Analyzing matrix models 08

Analyzing matrix models 08 - trouble! We might not have...

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Analyzing Matrix Models ESM 211 Feb. 27, 2008
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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
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0 5 10 15 20 1 e-03 e-02 e-01 e+00 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
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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 = = 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
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Unformatted text preview: trouble! We might not have info on temporal variability Insights from sensitivity analysis (next) carry over to stochastic case Sensitivity &amp; Elasticity Sensitivity: absolute rate of change of 1 with respect to absolute change in a matrix element Elasticity: relative rate of change of 1 with respect to relative change in a matrix element 1 ij ij S a = 1 1 1 1 1 1 1 ij ij ij ij ij ij ij a a E S a a a = = = Sensitivity &amp; 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. reduce mortality of adults) Estimate cost per unit effort Use sensitivity or elasticity to find improvement in lambda per unit cost...
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Analyzing matrix models 08 - trouble! We might not have...

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