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Age-structured models (continued):

Estimating from Leslie matrix models

Fish 458, Lecture 4

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The facts on

• Finite rate of population increase

• =er & r=ln(), therefore Nt=Nt

• A dimensionless number (no units)

• Associated with a particular time step

(Ex: =1.2/yr not the same as = 0.1/mo)

• >1: pop. ; <1 pop

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Matrix Population Models: Definitions

•Matrix- any rectangular array of symbols. When used to describe population change, they are called population projection matrices.

•Scalar- a number; a 1 X 1 matrix

•State variables- age or stage classes that define a matrix.

•State vector- non-matrix representation of individuals in age/stage classes.

•Projection interval- unit of time define by age/stage class width.

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4x1 + 3x2 + 2x3 = 0

2x1 - 2x2 + 5x3 = 6

x1 - x2 - 3x3 = 1

0

6

1

4 3 2

2 –2 5

1 –1 3

x1

x2

x3

=

Basic Matrix Multiplication

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What does this remind you of?

n(t + 1) = An(t)

Where:A is the transition/projection matrixn(t) is the state vectorn(t + 1) is the population at time t + 1

This is the basic equation of a matrix population model.

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Eigenvectors & Eigenvalues

Aw = w

v,w = Eigenvector = Eigenvalue

When matrix multiplication equals scalar multiplication

Note: “Eigen” is German for “self”.

vA = v

• Rate of Population Growth (): Dominant Eigenvalue

• Stable age distribution (w): Right Eigenvector

• Reproductive values (v): Left Eigenvector

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Example: Eigenvalue

3 -6

2 -5

3 -6

2 -5

4

1=

6

3

-3

-3

1

1=

No obvious relationship between x and y

A x = y A x = y

Obvious relationship between x and y:

x is multiplied by -3

Thus, A acts like a scalar multiplier.

How is this similar to ?

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Characteristic equations

From eigenvalues, we understand that Ax = xWe want to solve for , so

Ax - x = 0 (singularity)

or(A- I)x = 0

“I” represents an identity matrix that converts into a matrix on the same order as A.

Finding the determinant of (A- I) will allow one to solve for . The equation used to solve for is called the Characteristic Equation

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Solution of the Projection Solution of the Projection EquationEquation

n(t+1) = An(t)

4 - P1F2 2 - P1P2F3 - P1P2P3F4 = 0

or alternatively (divide by 4)

1 = P1F2 -2 + P1P2F3 -3 + P1P2P3F4 -4

- 1.25 1.20 0.03 0.80 - 0 0 0 0.625 - 0 0 0 0.2 -

This equation is just the matrix form of Euler’s equation:

1 = Σ lxmxe-rx

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Constructing an age-structured (Leslie) matrix

model

Build a life table Birth-flow vs. birth pulse Pre-breeding vs. post-breeding

census Survivorship Fertility

Build a transition matrix

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Birth-Flow vs. Birth-Pulse Birth-Flow (e.g humans)

Pattern of reproduction assuming continuous births. There must be approximations to l(x) and m(x); modeled as continuous, but entries in the projection matrix are discrete coefficients.

Birth-Pulse (many mammals, birds, fish)Maternity function and age distribution are discontinuous, matrix projection matrix very appropriate.

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Pre-breeding vs. Post-breeding Censuses

Pre-breeding (P1)

Populations are accounted for just before they breed.

Post-breeding (P0)

Populations are accounted for just after they breed

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Calculating Survivorship and Fertility Rates for Pre- and Post-Breeding

Censuses

x lx mx

0 1 01 0.8 02 0.5 23 0.1 64 0 3

class

1 0.8/1.0= 0.8 0.5/0.8= 0.6252 0.5/0.8= 0.625 0.1/0.5= 0.23 0.1/0.5= 0.2 04 0

1 0.8*0= 0 0.8*0= 02 0.625*2= 1.25 0.8*2= 1.6

3 0.2*6= 1.2 0.8*6= 4.8

4 0*3= 0 0.8*3= 2.4

Fertility

Survivorship

Birth pulsep-->0 p-->1

Different approaches, yet both ways produce a of

1.221.

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• 4 a g e c l a s s e s• F e r t i l i t y c o e f f .

F 2 = 1 . 2 5F 3 = 1 . 2 0F 4 = 0 . 0 3

• S u r v i v a l p r o b .P 1 = 0 . 8 0 0P 2 = 0 . 6 2 5P 3 = 0 . 2 0 0

1 2 3 4

0 1 . 2 5 0 0 1 . 2 0 0 0 0 . 0 3 0 00 . 8 0 0 0 0 0 0

0 0 . 6 2 5 0 0 00 0 0 . 2 0 0 0 0

0 F 1 F 2 F 3

P 1 0 0 0

0 P 2 0 0

0 0 P 3 0

The Transition/Population Projection Matrix

4 age class life cycle graph

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Example:Example: Shortfin Mako (Isurus oxyrinchus)

Software of choice: PopTools

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Mako Shark Data

Mortality: M1-6 = 0.17

M7- = 0.15

Fecundity: 12.5 pups/female

Age at female maturity: 7 years

Reproductive cycle: every other 2 years

Photo: Ron White

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Essential Characters of Population Models

Asymptotic analysis: A model that describes the long-term behavior of a population.

Ergodicity: A model whose asymptotic analyses are independent of initial conditions.

Transient analysis: The short-term behavior of a population; useful in perturbation analysis.

Perturbation (Sensitivity) analysis: The extent to which the population is sensitive to changes in the model.

Caswell 2001, pg. 18

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Uncertainty and hypothesis testing

Characterizing uncertainty

•Series approximation (“delta method”)

•Bootstrapping and Jackknifing

•Monte Carlo methods

Hypothesis testing

•Loglinear analysis of transition matrices

•Randomization/permutation tests Caswell 2001, Ch. 12

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References

Caswell, H. 2001. Matrix Population Models: Construction, Analysis, and Interpretation. Sunderland, MA, Sinauer Associates. 722 pp.

Ebert, T. A. 1999. Plant and Animal Populations: Methods in Demography. San Diego, CA, Academic Press. 312 pp.

Leslie, P. H. 1945. On the use of matrices in certain population mathematics. Biometrika 33: 183-212.

Mollet, H. F. and G. M. Cailliet. 2002. Comparative population demography of elasmobranch using life history tables, Leslie matrixes and stage-based models. Marine and Freshwater Research 53: 503-516.

PopTools: http://www.cse.csiro.au/poptools/