Hello,
I'm looking at elasticity values from a range of amphibian PVA's. There are lots of elasticities values within models that turn out to be the same, not similar, but exactly the same to 10 decimal places. I've re-ran some of the models and no matter how I structure them transitions turn out to be identical and stasis are all different. I've also recalculated them by hand and get the same results. I'm using a deterministic model, if that makes any difference. I'm new to this kind of modelling so I could totally be messing something up. Does anyone have any insight as to what might be going on here?
Please let me know if there is some specific additional information that would be helpful here.
Thanks in advance,
Paul
The first thing I would do is make sure all of your elasticities add up to 1. If not, you have an error somewhere. What program are you using to calculate the values?
Benjamin
Are these matrix entry or lower level elasticities? In either case, it is indeed common (depending on matrix structure) for different vital rates to have identical elasticities, if they contribute to population growth in the same way. For example, in the simple 2x2 matrix:
A= Fj Fa
Sj Sa
Sj and Fa will have identical elasticities.
Hey guys,
I'm using popbio in R, and I have used poptools in MS Excel.
Yes, the matrix elements sum to 1, and yes they are for matrix elements. I've done some lower level calculations too, and this changes some. The situation you described Jeffrey with the 2x2 matrix is exactly what I am dealing with.
As I mentioned, I'm learning the ropes here, so I'm not totally clued into all the theory, but do you think that different vital rates really could have identical elasticities? Any thoughts on this?
Cheers,
Paul
As Jeffrey pointed out, the off-diagonal elements of a 2x2 matrix will have identical elasticities; this is because in the formula for the eigenvalues (found by solving det(A - I*lambda) = 0), those two elements are multiplied together. Thus the same proportional change in either of them will have the same proportional effect on lambda.
More generally, if you do a "loop analysis" (identify any path that begins and ends in the same stage; in Jeffrey's 2x2 example there are 3 loops [j -> j, a -> a, and j-> a -> j), and find that, within a given loop, there are two or more matrix transitions that contribute only to that loop, then those transitions will have the same elasticities. See section 9.2.4.3 in Caswell's book, or papers by van Groenendael et al. (1994; Ecology 75:2410) or Wardle (1998; Ecology 79:2539).
This is really a mathematical property of matrices; but if you think of a loop as representing a "contribution to lambda," then it's (somewhat) intuitive that the individual transitions on that loop should have identical elasticities (since the transitions are multiplied in the process of going around the loop, a proportional change of any of them will have the same effect on the loop's contribution to lambda).
I am not, in general, a big fan of loop analysis; but it is conceptually helpful in this case.
Thanks for providing this additional explanation Bruce, I appreciate it.
For comparing life stage sensitivity (in the general sense) within and between species, do you guys thank that elasticities are still considered the best option? Many authors report them, but are there any new techniques that have supplanted them?
Thanks again for your help,
Paul
I don't know of anything that supersedes elasticities and sensitivities. The decision of which of those two is more relevant depends on the type of question you are asking.
One thing to consider, especially if you are asking management questions, is whether you should be looking at responses to survival or mortality. The two parameters will have sensitivities of identical magnitudes, but the elasticities will be very different. It is often easier to see how a potential management action will proportionally impact mortality than how it will impact survival; the focus on mortality reduction will also prevent you from accidentally looking at scenarios that raise survival above 1!
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