458 Age-structured models (continued): Estimating from Leslie matrix models Fish 458, Lecture 4.
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Transcript of 458 Age-structured models (continued): Estimating from Leslie matrix models Fish 458, Lecture 4.
458
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/