16 leslie matrix

7/27/2019 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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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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0

5

10

15

20

25

0 5 10 15 20 25

Time

PercentinAge1


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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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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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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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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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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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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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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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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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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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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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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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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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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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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)



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PERCH



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PERCH



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PERCH



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PERCH



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PERCH



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PERCH



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PERCH



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PERCH



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PERCH

16 leslie matrix

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