Estimating the average effect size of Jensen’s Inequality
In the presence of truly concave-up and concave-down relationships (such as in Fig. 1B), the average effect size of Jensen’s Inequality might be strongly underestimated when based on raw deviations, i.e. naive differences or log-response ratios between the functional mean and the functional value at the mean, because positive and negative effect sizes may cancel each other. In this case, a robust estimate has to account for potential changes in the sign of the effect of Jensen’s Inequality due to different curvatures, i.e. a mixture of concave-up or concave-down shapes. Hence, the average effect size of Jensen’s Inequality might be more reliably estimated from the absolute deviations (note that the use of absolute means does not allow to estimate the direction of the effect size).

To test this approach, we compared two different estimates of the average effect size of Jensen’s Inequality based on a linear versus strongly non-linear relationship between herbivore performance and plant defense level, by including both a set of concave-up and concave-down 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:

(1)

The parameter n determines the shape of the herbivores’ functional response to plant defense level. The functional relationship is concave-up for n > 1 and concave-down 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 non-linear (~linear) functions, by drawing 1000 values of n from a log-normal 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 non-linear concave-up and concave-down functions, by drawing 500 values of n from a log-normal distribution with µ equal to one and σ equal to 0.1 (concave-up) and 500 values of n from a log-normal 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 1.

Covariance effects
We illustrate the non-linear 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 (Raubenheimer et al. 2005) and we model it using a quadratic function. For simplicity, we assume that the relationship between performance and chemical defense concentration is linear. The herbivore performance is modeled as:

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 (Simpson and Raubenheimer 2001; Tao and Hunter 2012; Hunter 2016a). Under certain assumptions (see (Koussoroplis et al. 2017)), the Jensen’s effect can be approximated by

(3)

where are the partial or cross-partial second order derivatives of F which quantify the various non-linearities of the herbivore’s reaction norms to the two factors. Note that the term quantifies the non-additivity 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

(4)

In the second example, we consider a case of two co-occurring 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:

(6)

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 2.