TL;DR No.

I’m going out on a bit of a limb here with that assertion. I don’t have The Gospel According to Caswell close at hand to check. But I think the following argument is sufficient.1

This was stimulated by a question to my boss who passed it on to me. Here’s the question:

Can you sum sensitivities if they are on the same scale (e.g., survival rates)? I know you can’t sum fecundity and survival sensitive because they are on different scales, but we were debating today whether you can sum or average sensitivities on stage transition rates.

## What’s the sensitivity?

In the analysis of structured population models, the sensitivity of a matrix element is just the partial derivative of the population growth rate $\lambda$ with respect to that matrix element. So the question is, does this make any sense?

## The total differential

Think of what’s behind this thing. The population growth rate is a function of the matrix elements, $\lambda = f(a_{ij}, a_{kl})$ and let’s pretend for the moment that there are only two. The change in $\lambda$ caused by changing the inputs to the function can be approximated using the total differential

which gets more accurate as the changes in the matrix elements get small. If $\Delta a_{ij} = \Delta a_{kl} = 1$, then this is just the sum of the sensitivities. So summing the sensitivities is the amount $\lambda$ changes if you change the matrix elements by 1 unit. But for survival that makes no sense biologically; survival is always less than 1.

OK, choose $\Delta a_{ij}$ to be something smaller, like, a 1% change. Fine. Guess what? That’s called the elasticity. Those CAN be summed up.

## Conclusion

You can sum up the sensitivities, but the result is biologically meaningless. So don’t.

1. All the code for this post, including that not shown, can be found here