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Analyzing matrix models 06

# Analyzing matrix models 06 - Analyzing Matrix Models ESM...

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Analyzing Matrix Models ESM 211 Nov. 8, 2006

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Projecting the model Semipalmated Sandpiper in Manitoba Three age classes Initial age distribution (#/ha): Projection matrix: Iterate the model: = 3 . 7 2 . 14 5 . 23 ) 0 ( N = 563 . 0 563 . 0 0 0 0 563 . 0 0846 . 0 074 . 0 02115 . 0 A ) ( ) 1 ( t t AN N = +

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0 5 10 15 20 0 5 10 15 20 Year N 1 2 3
0 5 10 15 20 1 e-03 1 e-02 1 e-01 1 e+00 1 e+01 Year N 1 2 3

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Asymptotic growth rate When population reaches stable age (or size or stage) distribution then all classes grow (or decline) at the same rate: 1 ( 1) ( ) t t λ + = N N lambda_1 = 0.6389479 Class Stable distribution Reproductive value 1 0.1188640 1 2 0.1047353 1.097332 3 0.7764007 1.113922
Matrices and growth rate The asymptotic growth rate ( λ 1 ) is the dominant eigenvalue of the projection matrix The stable age distribution ( w 1 ) is the associated right eigenvector The reproductive value distribution ( v 1 ) is the associated left eigenvector 1 1 1 1 1 1 λ λ = = w Aw v v A

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What good is a deterministic matrix model? Assumes that the environment is constant – unrealistic! But: If lambda < 1, population is really in trouble! We might not have info on temporal variability Insights from sensitivity analysis (next) carry over to stochastic case
Sensitivity & Elasticity Sensitivity: absolute rate of change of λ 1 with respect to absolute change in a matrix element Elasticity: relative rate of change of λ 1 with respect to relative change in a matrix element 1 ij ij S a λ = 1 1 1 1 1 1 1 ij ij ij ij ij ij ij a a E S a a a λ λ λ λ λ = = =

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Sensitivity & Elasticity of vital rates 1 1 k s s ij r ij i j k a S S r = = = �� 1 1 k s s ij k r ij i j ij k a r E E a r = = = ��
Sensitivity in management For each potential mgmt action Define its effect on demography (e.g.

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Analyzing matrix models 06 - Analyzing Matrix Models ESM...

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