Mark Rees and Stephen P. Ellner. 2009. Integral projection models for
populations in temporally varying environments. Ecological Monographs 79
:575–594.
Appendix C. Derivation of the small variance approximation.
Approximating λS for kernels with small variability requires little more than translating
matrix model derivations into operator notation. For notational consistency with those (Caswell
(2001), Tuljapurkar and Haridas (2006)) define
H t = Kt − K , M t = H t / λ1 , R = K / λ1 .
Note that R m → P0 , the projection onto the subspace of L2 spanned by w (Ellner and Rees 2006,
Appendix C) given by P0 n = v, n w . The asymptotic rate of convergence is given by the
damping ratio of K, the ratio between the second-largest and largest elements in the spectrum of
K, which is well defined because K is compact as on operator on L2 .
We assume that H t is order ε 1 and expand log λs to second order in ε . Picking up at
Eq. (B4) in Tuljapurkar and Haridas (2006),
log λS ≈ log λ1 + lim 1t E ⎡⎣ S 2t − 12 S12t ⎤⎦ , where
t →∞
t
t −1 t − j
j =1
j =1 l =1
S1t = ∑ v, M j w , S 2t = ∑∑ v, M j +l R l −1M j w .
(C.1)
(C.2)
Still following Tuljapurkar and Haridas (2006), we separate (C.1) into two sums depending on
environmental variance and covariance. Expand
t
t
t− j
S12t = ∑ v, M j w + 2∑∑ v, M j +l w v, M j w .
j =1
2
(C.3)
j =1 l =1
Writing out v, M j +l w v, M j w in terms of integrals and doing all but the left-most (the one to
compute v, M j +l w ) gives
v, M j +l w v, M j w = v, M j +l P0 M j w .
(C.4)
Then substituting (C.4) into (C.3),
t −1 t − j
t −1 t − j
j =1 l =1
j =1 l =1
t
S 2t − 12 S12t = ∑∑ v, M j +l R l −1M j w − ∑∑ v, M j +l P0 M j w − 12 ∑ v, M j w
2
j =1
t −1 t − j
t
j =1 l =1
j =1
= ∑∑ v, M j +l ( R l −1 − P0 ) M j w − 12 ∑ v, M j w
2
(C.5)
= W2t + W1t .
Next we need to compute the limiting expectations in equation (C.1). W1t is straightforward.
Each term has the same expectation: E v, M j w
2
= Var v, M j w = Var v, K j w / λ12 , because
E v, M j w = 0 . So the contribution of W1t to the right-hand side of (C.1) is the familiar
−Var v, K t w / 2λ12 , which is written as −τ 2 / 2λ12 in equation 14.65 in Caswell (2001).
Terms in W2t with a given value of l correspond to perturbations separated by l years
( l ≥ 1 ), and each of these has the same expectation. Define D = R − P0 . Using the properties of
the projection operators P0 and Q0 = I − P0 (Ellner and Rees 2006, Appendix C) we have
D m = R m − P0 for m ≥ 1 . In a serious abuse of notation write D 0 = I − P0 . Then any term in W2t
with an l-year separation has expectation
cl = E v, M l Dl −1M 0 w .
(C.6)
The convergence of R m to P0 implies that D m → 0 at an asympotically geometric rate bounded
by the damping ratio for K, so (C.6) implies an upper bound cl ≤ C ρ l where 0 < ρ < 1 is the
damping ratio.
Up to time t, there are t-1 terms with a 1-year separation, t-2 terms with a 2-year
separation, etc. We therefore have
EW2t = t ( c1 + c2 + " ct ) − ( c1 + 2c2 + 3c3 + " (t − 1)ct −1 ) .
(C.7)
The second term on the right-hand side of (C.7) is bounded in magnitude by
∞
∞
k =1
j =1
C ∑ k ρ k = C (1 − ρ ) −2 . So lim 1t EW2t = ∑ c j , the sum being absolutely convergent and therefore
t →∞
convergent. Combining all the pieces, the small-variance approximation is then
log λS ≈ log λ1 −
∞
Var v, K t w
+
cj ,
∑
2λ12
j =1
(C.8)
with the three terms on the right-hand side representing the average environment, the effect of
variance, and a sum of effects for autocorrelations at lag j.
LITERATURE CITED
Caswell, H. 2001. Matrix population models. Construction, analysis and interpretation. Second
edition. Sinauer Associates, Sunderland, Massachusetts, USA.
Ellner, S. P., and M. Rees. 2006. Integral projection models for species with complex
demography. American Naturalist 167:410–428.
Tuljapurkar, S., and C. V. Haridas. 2006. Temporal autocorrelation and stochastic population
growth. Ecology Letters 9:324–334.