Technical Comments

Response to Comment on "Extinction Debt and Windows of Conservation Opportunity in the Brazilian Amazon"

Science  18 Jan 2013:
Vol. 339, Issue 6117, pp. 271
DOI: 10.1126/science.1231618


Halley et al. purport to show a power-law relationship between fragment size and relaxation rates. We use a much more extensive data set to show that area dependence of relaxation rates exists only for very small fragment sizes (<60 hectares), which has limited relevance for our analyses conducted using 250,000-hectare grid squares. We also show that the example of Halley et al. is based on an unrealistic fragmentation model with an infinite number of fragments that have average size of zero hectares. A more realistic formulation of the model shows that relaxation is much less dependent on fragmentation than Halley et al. present.

Halley et al. (1) draw the conclusion that relaxation rates are dependent on fragment area raised to an exponent and apply this premise to an idealized landscape of fragments to demonstrate a strong impact on relaxation rates expected within the landscape. Their premise was based on an analysis of 41 observations of relaxation from seven study sites (2), of which a large proportion (40%) are from inferred post-Pleistocene relaxation on islands (3) rather than relaxation from anthropogenic habitat loss. We tested for area effects on k, the relaxation constant, using our much larger data set that includes a total of 374 observations from 53 data sources and which fully subsumes the data of Halley et al. (4). Unlike Halley et al., we controlled for the unbalanced and autocorrelated nature of these data, using a stratified random resampling approach to ensure that data points could be considered to be statistically independent. We also competed additional regression models to explain variation in k (linear, nonlinear power, semilogarithmic, and breakpoint), including some that accounted for broad latitudinal or taxonomic differences in relaxation rate. The power model, which Halley et al. fit to their data, was almost always bettered by a breakpoint model that was ranked as the best model in almost 60% of cases (Table 1). In all cases, this was a special case of the breakpoint model, with the value of k unvarying with fragment area at large scales but exhibiting a positive relationship below a threshold value of < 60 ha (95% CI: 4.0 to 200 ha). We therefore agree with Halley et al. that relaxation rate is empirically dependent on area, but only for very small fragment sizes. We had omitted relaxation rates obtained from fragments <100 ha in our analysis (4), which was based on units of 250,000-ha grid squares. By 2020, even in the worst-case Business as Usual scenario that we modeled (4), just 0.1% of units contained areas of forest less than the threshold for area dependence. Thus, omitting threshold changes in k from our models exerts negligible influence on our final results.

Table 1

Results of model selection by Akaike Information Criterion (corrected for small sample size), and averaged over bootstraps (n = 10,000) stratified by study and taxon, where AICci is the bootstrapped model selection frequency and MDE is the mean proportion of the total deviance explained by the model. Latitude and taxon are unordered factors with the levels tropical and nontropical (latitude) or vertebrate, invertebrate, and plant (taxon).

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Even if the power-law relationship between relaxation and area that Halley et al. describe were valid, unrealistic assumptions in their mathematical model still lead to exaggerated conclusions. Their scenario had habitat of area AT divided into patches such that there are 2j patches of area ρja0 for all nonnegative integers j. In this model, there are infinitely many patches and the average patch size is 0. Using a more realistic model in which area AT is divided into patches such that there are 2j patches of area ρja0 for all j between 0 and n – 1, it is easy to show that E(th)=τ01(2ρβ+1)n12ρβ+112ρ1(2ρ)n. Plotting E(th) against ρ indicates that the limiting case that Halley et al. considered provides an extreme account of the degree to which E(th) depends on the fragmentation measure ρ (Fig. 1A). Large values of n ≥ 20 produce a dependence that is comparable to the limiting case, but these values correspond to a landscape in which more than half the patches have area smaller than or close to 1 m2 (Fig. 1B), which are clearly irrelevant for the vertebrate species we examined (4). Halley et al. provided no guidance on what values of ρ are realistic, so we computed real analogues of ρ in 2° by 2° grid cells across the Brazilian Amazon, obtaining a median of 0.33 and 95% quantiles 0.22 and 0.45. The most extreme fragmentation-dependence of E(th) modeled by Halley et al. occurs at values of ρ that fall outside of this range. So even if Halley et al.’s premise of power-law dependence of relaxation were valid, real systems have ρ and n values that ensure their conclusions about the sensitivity of E(th) to fragmentation have been exaggerated.

Fig. 1

(A) E(th) as a function of ρ for different values of n (colors) and for the limiting case (n → ∞) that Halley et al. considered (black). Tick marks on the horizontal axis show real analogs of ρ estimated in the Brazilian Amazonian. (B) The size of the smallest patches as a function of ρ and n. We here used β = 0.6525, c = 4.3408, and AT = 400 km2.


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