Technical Comments

Response to Comment on "Foraging Adaptation and the Relationship Between Food-Web Complexity and Stability"

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Science  15 Aug 2003:
Vol. 301, Issue 5635, pp. 918
DOI: 10.1126/science.1087539

In their comment (1), Brose et al. analyzed extended models of adaptive food webs and found that, under their alternative assumptions, a positive complexity–stability relationship does not emerge. As they mention, my original model (2) made a number of simplifying assumptions, and its reexamination under alternative assumptions is indeed necessary to check the robustness of the theory, to identify possible countering effects that may obscure the predicted pattern, and to examine the environmental conditions under which inversion is more likely to be observed. However, I disagree with their comment's implication that positive complexity–stability relationships are unlikely to occur in nature.

Brose et al.(1) focused on three aspects of the original model: (i) the cascade and random models used to construct food-web architecture, (ii) the type I response of trophic interaction, and (iii) and the positive intrinsic growth rate for all species. In their model analysis, altering (i) and (ii) does not appear to change qualitatively the result of the earlier analysis (2). The cascade model, when both type II response and the distinction of autotrophs and heterotrophs are included, still predicts positive complexity-stability relationship at high levels of potential connectance (C) and adaptation (F, g) (1).

The main difference between the analysis of Brose et al. (1) and Kondoh (2) is that, in Brose et al.'s model, the inversion is restricted to the higher (> 0.2) region of C. They point out that the C values that result in a positive complexity–stability relationship are “higher than the large majority of empirical values” (1). However, 0.2 < C < 0.45 cannot possibly be regarded as unrealistically high, because C in the adaptive food-web model represents “potential connectance” (2), a fraction of any possible links, and may thus also include many nonrealized links. Because adaptive foragers consume only a small fraction of their potential resource species (2), high potential connectance does not necessarily imply high realized connectance. For example, in the original model (2), realized connectance (L/S2) is lower than 0.132 even if C has its maximum value (0.45) in the adaptive cascade model food web with 10 species. This value is not unrealistic (3), although from the existing empirical data it is still unclear how realized connectance of real food webs depends on time scale in nature.

Aside from asking which network structure is the more “realistic,” it is an important finding that adaptation may not invert complexity–stability relationships of food webs with certain network structures or dynamics. Comparison of complexity–stability relationships of adaptive food webs with different basic frameworks provides an interesting opportunity to identify the precise factors that influence the inversion. However, although the models presented by Brose et al. (1) incorporate biological reality into the original model (2), the result does not necessarily imply that increasing biological reality reverses or weakens the positive complexity–stability relationship. I propose that, by changing C, Brose et al. (1) have incorporated a “hidden treatment” that causes the reversal of the positive relationship.

It is unclear how producer species fraction is determined in the models of Brose et al. (1). If the fraction of producer changes with increasing C, then it would influence the system primary productivity and would mask the real complexity–stability relationship (2). To separate the complexity effect from this producer fraction effect, I reanalyzed the Brose et al. cascade model, keeping the number of producer species constant. In contrast to the results presented by Brose et al. (1), the reanalysis shows that, with a sufficiently high adaptive rate, the complexity–stability relationship is positive for the entire range of C (Fig. 1). Further, decreasing producers appears to destabilize food webs at low C. Presumably, in the cascade model presented by Brose et al. (1), C is increased in a way that decreases the fraction of producer species, and this restricts the range of C in which the positive relationship occurs. Because the fraction of producer species decreases with increasing C in the niche model (4), it is likely to contribute to the low stability at high connectance. The negative complexity–stability relationship in the Brose et al. model (1) is probably due not to the inclusion of biological reality, but to the decreasing fraction of producer species with increasing C.

Fig. 1.

Relationship between connectance and robustness in cascade model food webs with (A, C, and E) different fractions of adaptive consumer, F, or (B, D, and F) different adaptation rates, g. The number of producer species, B, is constant, at B = 1 (A and B), B = 2 (C and D) and B = 3(E and F). All nonproducer species have at least one resource; producer species have no resources; each of the other potential links is connected with probability c. Connectance (L/S2) of a food web with 10 species and connection probability c is given by [(10-B){c(B+7)+2}/200]. Parameters are g = 0.25 in (A), (C), and (E) and F = 1 in (B), (D), and (F) r = 1.0. The other parameters, model settings, and analysis procedure are the same as used by Brose et al. (1).

I also dispute the contention by Brose et al. (1) that my hypothesis is flawed because it does not hold in the niche model (although they did not provide a complexity-stability relationship for C > 0.45) (1). The niche model is known to predict some food web characteristics for a given species number and connectance (5). That is why Brose et al. (1) suppose that the niche model is a more realistic assumption. However, I would argue that a niche model that incorporates adaptation is unlikely to generate food web structures as realistic as the non-adaptive niche model (5) and that it thus may not serve as an appropriate benchmark. That is because with foraging adaptation, a network would be reconstructed to maximize the energy gain of each species depending on evolutionary constraints such as foraging efficiency or the prey's palatability. Realized connectance should be lower than potential connectance, and distribution of trophic links would be biased by biomass distribution pattern over trophic systems, which is likely to conflict the basic assumption of niche model (5).

In theoretical food web studies that test the connectance–stability relationship, the primary “treatment” is the density of trophic links within a web. However, changing link density is usually accompanied by simultaneous changes in a number of food web properties (for example, chain length, loop number, cannibalism, or fraction of producer species), each of which has qualitatively or quantitatively different impacts on food web stability. Evaluation of the connectance–stability relationship should therefore be carried out with sufficient clarification of which consequences of changing “complexity” we are focusing upon and also which properties are kept unchanged for this purpose. If the food web property is carefully controlled, comparison of food web models with different predictions can be a powerful method to identify the key factors behind the predicted patterns, and would tell us in which condition the patterns should emerge and can be tested in nature.

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