Download Leslie and Lefkovitch matrix methods

There is one last variable to calculate. Vx , or the reproductive
value, measures the relative importance of different age
classes to the future reproduction of a population.
If many die before reaching the age at which reproduction
begins, those organisms are less important to future growth
than are those that have begun to reproduce.
Assume you are a manager trying to determine how much
harvesting to allow, while optimizing the catch available to be
harvested. MacArthur worked out the math, and suggested the
following: The permissible harvest from an age classis
inversely related to the reproductive value of that class.
A definition: Vx is defined as the age specific expectation of
future offspring (for an equilibrium population) or age
specific contribution to future population growth (if
population size is changing).
For equilibrium populations (and thus in a stable age
distribution) the calculation of Vx is simple:
Vx =  limi/lx
For populations whose size is changing, the formula and
calcuations are more complex:
rx
Vx e

V0
lx

 ri
e
 li mi
i x
V0 is 1 in our formulation of a life table and reproductive
value, and can be dropped from the equation (but the
approach permits comparing other ages).
Reproductive value normally peaks at the age of first
reproduction, and will always rise from birth to . It could
continue to rise if survivorship during initial bouts of
reproduction was high and litter size increased during those
years as experience increased, but that rarely, if ever, happens.
The reproductive values for our sample life table are:
Age x lx
mx
app.Vx Vx
0
1
2
3
4
5
1.000
.800
.600
.400
.200
0
0
0
1
2
0
-
1.4
1.75
2.33
2.00
0
-
1.0
1.426
2.168
1.999
0
-
The reproductive values tell us why young age classes can be
harvested more heavily (at least according to theory) – their
reproductive values are low, and thus their relative
contribution to population growth. The values also indicate
why harvesting prime, reproductive animals must be so
limited - they are the ones most critical to future population
growth.
The Leslie Matrix
The Leslie matrix is a simpler way to organize the life table
calculations we've done. The arithmetic is really the same,
and what we're really doing is setting the numbers up in a
form most easily computerized. The matrix (sometimes also
called the 'population projection matrix') is simply a catalog of
age-dependent fecundities and proportional survivorships.
The first row of the matrix is simply the fecundities F0 --> Fn.
In each succeeding row (forming a sub-diagonal) the x+1th
column is filled with px (i.e. column 1 for newborns, age 0 and
age class 1, has p0 in column 1 of row 2).
Here’s what the matrix looks like in abstract terms…
F0 F1 F2
p0 0 0
0 p1 0
0 0 p2
0 0 0
0 0 0
F3
0
0
0
p3
0
F4
0
0
0
0
p4
And for the sample life table:
0
0 .66 1
.8 0 0 0
0 .75 0 0
0 0 .66 0
0
0
0
0
0
0
0
0
0
0
0 .50
0 0
F5
0
0
0
0
0
Now if we post-multiply this matrix by a matrix (really a
column vector) which is the age-class structure at time t, we'll
get a new column vector which is the age structure at time t+1.
_ _
_
L x Nt = Nt+1
_
_ _
_
_ _
_ _
Nt+2 = L x Nt+1 = L x (L x Nt) = L2Nt
and
Using the sample data, let’s project a simple age structure 1
time unit forward:
L
0 .66
0
1
.8 0 0 0
0 .75 0 0
0
0 .66 0
0
0 0 .50
x
0
0
0
0
0
Nt
100
=
Nt+1
166
100
100
100
80
75
66
100
50
Everything we’ve done thus far has been in a world of
exponential growth. How can we make the calculations and/or
matrix projection reflect the density dependence we know
exists in the real world?
Obviously, we need to find a way to have the mx and px
respond to density, rather than remaining fixed.
To add density dependence to the dynamics we construct
another matrix to represent density effects. While it is
possible to separate density effects on fecundity and
survivorship, that's unnecessarily complicated for our
purposes.
The matrix we're constructing Qt, recognizing, in principle at
least, that there may be time dependence, as well as pure
density dependence.
While the t will remain in the model structure, calculations
typically end up assuming a time-invariant densitydependence.
We also recognize that density effects will predominantly be
of 2 kinds: effects resulting from crowding (density) at birth
(those effects are typically manifest in reduced size,
survivorship and fecundity throughout life) and effects due to
density at the present time (which may compound birthdensity effects).
W e assume that effects are directly proportional to
appropriate densities, and that the two sources of effect are
additive.
To reduce fecundity and survivorship according to density we
define a 'density-dependence‘ term q(x,t):
q(x,t) = 1 + bNt-x-1 + aNt
b measures the effects of crowding at birth for each age class
x, and a measures the effects of current density.
In this simplification of reality, a and b are not only time
invariant, but also invariant with age.
The q varies with time and with density. We generally
consider the time variation to result solely from density
change.
The result is a matrix:
Qt = |q(0,t) 0 0 ...|
| 0 q(1,t) 0
...|
| 0 0 q(2,t) ...|
|...
|
In practice we use the inverse of this matrix, so that each term
in the multiplier is not q(x,t), but 1/q(x,t).
Values of q(x,t) > 1then indicate density-dependent reductions
in fecundity and survivorship. What we produce is a modified
Leslie matrix L' for use in population projection.
L' = L x Q-1 =
|F0/q(0,t) F1/q(1,t) F2/q(2,t)
|p0/q(0,t)
0
0
| 0
p1/q(1,t)
0
| 0
0
p2/q(2,t)
| ...
...|
...|
...|
...|
|
Population projection using L’ produces density-dependent,
logistic population growth, first leading to a stable age
distribution, then leading to the population reaching its
carrying capacity, K.
K is related to population growth and density dependence in
the following way:
K = (- 1)/(a + b)
The Lefkovitch or Stage Projection Matrix
The Leslie matrix is the correct approach for animals, but not
for many plants. For them size or stage is more important than
age.
Think back to the diagrammatic model for teasel – what
determined whether the plants flowered or not was the size of
the rosette of basal leaves. And look back at the diagram …
some plants did not advance to the next size class during one
measurement interval (a year); they remained in the same
class. That would not fit into a Leslie matrix model.
Instead, plants (and things like corals) are better described
using a transition matrix approach.
Nt+1 = ANt
A is a Lefkovitch or
transition matrix
It contains all stagespecific rates of
transition and seed
production
There are i columns and j
(= i) rows for a
population with i stages
A This year (t)
t+1
stage
1
stage
2
stage
i
stage 1
a11
a12
a1i
stage 2
a21
a22
a2i
stage j
aj1
aj2
aj3
If i = j the value indicates the
fraction of organisms that
remain in the same stage.
The elements of the Lefkovitch matrix are not proportional
survivorships, but the probabilities of making transitions from
one stage (or size) class to others.
We could use the teasel data to produce a Lefkovitch matrix,
but we’ll use a much simpler case of Coryphanthum
robbinsorum, an endangered cactus from the Sonoran desert.
This cactus has no seed bank, so that annual census would not
detect seeds, and we can use a fecundity measure that
incorporates not just seeds produced, but probability of seeds
germinating and surviving to the time of the census. The
result is this simplified life cycle graph, which is the
equivalent of the diagrammatic life tables you saw earlier…
Since this model doesn’t include a seed stage, consider how
we determine the number of small juveniles at the ‘next’
census:
n1(t+1) = p11n1 + n3(t)F
This formulation means that, unlike the Leslie matrix, the first
row of this Lefkovitch matrix includes both the probability of
remaining in the small juvenile class in row1,column1 and the
fecundity of adults in row1,column3. Here’s the basic matrix
projection equation:
n1(t+1)
p11 0 F
n1(t)
n2(t+1) = p21 p22 0 x n2(t)
n3(t+1)
0 p32 p33
n3(t)
Here are the values for the Lefkovitch matrix for one of the
three sites collected by Schmalzel (1995):
p11 0 F
p21 p22 0
0 p32 p33
=
0.434
0 0.560
0.333 0.610 0
0 0.304 0.956
From this matrix you can calculate various important things
about the population: the stable age distribution – in the figure
below this matrix describes the population at site C, the only
one where a fairly large proportion of small juveniles grow to
large juveniles and mature to reproduce.
That difference in the proportion growing and maturing also
produces a larger  for site C (1.12, indicating a growing
population, versus 0.998 and 0.997, indicating declining
populations at the other sites).
 is the dominant (largest) eigenvalue of the matrix. There are
additional, smaller eigenvalues as solutions to the polynomial
equation you saw in the matrix algebra section of the notes.
Even before the population has reached a SAD and a stable
growth rate, the growth can be predicted from eigenvalues and
eigenvectors (Caswell, 2001) according to the formula:
N
n( t ) 

i 1
ci it xi
where the c values are constants set by initial population
structure.
For site C the eigenvalues and eigenvectors can be plotted
over time. The plots show that the observed population
growth is explained by the dominant eigenvalue (component
1) after only a very few years. The other components
(eigenvalues) converge on 0 contribution…
Remember (from the matrix algebra section) that the left
eigenvector gives proportions in each ‘age class’ in a SAD
when age classes are appropriate, or here the stable relative
stage abundances, and the right eigenvector gives
reproductive values. The reproductive values for site C
Coryphantha robbinsorum are:
small juveniles
large juveniles
adults
1
2.06
3.44
How sensitive are these results to small changes in
survivorship or transition probabilities? The matrices can also
be used to determine that. Without going into the means of
calculation, sensitivity measures how much population growth
changes for a unit change in each measure.
Sensitivity as a measure therefore has the problem that a unit
change in fecundity is much different than a unit change in
transition probability.
To deal with that problem, experts developed a better
measure, called elasticity. Elasticities for a Lefkovitch (or a
Leslie) matrix sum to 1. They therefore measure the relative
contributions of each element of the matrix to the growth rate
(), tested by unit change in each, all other elements held
constant. As is evident in the figure, it is survivorship as
adults that largely determines growth rate in this cactus at all
sites…
Appendix – the Lefkovitch matrix treatment of teasel data
Let’s look at the matrix for teasel, using the same data as in the
diagrammatic life table. Remember the classes into which the
population was divided:
n1 - dormant seeds in year 1
n2 - dormant seeds in year 2
n3 - small rosettes
n4 - medium rosettes
n5 - large rosettes
n6 - flowering plants
Here’s the projection matrix for that field of teasel:
0
0
0
.966
0
0
0.013 0.010 0.125
0.07
0
0.125
0.002 0
0.038
0
0
0
0
0
0
0
0
0
0.238
0
0.245 0.167
0.023 0.750
322.38
0
3.448
30.170
0.862
0
And here’s what some of those numbers mean:
a) 0.966 of the seeds which were dormant in the first year remain
dormant until at least the beginning of the second year.
b) 0.013 of the seeds which were dormant through the winter
germinate and become small rosettes in the first year.
c) 0.007 of the seeds which were dormant through their first winter
grow in the next year to become medium rosettes, and the
next number, 0.008, represents the fraction which grew to become
large rosettes in their first growth season.
d) Now look at the 3rd column. The two 0.125s represent the fraction
which remain in the small rosette category and the fraction which
advance to become medium rosettes. The 0.036 represents the
fraction which grow so rapidly that they become large rosettes in
their first year of growth.
e) Similarly, the 4th column includes values for remaining in the
medium rosette category, growing to become large rosettes, and
the small fraction which reach the large category so early in
the season that they proceed to flower.
f) The last column may be initially confusing. If we wrote the matrix
without correction, there would be only one non-zero value in it:
the average number of seeds produced by a flowering plant,
since teasel dies after flowering. That number would have been 431.
However, correction recognizes that some seeds are not dormant in
their first year. The fraction which enter dormancy is 0.749. Thus
the proper value for seeds produced to enter the first category, seeds
dormant in their first year, is 431 x 0.749 = 322.38.
f) (cont.) The other numbers reflect seeds which germinate and begin
growth in their first year, rather than becoming dormant. The
fraction germinating and becoming small rosettes is 0.008, which,
multiplied by the total seed production (431), gives 3.448. The
number which grow to medium rosettes is 30.170, and the number
which grow all the way to large rosettes is 0.862. As you might
expect of biennials, none flower in their first year.
If you do a standard matrix multiplication L x Nt with the population
vector being numbers in the various stages, the product is the number
in those stages at time t+1.