untitled

that might be taken. In the case of loggerheads, for example, it might be possible to

protect nesting sites on beaches or alternatively devote larger effort to improving

survival while animals are out at sea. How can we evaluate these options?

There is an easy (but not very exact) way to do this. We substitute different

values for each parameter in the transition matrix and see which one has the biggest

effect. You should do so for yourself, to determine which parameter is most con-

ducive to improving the growth rate λ. A more elegant answer can be obtained by

using calculus to determine how sensitive λis to a proportionate change in each

parameter (Aij) in the transition matrix, where irefers to row and jrefers to column.

This is termed by ecologists the elasticityof λ, and the derivation is presented in

Box 14.2 (based on Caswell 1978; De Kroon et al. 1986):

·

where Wis the vector corresponding to the stable age distribution and Vis the

vector of reproductive values (defined below).

We have already introduced two key concepts in our discussion of age-structured

(Leslie matrix) models: the dominant eigenvalue and right eigenvectors. These

useful tools also apply to stage-structured (Lefkovitch matrix) models. There is another

useful string of numbers, needed to calculate elasticity, called the vector of

reproductive values. It is the left eigenvector (as opposed to the stable age distribu-

tion, which is the right eigenvector) that corresponds to the dominant eigenvalue λ.

Left eigenvector just means that the order of the multiplication shown above is reversed.

The left eigenvector is a string of numbers that satisfies the following equality:

VT·A=λ·VT

D

F

Vi·Wj

V·W

A

C

Aij

λ

AGE AND STAGE STRUCTURE 249

For small changes in λas a function of small changes in any element of the transition matrix (Aij),

elasticity is defined as:

and

So elasticity equals:

where V×Win the denominator refers to the scalar or dot product obtained by multiplying together

the column vector representing the stable stage distribution (W) and the row vector of reproductive

values (V).

AVW

VW

ij i j

λ

⋅

⋅

⋅

⎛

⎝⎜

⎞

⎠⎟

A

A

AVW

VW

ij

ij

ij i j

λ

λ

λ

⋅

⎛

⎝

⎜

⎞

⎠

⎟=⋅

⋅

⋅

⎛

⎝⎜

⎞

⎠⎟

d

d

d

d

d

log d

log

eij

e

ij

A ij

A

A

λ

λ

=⋅λ

Box 14.2Deriving the
elasticity of matrix
models (Caswell 1978).