what is the leslie matrix? method for representing dynamics of age or size structured populations...

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

N0 = N1F1 + N2F2 +N3F3 ….+FsNs

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.

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)

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

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

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

Projection with the Leslie Matrix

Nt+1 = ANt

Nt+2 = AANt

Nt+3 = AAANt

Nt+4 = 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 ModelGraphical Illustration

0

5000

10000

15000

20000

25000

0 5 10 15 20 25

Time

N

Age 0

Age 1


0

1

2

3

4

5

6

0 5 10 15 20 25

Time

Lam

bd

a

Lambda = Nt+1 / Nt

Thus,

Nt+1 = λ Nt


0

5

10

15

20

25

0 5 10 15 20 25

Time

Per

cen

t in

Ag

e 1


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


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 MatrixEigenvalues

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), whichbasically means that these give us a single equation for the population growth over time

Projection with the Leslie MatrixCharacteristic 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 MatrixEigenvectors

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 ModelsLefkovitch 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

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

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

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

Example Application:Sustainable Fishing Mortality

• An important question in fisheries management is “How much fishing pressure or mortality can a population support?”


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 = e-(M+F)

Knife-edge recruitment, meaningthat fish at a given age are eithernot exposed to fishing mortality or are fully vulnerable


0.0 0.5 1.0 1.5 2.0 2.50.2

0.4

0.6

0.8

1.0

1.2

1.4

Instantaneous fishing mortality

Rat

e of

incr

ease

(la

mbd

a)

1

2

3

4

Age at entry

Maintenancelevel

5

Current


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

Fre

quen

cy


0.6 0.8 1.0 1.2 1.4 1.6 1.8 2

02,000

4,000

6,000

0

2,000

0400800

1,200

0

400

800

0

400

800

0

400

800

0400800

0

400

800

0

400

800

4,000

Population growth rate

Fre

quen

cy

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


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 1.40

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

Fre

quen

cy


0.0 0.5 1.0 1.5 2.0 2.50.2

0.4

0.6

0.8

1.0

1.2

1.4

Instantaneous fishing mortality

Rat

e of

incr

ease

(la

mbd

a)

1

2

3

4

Age at entry

Maintenancelevel

5

Current

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

what is the leslie matrix? method for representing dynamics of age or size structured populations...

Documents