16 Leslie Matrix
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Transcript of 16 Leslie Matrix
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APAKAH LESLIE MATRIX?
Kaedah yang dinamik untuk mewakilkan usia
atau struktur saiz populasi.
Gabungan proses populasi (kelahiran dan
kematian) untuk membentuk model tunggal.
Biasanya digunakan pada populasi kitaran
pembiakan tahunan.
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APA YG PERLU UTK LESLIE MATRIX?
Konsep VEKTOR POPULASI Kelahiran
Kematian
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POPULATION VECTOR
N0
N1
N2
N3.
Ns
s+1 baris dengan 1 colum
(s+1) x 1
s= umur maksimum
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KELAHIRAN/BIRT
H
N0 = N1F1 + N2F2 +N3F3 .+FsNs
BARU LAHIR = (Number of age 1 females) times (Fecundity of age 1 females) plu
(Number of age 2 females) times (Fecundity of age 2 females) plus
..
Note: fecundity here is defined as number of female offspringAlso, the term newborns may be flexibly defined (e.g., as eggs, newly
hatched fry, fry that survive past yolk sac stage, etc.
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KEMATIAN/MORTALITY
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) +
Number at age in next year = (Number at previous age in prior year) times(Survival from previous age to current age)
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LESLIE MATRIX
N0
N1
N2
N3.
Ns
F0 F1 F2 F3 . Fs
S0 0 0 0 . 0
0 S1 0 0 . 0
0 0 S2
0 . 0
.
0 0 0 0 Ss-1 0
=
N0
N1
N2
N3.
Ns
(s+1) x 1 (s+1) x (s+1) (s+1) x 1
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LESLIE MATRIX
N0
N1
N2
N3.
Ns
F0 F1 F2 F3 . Fs
S0 0 0 0 . 0
0 S1 0 0 . 0
0 0 S2
0 . 0
.
0 0 0 0 Ss-1 0
=
N0
N1
N2
N3.
Ns
s x 1 s x s s x 1
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LESLIE MATRIX
N0
N1
N2
N3.
Ns
F0 F1 F2 F3 . Fs
S0 0 0 0 . 0
0 S1 0 0 . 0
0 0 S2
0 . 0
.
0 0 0 0 Ss-1 0
=
N0
N1
N2
N3.
Ns
Nt+1 = A Nt
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PROJECTION WITH THE LESLIE MATRIX
Nt+1 = ANtNt+2 = AANt
Nt+3 = AAANtNt+4 = AAAANt
Nt+n = AnNt
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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
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PROPERTIES OF THIS MODELGRAPHICAL ILLUSTRATION
0
5000
10000
15000
20000
25000
0 5 10 15 20 25
Time
NAge 0
Age 1
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PROPERTIES OF THIS MODELGRAPHICAL ILLUSTRATION
0
1
2
3
4
5
6
0 5 10 15 20 25
Time
Lambda
Lambda = Nt+1 / Nt
Thus,
Nt+1= Nt
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PROPERTIES OF THIS MODELGRAPHICAL ILLUSTRATION
0
5
10
15
20
25
0 5 10 15 20 25
Time
PercentinAge1
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PROJECTION WITH THE LESLIE MATRIX
Given that the population dynamics are ergodic,
we really dont even need to worry about theinitial starting population vector. We can base
our analysis on the matrix A itself
Nt+n
= AnNt
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PROJECTION WITH THE LESLIE MATRIX
Given the matrix A, we can
compute its eigenvalues andeigenvectors, which correspondto population growth rate,
stable age distribution, and
reproductive value
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PROJECTION WITH THE LESLIE MATRIX
EIGENVALUES
Whats an eigenvalue?
Cant really give you a plain English definition
(heaven knows Ive 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
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PROJECTION WITH THE LESLIE MATRIX
CHARACTERISTIC EQUATION
1= F1-1 + P1F2-2 + P1P2F3-3 + P1P2P3F4-4
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 oftheir solution)
The root () that has the largest absolute value is thedominant eigenvalue and will determine populationgrowth in the long run. The other eigenvalues will
determine transient dynamics of the population.
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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 wont worry how to compute this stuff in class computing the eigenvalues and eigenvectors can be abugger!)
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PROJECTING VS. FORECASTING OR PREDICTION
So far, Ive used the term projecting what does this mean intechnical 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 worryingabout transient dynamics.
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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 inone 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 systems dynamics.
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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 ortrees, the size of the individual is more important than its age.
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LEFKOVITCH MATRIX EXAMPLE
N0
N1
N2
N3.
Ns
F0 F1 F2 F3 . Fs
T0-1 T1-1 T2-1 T3-1 .. Ts-1
T0-2 T1-2 T2-2 T3-2 .. Ts-2
T0-3
T1-3
T2-3
T3-3 ..
Ts-3.
T0-s T1-s T2-s T3-s .. Ts-s
=
N0
N1
N2
N3.
Ns
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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 alsothat there is the possibility for an organism to regress instages (i.e., go to an earlier stage), whereas in the Leslie
matrix, everyone gets older if they survive, and they only
advance one age
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LEFKOVITCH MATRIX EXAMPLE
N0
N1
N2
N3
.
Ns
=
N0
N1
N2
N3
.
Ns
F0 F1 F2 F3 . Fs
T0-1 T1-1 T2-1 T3-1 .. Ts-1
T0-2 T1-2 T2-2 T3-2 .. Ts-2
T0-3 T1-3 T2-3 T3-3 .. Ts-3.
T0-s T1-s T2-s T3-s .. Ts-s
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
An important question in fisheries
management is How much fishing pressureor mortality can a population support?
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
N0
N1
N2
N3
.
Ns
F0 F1 F2 F3 . Fs
S0 0 0 0 . 0
0 S1 0 0 . 0
0 0 S2 0 . 0
.
0 0 0 0 Ss-1 0
=
N0
N1
N2
N3
.
Ns
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
S = e-(M+F)
Knife-edge recruitment, meaning
that fish at a given age are either
not exposed to fishing mortality orare fully vulnerable
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
0.0 0.5 1.0 1.5 2.0 2.5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Instantaneous fishing mortality
Rateofincrease(lambda)
1
2
3
4
Age at entry
Maintenance
level
5
Current
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
0250500750
0250500
0250500
0250500
0250500
0250500
0250500
0 5,000 10,000 15,000 20,000 25,0000
250500
Year 3
Year 4
Year 5
Year 6
Year 7
Year 8
Year 9
Year 10
Population size
Frequency
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2
0
2,000
4,000
6,000
0
2,000
0400800
1,200
0
400
800
0
400
800
0
400
800
0
400
800
0
400
800
0
400
800
4,000
Population growth rate
Frequency
Year 1-2
Year 2-3
Year 3-4
Year 4-5
Year 5-6
Year 6-7
Year 7-8
Year 8-9
Year 9-10
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
400
800
1,200
400
800
1,200
400
800
1,200
400
800
1,200
400
800
1,200
400
800
1,200
1.0 1.1 1.2 1.3 1.4
0
400
800
1,200
Year 10-20
Year 10-30
Year 10-40
Year 10-50
Year 10-60
Year 10-70
Year 10-150
Population growth rate
Frequency
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EXAMPLE APPLICATION:
SUSTAINABLE FISHING MORTALITY
0.0 0.5 1.0 1.5 2.0 2.5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Instantaneous fishing mortality
Rateofincrease(lambda)
1
2
3
4
Age at entry
Maintenance
level
5
Current
EXAMPLE APPLICATION
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EXAMPLE APPLICATION:
REPRODUCTIVE STRATEGY OF YELLOW
PERCHOne of main questions was whether
stunting, meaning very slow growth, ofyellow perch was caused by reproductive
strategy or if reproductive strategy resultedfrom adaptation to low prey abundance
Our goal was to understand what stunted fish
should do in terms of age at maturity
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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 largerthan 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
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH
REPRODUCTIVE STRATEGY OF YELLOW
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REPRODUCTIVE STRATEGY OF YELLOW
PERCH