Can you sum the sensitivities?

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?

$$\frac{\partial \lambda} {\partial a_{ij}} + \frac{\partial \lambda} {\partial a_{kl}} = ?$$

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

$$\Delta \lambda \approx \frac{\partial \lambda} {\partial a_{ij}} \Delta a_{ij} + \frac{\partial \lambda} {\partial a_{kl}} \Delta a_{kl}$$

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. ^ 