What is the Leslie Matrix?
• Method for representing dynamics of age
or size structured populations
• Combines population processes (births
and deaths) into a single model
• Generally applied to populations with
annual breeding cycle
• By convention, use only female part of
population
Setting up the Leslie Matrix
• Concept of population vector
• Births
• Deaths
Population Vector
N0
N1
N2
N3
….
Ns
s+1 rows by 1 column
(s+1) x 1
Where, s=maximum age
Births
Newborns = (Number of age 1 females) times (Fecundity of age 1 females) plus
(Number of age 2 females) times (Fecundity of age 2 females) plus
…..
Note: fecundity here is defined as number of female offspring
Also, the term “newborns” may be flexibly defined (e.g., as eggs, newly
hatched fry, fry that survive past yolk sac stage, etc.
N0 = N1F1 + N2F2 +N3F3 ….+FsNs
Mortality
Number at age in next year = (Number at previous age in prior year) times
(Survival from previous age to current age)
Na,t = Na-1,t-1Sa
Another way of putting this is, for age 1 for
example:
N1,t = N0,t-1S0-1 + N1,t-1 (0) + N2,t-1 (0) + N3,t-1 (0) + …
Leslie Matrix
N0
F0
F1 F2 F3 …. Fs
N0
N1
S0
0
N2
0
S1 0
N3
….
0
….
Ns
0
(s+1) x 1
=
0
0
….
0
N1
0 ….
0
N2
0
S2 0 ….
0
N3
….
0
0 0
Ss-1 0
Ns
(s+1) x (s+1)
(s+1) x 1
Leslie Matrix
N0
F0
F1 F2 F3 …. Fs
N0
N1
S0
0
N2
0
S1 0
N3
….
0
….
Ns
0
sx1
=
….
0
N1
0 ….
0
N2
0
S2 0 ….
0
N3
….
0
0 0
Ss-1 0
Ns
sxs
0
0
sx1
Leslie Matrix
N0
F0
F1 F2 F3 …. Fs
N0
N1
S0
0
N2
0
S1 0
N3
….
0
….
Ns
0
=
Nt+1 = A Nt
0
0
….
0
N1
0 ….
0
N2
0
S2 0 ….
0
N3
….
0
0 0
Ss-1 0
Ns
Projection with the Leslie Matrix
Nt+1
Nt+2
Nt+3
Nt+4
= ANt
= AANt
= AAANt
= AAAANt
Nt+n = AnNt
Properties of this Model
• Age composition initially has an effect on
population growth rate, but this disappears
over time (ergodicity)
• Over time, population generally
approaches a stable age distribution
• Population projection generally shows
exponential growth
Properties of this Model
Graphical Illustration
25000
20000
15000
N
Age 0
Age 1
10000
5000
0
0
5
10
15
Time
20
25
Properties of this Model
Graphical Illustration
Lambda
6
Lambda = Nt+1 / Nt
5
Thus,
4
Nt+1 = λ Nt
3
2
1
0
0
5
10
15
Time
20
25
Properties of this Model
Graphical Illustration
25
Percent in Age 1
20
15
10
5
0
0
5
10
15
Time
20
25
Projection with the Leslie Matrix
Given that the population dynamics are ergodic,
we really don’t even need to worry about the
initial starting population vector. We can base
our analysis on the matrix A itself
Nt+n = AnNt
Projection with the Leslie Matrix
Given the matrix A, we can
compute it’s eigenvalues and
eigenvectors, which correspond
to population growth rate,
stable age distribution, and
reproductive value
Projection with the Leslie Matrix
Eigenvalues
What’s an eigenvalue?
Can’t really give you a “plain English” definition
(heaven knows I’ve searched for one!)
Mathematically, these are the roots of the
characteristic equation (there are s+1 eigenvalues
for the Leslie matrix), which
basically means that these give us a single equation
for the population growth over time
Projection with the Leslie Matrix
Characteristic Equation
1= F1λ-1 + P1F2 λ-2 + P1P2F3 λ-3 + P1P2P3F4 λ-4 …
Note that this is a polynomial, and thus can be
solved to get several roots of the equation (some of
which may be “imaginary”, that is have √-1 as part of
their solution)
The root (λ) that has the largest absolute value is the
“dominant” eigenvalue and will determine population
growth in the long run. The other eigenvalues will
determine transient dynamics of the population.
Projection with the Leslie Matrix
Eigenvectors
Associated with the dominant eigenvalue is two sets of
eigenvectors
The right eigenvectors comprise the stable age distribution
The left eigenvectors comprise the reproductive value
(We won’t worry how to compute this stuff in class –
computing the eigenvalues and eigenvectors can be a
bugger!)
Projecting vs. Forecasting or Prediction
So far, I’ve used the term projecting – what does this mean in
technical terms, and how does it differ from a forecast or
prediction.
Basically, forecasting or prediction focuses on short-term
dynamics of the population, and thus on the transient
dynamics. Projection refers to determining the long-term
dynamics if things remained constant. Thus projection gives
us a basis for comparing different matrices without worrying
about transient dynamics.
Projecting vs. Forecasting or Prediction
Simple (?) Analogy: The speedometer of a car gives you an
instantaneous measure of a cars velocity. You can use to
compare the velocity of two cars and indicate which one is
going faster, at the moment. To predict where a car will be in
one hour, we need more information, such as initial
conditions: Where am I starting from? What is the road ahead
like? etc. Thus, projections provide a basis for comparison,
whereas forecasts are focusing on providing “accurate”
predictions of the system’s dynamics.
Stage-structured Models
Lefkovitch Matrix
Instead of using an age-structured approach, it may be more
appropriate to use a stage or size-structured approach.
Some organisms (e.g., many insects or plants) go through
stages that are discrete. In other organisms, such as fish or
trees, the size of the individual is more important than it’s age.
Lefkovitch Matrix Example
F1 F2 F3 …. Fs
N0
F0
N1
T0-1 T1-1 T2-1 T3-1
….. Ts-1
N1
N2
T0-2 T1-2 T2-2 T3-2
….. Ts-2
N2
N3
….
T0-3 T1-3 T2-3 T3-3
….
….. Ts-3
N3
….
Ns
T0-s T1-s T2-s T3-s
….. Ts-s
Ns
=
N0
Lefkovitch Matrix
Note now that each of the matrix elements do not correspond
simply to survival and fecundity, but rather to transition rates
(probabilities) between stages. These transition rates depend
in part on survival rate, but also on growth rates. Note also
that there is the possibility for an organism to “regress” in
stages (i.e., go to an earlier stage), whereas in the Leslie
matrix, everyone gets older if they survive, and they only
advance one age
Lefkovitch Matrix Example
F1 F2 F3 …. Fs
N0
F0
N1
T0-1 T1-1 T2-1 T3-1
….. Ts-1
N1
N2
T0-2 T1-2 T2-2 T3-2
….. Ts-2
N2
N3
….
T0-3 T1-3 T2-3 T3-3
….
….. Ts-3
N3
….
Ns
T0-s T1-s T2-s T3-s
….. Ts-s
Ns
=
N0
Example Application:
Sustainable Fishing Mortality
• An important question in fisheries
management is “How much fishing
pressure or mortality can a population
support?”
Example Application:
Sustainable Fishing Mortality
N0
F0
F1 F2 F3 …. Fs
N0
N1
S0
0
N2
0
S1 0
N3
….
0
….
Ns
0
=
0
0
….
0
N1
0 ….
0
N2
0
S2 0 ….
0
N3
….
0
0 0
Ss-1 0
Ns
Example Application:
Sustainable Fishing Mortality
S=
-(M+F)
e
Knife-edge recruitment, meaning
that fish at a given age are either
not exposed to fishing mortality or
are fully vulnerable
Example Application:
Sustainable Fishing Mortality
Rate of increase (lambda)
1.4
Age at entry
1.2
5
1.0
0.8
4
Current
3
0.6
2
0.4
1
Maintenance
level
0.2
0.0
0.5
1.0
1.5
2.0
Instantaneous fishing mortality
2.5
Example Application:
Sustainable Fishing Mortality
750
500
250
0
Year 3
Year 4
500
250
0
Year 5
Frequency
500
250
0
Year 6
500
250
0
Year 7
500
250
0
Year 8
500
250
0
500
250
0
Year 9
500
250
0
Year 10
0
5,000
10,000
15,000
Population size
20,000
25,000
Example Application:
Sustainable Fishing Mortality
6,000
4,000
2,000
0
4,000
Year 1-2
Year 2-3
Frequency
2,000
0
1,200
800
400
0
800
400
0
800
Year 3-4
Year 4-5
Year 5-6
400
0
800
400
0
800
400
0
800
400
0
800
400
0
0.6
Year 6-7
Year 7-8
Year 8-9
Year 9-10
0.8
1.0
1.2
1.4
1.6
Population growth rate
1.8
2
Example Application:
Sustainable Fishing Mortality
1,200
Year 10-20
800
400
1,200
Year 10-30
800
400
1,200
Year 10-40
800
Frequency
400
1,200
Year 10-50
800
400
1,200
Year 10-60
800
400
1,200
Year 10-70
800
400
1,200
Year 10-150
800
400
0
1.0
1.1
1.2
1.3
Population growth rate
1.4
Example Application:
Sustainable Fishing Mortality
Rate of increase (lambda)
1.4
Age at entry
1.2
5
1.0
0.8
4
Current
3
0.6
2
0.4
1
Maintenance
level
0.2
0.0
0.5
1.0
1.5
2.0
Instantaneous fishing mortality
2.5
Example Application:
Reproductive Strategy of Yellow Perch
• One of main questions was whether
“stunting”, meaning very slow growth, of
yellow perch was caused by reproductive
strategy or if reproductive strategy resulted
from adaptation to low prey abundance
• Our goal was to understand what stunted
fish “should” do in terms of age at maturity
Reproductive Strategy of Yellow Perch
• Basic model had a number of assumptions
– Energy intake is limited and depends on size of fish
• Yellow perch show an ontogenetic shift in diet where
indivduals less than 10 grams eat zooplankton, individuals 10
to 30 grams eat benthic invertebrates, and individuals larger
than 30 grams eat fish
– Net energy intake can only be partitioned to growth or
reproduction
– Reproduction is all or nothing
– Reproduction may have “survival costs (theta)”
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch
Reproductive Strategy of Yellow Perch