Corresponding author: ApostolosManuel Koussoroplis ( amanuel.koussoroplis@uca.fr ) Academic editor: Michael Rostás
© 2019 ApostolosManuel Koussoroplis, Toni Klauschies, Sylvain Pincebourde, David Giron, Alexander Wacker.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Koussoroplis AM, Klauschies T, Pincebourde S, Giron D, Wacker A (2019) A comment on “Variability in plant nutrients reduces insect herbivore performance”. Rethinking Ecology 4: 7987. https://doi.org/10.3897/rethinkingecology.4.32252

In their recent contribution, Wetzel et al. [
Plant trait variance can have important consequences for herbivore populations in natural systems via nonlinear averaging effects of herbivore performance (
Using an analogy of the Hedge’s d metric (
We argue that this study may have underestimated the potential of plant defense trait variance to generate Jensen’s effects. The conclusion is based on the average shape of the relationship between herbivore performance and plant defenses, i.e. . This approach seems to be valid when considering a group of random datasets with no true nonlinearities (Fig.
An illustration of the “fallacy of the averaged shape”. Mixtures of randomly generated quasilinear (A) or strongly nonlinear concaveup (blue) and concavedown (red) (B) relationships between herbivore performance and plant defense level. C, D Mean (open and closed black dots) and standard deviation (black bars) of the effect size of Jensen’s Inequality based on the individual values (colored dotes) of the corresponding relationships shown in A and B.
The choice of the most appropriate way to report and analyze such data depends on prior physiological knowledge. Growth limitation by macro or micronutrients involves similar physiological mechanisms: Anabolic rate limitation when nutrient supply is limiting (
Another important aspect that is not considered in the study is the possibility for interactive effects (i.e. synergy or antagonism) between traits which can also generate nonlinear averaging effects on insect performance (
In conclusion, we consider Wetzel et al. (
An illustration of the covariance effects. Plant nutrientdefense (A) or defensedefense (B) covariance effects on herbivore performance in the cases of a hypothetical antagonistic effect between the plant nutrient and chemical defense concentrations and two synergistically acting defense traits, respectively. When herbivores feed on a plant population with both nutritive and defense variance (the two types of plants on the xaxis), the Jensen’s effect on performance, , depends on whether the covariance between the traits is positive (blue), negative (red) or null (purple). See methods for procedure.
In the presence of truly concaveup and concavedown relationships (such as in Fig.
To test this approach, we compared two different estimates of the average effect size of Jensen’s Inequality based on a linear versus strongly nonlinear relationship between herbivore performance and plant defense level, by including both a set of concaveup and concavedown curvatures.
We used a general monotonic function to describe the impact of a plant defense trait x on the performance, i.e. maximum growth rate F of an herbivore:
The parameter n determines the shape of the herbivores’ functional response to plant defense level. The functional relationship is concaveup for n > 1 and concavedown for n < 1. In contrast, the functional relationship will be approximately linear for n≈1. To compare the two different estimates of an average effect size of Jensen’s Inequality under two different conditions we first generated a set of weakly nonlinear (~linear) functions, by drawing 1000 values of n from a lognormal distribution with µ equal to zero and σ equal to 0.1. Hence, in this case, randomly drawn values of n will be often very similar to 1. Afterwards, we generated a set of strongly nonlinear concaveup and concavedown functions, by drawing 500 values of n from a lognormal distribution with µ equal to one and σ equal to 0.1 (concaveup) and 500 values of n from a lognormal distribution with µ equal to minus one and σ equal to 0.1. In this case, randomly drawn values of n will often diverge strongly from 1.
We then calculated the average of the herbivore’s performance in the absence F (x‒) and in the presence of variation in the plant defense level. For the latter, we assumed the trait distribution to follow a beta distribution with mean equal to 0.5 and variance equal to 0.05. In line with the function defined above, the upper and lower limits of the plant defense trait were set to 1 and 0. Then, we calculated the average of the raw differences, i.e. and of the absolute differences, i.e. . The corresponding results are shown in Figure
We illustrate the nonlinear averaging effects on performance caused by the covariance of two interacting factors through two examples. In the first example, we consider a hypothetical interaction between a plant limiting nutrient and chemical defense concentrations (i.e. a secondary metabolite). We assume a typical response to nutrients that follows Bertrand’s rule (
F (N,D) = ND^{2} + bN + c (2)
where N and D are nutrient and chemical defense concentrations, respectively. b and c are arbitrarily chosen constants. In this model, the toxicity of the defense increases proportionally to the deviation of the plant nutrient concentration from the optimum, a commonly observed pattern (
where are the partial or crosspartial second order derivatives of F which quantify the various nonlinearities of the herbivore’s reaction norms to the two factors. Note that the term quantifies the nonadditivity of the interaction between the two factors. When the two factors act in synergy, then >0, whereas <0 when the two factors act antagonistically on performance. In the case of an additive effect of the two factors = 0 and the covariance effect vanishes. In our example, the herbivore response to defense is linear, F^{"}_{D} = 0 and equation (3) simplifies to
In the second example, we consider a case of two cooccurring chemical defenses. Their interactive effect on herbivore performance is modeled as
F (D_{1},D_{2}) = D_{1}D_{2} + c (5)
where D_{i} are the concentrations in the two defensive chemicals in plant tissues and c an arbitrarily chosen constant. The Jensen’s effect can be approximated using eq. 3. However, we assume that the herbivore performance relates linearly to each chemical alone, i.e. F^{"}_{D}_{1} = 0 and F^{"}_{D}_{2} = 0. So the equation collapses to:
The equations 4 and 6 demonstrate that even when the relationship between herbivore performance and plant defense traits is linear, defense variance can affect herbivore performance. This situation is realized when (1) defense covaries with another limiting plant nutritional or defensive trait and (2) the two covarying traits interactively affect herbivore performance. A simple graphical illustration of this phenomenon is provided in Figure
AMK and TK conceived and wrote the manuscript. TK performed the analyses for Fig.
Authors  Contribution  ACI 

AMK  0.45  3.273 
TK  0.40  2.667 
SP  0.05  0.211 
DG  0.05  0.211 
AW  0.05  0.211 
We thank the editor M.Rostás and the anonymous reviewer for their constructive comments on a previous version of the manuscript.