A consistent equation for ecological sensitivity in matrix population analysis

CORRESPONDENCE

A consistent equation for ecological sensitivity in matrix population analysis Matrix population analysis is now accepted widely as an important tool in ecological studies, as reaffirmed in the excellent recent TREE review by Benton and Grant1. The controversy over whether elasticity or sensitivity is a better measure of the effect of a change in one vital rate on a population’s growth rate, has been largely settled by accepting that each measure provides an accurate answer to a different question2,3. Nevertheless, Benton and Grant’s reaffirmation inadvertently perpetuates a commonly used, but inconsistent, equation for sensitivity. Specifically, Benton and Grant – as well as previous authors2,3 – assert that the sensitivity matrix is invariably equal to the outer product of the dominant left and right eigenvectors of the population projection matrix (Box 1). Unfortunately, in general, this equation is not consistent with the life cycle graph that generates the projection matrix: it sometimes implies flows along nonexistent arcs. For example, Benton and Grant1 decompose the life cycle of the killer whale, Orcinus orca, into three stages: yearlings (stage 1), juveniles or young adults (stage 2), and reproductive adults (stage 3). On using the inconsistent equation, they infer that the sensitivities with respect to paths from stage 1 to itself or stage 3, and from stage 3 to stage 2 are all positive, which implies that the population growth rate is affected by the rate per period at which yearlings reproduce or develop directly into reproductive adults and at which reproductive adults revert to juveniles. But each of these transitions is assumed impossible, because the life cycle graph contains arcs neither from node 1 to itself or node 3, nor from node 3 to node 2; in other words, because the corresponding terms in the projection matrix all are assumed to be identically zero. Why is an equation that is so obviously ecologically inconsistent, believed to be a mathematical consequence of the definition of sensitivity as a partial derivative of growth rate (Box 1)? The answer, I suspect, is that the indirect method usually used to obtain this equation2 obscures the fact that it is just as inconsistent mathematically as ecologically. Therefore, in establishing the consistent equation (Box 1) – according to which growth rate cannot be affected by flow along nonexistent arcs – I use an alternative method that has the virtue of being direct. However, regardless of whether one uses the inconsistent or the consistent equation for sensitivity, one obtains the same equation for elasticity, which automatically eliminates any flows along nonexistent arcs. This corrective feature is suggested sometimes, if only implicitly (e.g. by Horvitz et al.3), as a partial justification for preferring elasticity to sensitivity as an impact measure in TREE vol. 15, no. 3 March 2000

Box 1. A consistent equation for sensitivity Let A be the n 3 n projection matrix for the population of interest and l its growth rate, so that sensitivity sij and elasticity eij of l with respect to aij are defined by: sij =

eij =

∂l ∂aij

(1)

aij sij

(2)

l

Then, it is stated by Benton and Grant1, and others2,3, that: sij = vi w j

(3)

where v and w, respectively, are the left and right eigenvectors of A, associated with the dominant eigenvalue l; in other words, where: n

n

n

∑ vkakm = lvm

∑ akmwm = lwk

k=1

∑ vkwk = 1

(4)

k=1

m=1

Here, n is the number of life-cycle stages; k and m are arbitrary stage indices. But (3) is not in general correct, because (4) implies:

l=

n

n

∑ ∑ akmvkwm

(5)

k=1 m=1

and ∂l/∂aij ­ 0 if aij ­ 0, because l is then independent of aij. For aij ± 0, however: n n n n n n ∂l ∂v ∂wm ∂a = ∑ ∑ km vkwm + ∑ ∑ akm k wm + ∑ ∑ akmvk ∂aij k=1 m=1 ∂aij ∂aij ∂aij k=1 m=1 k=1 m=1

(6)

In view of (4), the second and third terms of (6) reduce to: n  n ∂v  ∂ ∂w l ∑ k wk + ∑ m vm  = l ∂a ∂a  k=1 ∂aij  ij ij m=1

n

∑ vkwk = 0

(7)

k=1

Furthermore, ∂akm 1 if i = k AND j = m = ∂aij 0 otherwise

(8)

Thus, it follows from (1), (6) and (7) that: vi w j if aij ≡/ 0 sij =   0 if aij ≡ 0

(9)

The key point is that, if the life cycle graph has only q arcs, then the dominant eigenvalue, defined implicitly by A2l I50 (where I is the identity matrix), is a function of only q parameters, and thus only q of its partial derivatives can be non-zero.

ecological (as opposed to evolutionary) studies. But, no corrective action is necessary in the first place if one uses the equation for sensitivity that actually is implied by the model that generates the population projection matrix (Box 1). Use of this equation in future ecological studies will mean that the relative merits of elasticity versus sensitivity can be decided solely according to whether it is more appropriate, for the purpose at hand, to use a relative or an absolute measure of the impact of vital rates on population growth rate.

Michael Mesterton-Gibbons Dept of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA ([email protected])

References 1 Benton, T.G. and Grant, A. (1999) Elasticity analysis as an important tool in evolutionary and population ecology. Trends Ecol. Evol. 14, 467–471 2 Caswell, H. (1997) Matrix methods for population analysis. In Structuredpopulation Models in Marine, Terrestrial, and Freshwater Systems (Tuljapurkar, S. and Caswell, H., eds), pp. 19–58, Chapman & Hall 3 Horvitz, C. et al. (1997) The relative ‘importance’ of life-history stages to population growth: prospective and retrospective analyses. In Structuredpopulation Models in Marine, Terrestrial, and Freshwater Systems (Tuljapurkar, S. and Caswell, H., eds), pp. 247–271, Chapman & Hall

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