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The stable age distribution for the loggerhead turtle can be readily calculated from the right eigen-

vector of the transition matrix A, where 0.947 is the dominant eigenvalue of A:

Wraw=eigenvec(A, 0.947)

In order to make these values easier to interpret, we convert them to the proportion in each age group,

by dividing each element by the sum of all of the elements:

This yields the following stable age distribution:

In other words, after the loggerhead turtle model has proceeded for a number of years, we would

expect to see 21% of the population being composed of eggs or hatchlings, 67% juveniles, and the

rest (12%) subadults and adults.

In the case of the loggerhead sea turtle, the vector Vdepicting stage-specific reproductive value is

as follows:

We can doublecheck to verify that this is indeed the left eigenvector by performing the two different

multiplications alluded to in the text:

VT=(1 1.4 6 116 569 507 588)

VT·A =(0.95 1.33 5.71 109.74 537.16 479.69 555.69)

λ·VT=(0.95 1.33 5.68 109.85 538.84 480.13 556.84)

With these values we can estimate elasticities for every parameter in the transition matrix:

S

...

...

...

...

.

.

..

=

⎛ ×

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎜⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

0 0 0 0 0 01 3 27 10 − 0 04 1

005 018 0 0 0 0 0 037

00050120 0 0 0025

0 0 0 05 0 14 0 0 0 0 09

0000050 0 00

0000004 0 00

00000 004 0230

4

⎟⎟⎟

S

AVW

ij VW

=⋅ij i⋅ j

⋅

⎛

⎝⎜

⎞

λ ⎠⎟

V =

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

.

1

14

6

116

569

507

588

W

.......

= ×

×

×

×

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

−

−

−

−

021

067

011

673 10

369 10

316 10

185 10

3

4

4

3

W W

W

=

∑

raw

rawj

i

Wraw

.......

= ×

×

×

×

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

−

−

−

−

029

094

016

948 10

521 10

445 10

262 10

3

4

4

3

Box 14.3Calculation
of elasticities of the
Lefkovitch matrix for
the loggerhead sea turtle
(based on data from
Crouse et al.1987).