and the eigenvector for the eigenvalue  = 1 is (1,-1)'.
Thus,
|2-  1 | |x1| = |0|
| 3 4-  | |x2|
|0|
Assuming you are now mostly confused, recognize that this is
not a course in linear algebra, and you do not have to
understand all this. It is time to move on to…
The Leslie Matrix