Sensitivity and Elasticity Analyses

One approach to determining how much various life-history stage transitions affect the population dynamics is by examining how changes in a particular stage affect the magnitude of the leading eigenvalue. The advantage of this approach is that once you've done all the work to measure the transition rates you don't have to experimentally manipulate the rates and watch for the effect, you can mathematically manipulate the rates and determine the effect.

Consider the following simple Leslie matrix

$$\left( \begin{array}{ccc} 0 & 1 & 4 \\ .7 & 0 & 0 \\ 0 & .5 & 0 \\ \end{array}\right)$$

For this matrix $\lambda_1 = 1.32528$. Suppose we subtract 0.1 from $a_{12}$ making it 0.9. Then $\lambda_1 = 1.30493$, a change of 0.02035.

Suppose we subtract 0.1 from $a_{21}$ making it 0.6. Then $\lambda_1 = 1.24923$, a change of 0.07605.

It appears that the population growth rate, $\lambda_1$, is more sensitive to changes in first year-survivorship, $a_{21}$, than to changes in second-year reproduction, $a_{12}$.

We can formalize this by saying that the sensitivity coefficient of $a_{ij}$ is

$$\frac{\partial\lambda}{\partial a_{ij}}$$

For the above matrix

Coefficient	Sensitivity
$a_{12}$	0.2030	$\approx 0.02035/0.1$
$a_{13}$	0.0766
$a_{21}$	0.7278	$\approx 0.07605/0.1$
$a_{32}$	0.6128

This analysis suggests that survivorship from age 1 to age 2 ($a_{21}$) and from age 2 to age 3 ($a_{32}$) are the most important stages in the life history. A unit change in either of these produces a corresponding change three or more times as great as increasing the fecundity of individuals in age-class 2 and nine to ten times as great as increasing the fecundity of individuals in age-class 3.

One problem with this approach is that some of the variables, i.e., survival rates, are intrinsically restricted in their range to values between 0 and 1, while others, i.e., fecundities, may be very large. Elasticity is a measure of ``proportional'' effect, i.e., the effect that a change in a given matrix element has as a proportional to the change in that element:

$$\frac{a_{ij}}{\lambda}\frac{\partial\lambda}{\partial a_{ij}}$$

Coefficient	Sensitivity	Elasticity12
$a_{12}$	0.2030	0.1532
$a_{13}$	0.0766	0.2312
$a_{21}$	0.7278	0.3844
$a_{32}$	0.6128	0.2312

The importance of some measure of proportional effect is illustrated by this very similar matrix:

$$\left( \begin{array}{ccc} 0 & 1000 & 4000 \\ .7 & 0 & 0 \\ 0 & .5 & 0 \\ \end{array}\right)$$

Coefficient	Sensitivity	Elasticity
$a_{12}$	0.0124	0.4507
$a_{13}$	0.0002	0.0329
$a_{21}$	18.9318	0.4836
$a_{32}$	1.8027	0.0329