Bimodal trait distributions with large variances question the reliability of trait-based aggregate models

Abstract

Functionally diverse communities can adjust their species composition to altered environmental conditions, which may influence food web dynamics. Trait-based aggregate models cope with this complexity by ignoring details about species identities and focusing on their functional characteristics (traits). They describe the temporal changes of the aggregate properties of entire communities, including their total biomasses, mean trait values, and trait variances. The applicability of aggregate models depends on the validity of their underlying assumptions that trait distributions are normal and exhibit small variances. We investigated to what extent this can be expected to work by comparing an innovative model that accounts for the full trait distributions of predator and prey communities to a corresponding aggregate model. We used a food web structure with well-established trade-offs among traits promoting mutual adjustments between prey edibility and predator selectivity in response to selection. We altered the shape of the trade-offs to compare the outcome of the two models under different selection regimes, leading to trait distributions increasingly deviating from normality. Their biomass and trait dynamics agreed very well for stabilizing selection and reasonably well for directional selection, under which different trait values are favored at different times. However, for disruptive selection, the results of the aggregate model strongly deviated from the full trait distribution model that showed bimodal trait distributions with large variances. Hence, the outcome of aggregate models is reliable under ideal conditions but has to be questioned when confronted with more complex selection regimes and trait distributions, which are commonly observed in nature.

Appendices

Appendix A: Derivation of the aggregate model

The FTD model describes the temporal changes in the biomass density of each particular trait value and thus yields, at every moment in time, the biomass densities of the prey and predator communities (or functional groups) as a function of their respective traits ϕ and ω. As a consequence, the magnitude and shape of the biomass-trait distributions may change in time. Instead of resolving the rate of change of the biomass density for each particular trait value, one may directly describe the temporal changes in the aggregate properties of the prey and predator communities using moment approximation methods. In this case, the dynamics of the FTD model is approximated by a corresponding aggregate model which tracks only the temporal changes in the total biomasses of the prey and predator communities and the locations (mean trait values) and widths (trait variances) of the corresponding trait distributions. These aggregate quantities are related to the trait distribution by integrals over the ecologically feasible trait range, in accordance with Eq. 5. In this way, aggregate models explicitly represent the trait-dependent growth and loss terms that determine the dynamics of the aggregate properties of the trait distribution (cf. Tirok et al. 2011).

To derive the aggregate model (6–8) from the FTD model (1–3), one has to write down the rate of change of the aggregate properties using Eqs. 1–3 and expand all functions in Taylor series around the mean trait value, since the average over the distribution of the per-capita net growth rates determines the overall trait and biomass dynamics. This leads to expressions written in terms of (central) moments of the trait distributions. Afterwards, a moment approximation is performed by truncating these series up to the second order. This will be a good approximation when the growth and grazing functions have only weak non-linearities and thus all higher-order derivatives quickly vanish, or when the trait distribution is very narrow, that is, has a small variance. Hence, the error in the biomass and trait dynamics will be most pronounced if both the higher-order derivatives of the fitness landscape, i.e., the per-capita net-growth rate as a function of the trait, evaluated at the mean trait value and the higher-order moments of the trait distribution obtain large absolute values at the same time. For example, if trait distributions with a sufficiently large variance exhibit large values of skewness or kurtosis, the quality of the approximation made by the corresponding aggregate model will be rather poor when the fitness landscape is highly non-linear. This holds since the performance of species with trait values away from the community average $\bar {\phi }$ will likely differ from the performance of a species with a trait value equal to $\bar {\phi }$ (cf. Ruel and Ayres 1999; Tirok et al. 2011).

For sake of brevity, we will derive the prey equations in detail below, as the derivation of the equations for the predator are completely analogous. The Taylor expansion of a function f around the prey’s mean trait value $\bar {\phi }$ is given by:

$$ f(\phi) = \sum\limits_{n=0}^{\infty} \frac{(\phi-\bar{\phi})^{n}}{n!} \frac{d^{n}f(\bar{\phi})}{d\phi^{n}} \,. $$

(13)

Since we are dealing with aggregate quantities such as prey biomass summed over the whole trait range, we will need the expansion of integrals over the prey’s trait distribution A(ϕ), such as:

where M _n is the nth central moment of the distribution A(ϕ) (in particular, M ₂ = ν _ϕ ), defined by

$$ M_{n} = \frac{1}{A_{T}}\int (\phi-\bar{\phi})^{n}A(\phi)d\phi \,. $$

(14)

The rate of change of the total prey biomass A _T is

$$\begin{array}{@{}rcl@{}} \frac{dA_{T}}{dt} &=& \frac{d}{dt} \int A(\phi)d\phi = \int \frac{dA(\phi)}{dt} d\phi\\ &=& \int \left[A(\phi) \tilde{R}(\phi) + I_{A} \right] d\phi, \end{array} $$

(15)

where $\tilde {R}$ is the per capita net growth rate from Eq. 1:

$$ \tilde{R}(\phi) = r(\phi) + B(\phi) - \frac{1}{A(\phi)}\int g(\phi, \omega) C(\omega) d\omega . $$

(16)

Now, the grazing term already contains an integral over $\phi ^{\prime }$, independent of ϕ, so we expand it first:

$$\begin{array}{@{}rcl@{}} &&\frac{1}{A(\phi)} \int g(\phi, \omega) C(\omega) d\omega\\ &=& \frac{g_{m}\, q(\phi, \omega)}{M(\omega) + \int q(\phi^{\prime}, \omega) A(\phi^{\prime})d\phi^{\prime}}\\ &=& \frac{g_{m}\, q(\phi, \omega)}{M(\omega) + A_{T}\left[q(\bar{\phi}, \omega) + \left.\frac{\nu_{\phi}}{2} \frac{\partial^{2}q(\phi,\omega)} {\partial\phi^{2}}\right|_{\phi=\bar{\phi}} + \dots\right]} , \end{array} $$

(17)

and here we drop terms of order higher than two (denoted by the dots), which gives us simply G(ϕ,ω) defined by Eq. 8. We now proceed expanding the integral in terms of ω:

$$\begin{array}{@{}rcl@{}} \tilde{R}(\phi)&=& r(\phi) + B(\phi) - \int G(\phi, \omega) C(\omega) d\omega\\ &=& r(\phi) + B(\phi) \\ &&+ C_{T} \left[ G(\phi, \bar{\omega}) +\left. \frac{\nu_{\omega}}{2}\frac{\partial^{2}G(\phi, \omega)}{\partial\omega^{2}}\right|_{\omega=\bar{\omega}} + \dots\right],\\ \end{array} $$

(18)

and, stopping once again at the second order, we arrive at the expression for R _A (ϕ) (7). Substituting it back into Eq. 15 and expanding:

$$\begin{array}{@{}rcl@{}} \frac{dA_{T}}{dt} &=& \int \left[A(\phi) R_{A}(\phi) + I_{A} \right] d\phi\\ &=& A_{T} \left( R_{A}(\bar{\phi}) + \left.\frac{\nu_{\phi}}{2}\frac{\partial^{2}R_{A}}{\partial \phi^{2}}\right|_{\phi=\bar{\phi}} + {\cdots} \right) + I_{A} , \end{array} $$

(19)

which yields Eq. 6 when we truncate at the second order.

The rate of change of the mean trait value is derived in a similar fashion:

$$\begin{array}{@{}rcl@{}} \frac{d\bar{\phi}}{dt} &=& \frac{1}{A_{T}} \left[\frac{d}{dt}\int \phi A(\phi) d\phi - \frac{1}{A_{T}} \frac{dA_{T}}{dt} \int \phi A(\phi) d\phi\right]\\ &=& \frac{1}{A_{T}} \left[\int \phi \frac{dA(\phi)}{dt} d\phi - \bar{\phi} \frac{dA_{T}}{dt} \right]\\ &=& \frac{1}{A_{T}} \int (\phi-\bar{\phi}) \frac{dA(\phi)}{dt} d\phi\\ &=&\frac{1}{A_{T}} \int (\phi-\bar{\phi}) \left( A(\phi) R_{A}(\phi) + I_{A}\right) d\phi\\ &=& \frac{1}{A_{T}} \int (\phi-\bar{\phi}) A(\phi) R_{A}(\phi) d\phi \\&&+ \frac{1}{A_{T}} \int (\phi-\bar{\phi}) I_{A} d\phi\\ &=&\left( \left.\nu_{\phi} \frac{\partial R_{A}}{\partial\phi}\right|_{\phi=\bar{\phi}} + \left.\frac{M_{3}}{2}\frac{\partial^{2} R_{A}}{\partial\phi^{2}}\right|_{\phi=\bar{\phi}} + \dots\right)\\ &&+ \frac{I_{A}}{A_{T}} \left( \frac{1}{2} - \bar{\phi} \right) . \end{array} $$

(20)

Finally, the trait variance will be expressed by:

$$\begin{array}{@{}rcl@{}} \frac{d\nu_{\phi}}{dt} &=& \frac{1}{A_{T}} \left[\frac{d}{dt}\int (\phi-\bar{\phi})^{2} A(\phi) d\phi \right. \\ && \qquad\left.- \frac{1}{A_{T}} \frac{dA_{T}}{dt} \int (\phi-\bar{\phi})^{2} A(\phi) d\phi\right]\\ &=& \frac{1}{A_{T}} \left[\int (\phi-\bar{\phi})^{2}\, \frac{dA(\phi)}{dt} d\phi - \nu_{\phi} \frac{dA_{T}}{dt} \right]. \end{array} $$

(21)

The second term can be expanded directly using Eq. 19

$$ \frac{\nu_{\phi}}{A_{T}} \frac{dA_{T}}{dt} = \nu_{\phi} \left[\left( R_{A}(\bar{\phi}) + \left.\frac{\nu_{\phi}}{2}\frac{\partial^{2}R_{A}}{\partial\phi^{2}}\right|_{\phi=\bar{\phi}} \,+\, {\cdots} \right) \,+\,\frac{I_{A}}{A_{T}} \right] , $$

(22)

while the first term can be written as:

$$\begin{array}{@{}rcl@{}} && \frac{1}{A_{T}} \int (\phi-\bar{\phi})^{2} \left( A(\phi) R_{A}(\phi) + I_{A} \right) d\phi = \nu_{\phi} R_{A}(\bar{\phi})\\ &&+\left. M_{3} \frac{\partial R_{A}}{\partial\phi}\right|_{\phi=\bar{\phi}} + \left.\frac{M_{4}}{2} \frac{\partial^{2} R_{A}}{\partial\phi^{2}}\right|_{\phi=\bar{\phi}}\\ &&+ \frac{I_{A}}{A_{T}} \left( \frac{1}{3}+\bar{\phi}^{2}-\bar{\phi}\right). \end{array} $$

(23)

Given the fact that the terms proportional to ν _ϕ in Eq. 23 cancel with the ones from Eq. 22 and that the terms proportional to $\nu _{\phi }^{2}$ combine with those from Eq. 22, the rate of change of the trait variance can be approximated by:

$$\begin{array}{@{}rcl@{}} \frac{d\nu_{\phi}}{dt} &=& M_{3} \left.\frac{\partial R_{A}}{\partial\phi}\right|_{\phi=\bar{\phi}} + \frac{\left( M_{4} - {\nu_{\phi}}^{2}\right)}{2} \left.\frac{\partial^{2} R_{A}}{\partial\phi^{2}}\right|_{\phi=\bar{\phi}}\\ &&+ \frac{I_{A}}{A_{T}} \left[\frac{1}{12} - \nu_{\phi} + \left( \frac{1}{2}-\bar{\phi}\right)^{2}\right] . \end{array} $$

(24)

As seen in Eqs. 20 and 24, temporal changes in the aggregate properties (lower-order moments) generally depend on higher-order moments. To derive self-contained expressions for the evolution of the aggregate properties, and thus to close the system of differential equations, one has to describe the higher-order moments in the corresponding equations by lower-order moments. Different moment closure techniques were described in detail by Norberg et al. (2001) and Merico et al. (2009). For instance, Norberg et al. (2001) described the third and fourth central moments by power-functions of the first (mean trait) and second (variance) central moments and the optimal trait value. The parameters were estimated from the trait distributions obtained from simulations of the underlying multispecies model. Another approach is to assume the trait distributions to be well represented by a Gaussian (Wirtz and Eckhardt 1996; Merico et al. 2009; Tirok et al. 2011) for which the higher-order moments are either zero (odd moments) or fully determined by the variance alone (even moments).

Following Wirtz and Eckhardt (1996) and Merico et al. (2009), we assume traits to be normally distributed. Using the fact that the third central moment of a Gaussian distribution equals zero (M ₃=0; the normal distribution is symmetric), and that its fourth central moment is proportional to the square of its variance ($M_{4} = 3 {M_{2}^{2}}$), that is, its excess kurtosis is 0, we can approximate the rate of change of the mean trait value Eq. 20 and of the trait variance Eq. 24 by Eq. 6.

Appendix: B: Model version with mutation

The immigration term in Eqs. 1 and 6 prevents the extinction of species by providing a very small constant influx of immigrants of all trait values. This is generally realistic and necessary to ensure that some variance is retained in the aggregate model, but may in principle strongly influence the dynamics of both types of models. Here, we build versions of the FTD model and the aggregate model without immigration, but with mutation instead. Mutation can be an important process when dealing with narrowly defined functional groups, where the trait variance may originate mostly from intra-populational processes rather than standing trait variance among different species.

Here, mutation is understood as a small probability of offspring to have trait values slightly different then their parents (we do not account for sexual mixing). It is modeled as a (Fokker-Planck) diffusive term, corresponding to the second derivative of the population density times the birth rate with respect to the trait value. In the case of prey, we assign all density-dependent terms to the mortality, and thus their birth rate, ρ _A , is simply the intrinsic growth rate ($r^{\prime }$). The birth rate ρ _C of predators is their grazing rate. This leads to the modified FTD model:

$$\begin{array}{@{}rcl@{}} \frac{\partial A(\phi, t)}{\partial t} &=& \left( r(\phi) + B(\phi)\right) A - \int g(\phi, \omega) C d\omega \\ &&+ \mu_{A} \frac{\partial^{2} \left[\rho_{A} A(\phi)\right]}{\partial \phi^{2}}\\ \frac{\partial C(\omega, t)}{\partial t} &=& \left( e \int g(\phi, \omega)d\phi - d + B(\omega)\right) C\\ &&+ \mu_{C} \frac{\partial^{2} \left[\rho_{C} C(\omega)\right]}{\partial \omega^{2}}\\ \rho_{A}(\phi) &=& r^{\prime}(\phi) \qquad \rho_{C}(\omega) = e \int g(\phi, \omega)d\phi , \end{array} $$

(25)

where μ _A and μ _C are the mutation rates for prey and predator, respectively, and all other terms and parameters are as before. We derive the aggregate model from the FTD model from this equation together with Eq. 5. Due to the boundary functions imposed, the population densities go to zero very quickly outside the range of traits from 0 to 1. Therefore, we assume that A(ϕ), as well as its derivatives, go fast enough to 0 as ϕ goes to (plus or minus) infinity, which guarantees that terms calculated at infinity appearing via integration by parts vanish.

The derivation proceeds exactly as in “AppendixA:Derivation of the aggregate model”. We begin by calculating the rate of change of total biomass, $\frac {dA_{T}}{dt}$. The new term corresponding to mutation in Eq. 15 will be:

$$ \mu_{A} \int \frac{\partial^{2} \left[\rho_{A} A(\phi)\right]}{\partial \phi^{2}} d\phi= \left.\mu_{A}\frac{\partial \left[\rho_{A} A(\phi)\right]}{\partial \phi} \right|_{-\infty}^{\infty} = 0 , $$

(26)

which means that mutation does not contribute any extra term to the total biomass equation. Proceeding to the mean trait equation, $\frac {d\bar {\phi }}{dt}$, we now get the following contribution of the mutation term:

$$\begin{array}{@{}rcl@{}} \frac{\mu_{A}}{A_{T}} \int \phi \frac{\partial^{2} \left[\rho_{A} A(\phi)\right]}{\partial \phi^{2}} d\phi &=&\frac{\mu_{A}}{A_{T}} \left.\left[\phi \frac{\partial \left[\rho_{A} A(\phi)\right]}{\partial \phi} \right] \right|_{-\infty}^{\infty} \\ &&- \frac{\mu_{A}}{A_{T}} \int \frac{\partial \left[\rho_{A} A(\phi)\right]}{\partial \phi} d\phi\\ &=& \left.0 - \frac{\mu_{A}}{A_{T}} \left[\rho_{A} A(\phi)\right] \right|_{-\infty}^{\infty}\\ &=& 0 , \end{array} $$

(27)

and hence mutation does not affect the mean trait value either. Finally, the equation for the rate of change of the trait variance, $\frac {d\nu _{\phi }}{dt}$, has a term due to mutation given by:

$$\begin{array}{@{}rcl@{}} &&\frac{\mu_{A}}{A_{T}} \int \phi^{2} \frac{\partial^{2} \left[\rho_{A} A(\phi)\right]}{\partial \phi^{2}}d\phi = \frac{\mu_{A}}{A_{T}} \left.\left[\phi^{2} \frac{\partial \left[\rho_{A} A(\phi)\right]}{\partial \phi} \right] \right|_{-\infty}^{\infty}\\ &&- 2 \frac{\mu_{A}}{A_{T}} \int \phi \frac{\partial \left[\rho_{A} A(\phi)\right]}{\partial \phi} d\phi \\ &\,=\,&0 - 2 \frac{\mu_{A}}{A_{T}} \left\{\left.{\vphantom{\frac{\mu_{A}}{A_{T}}}}\left[\phi \rho_{A} A(\phi)\right] \right|_{-\infty}^{\infty}- \frac{\mu_{A}}{A_{T}} \int \rho_{A} A(\phi) d\phi \right\}\\ &\,=\,& 2 \frac{\mu_{A}}{A_{T}} \int \rho_{A} A(\phi) d\phi\\ &\,=\,& 2 \mu_{A} \left[\rho_{A}(\bar{\phi}) + \left.\frac{\nu_{\phi}}{2} \frac{\partial^{2} \rho_{A}(\phi)}{\partial \phi^{2}}\right|_{\phi=\bar{\phi}} \,+\, {\dots} \right] . \end{array} $$

(28)

Thus the equation for trait variance becomes:

$$ \frac{d\nu_{\phi}}{dt} = \nu_{\phi}^{2} \left.\frac{\partial^{2} R_{A}}{\partial\phi^{2}}\right|_{\phi=\bar{\phi}} + 2 \mu_{A} \rho_{A}(\bar{\phi}) + \nu_{\phi} \mu_{A} \left.\frac{\partial^{2} \rho_{A}(\phi)}{\partial \phi^{2}}\right|_{\phi=\bar{\phi}} . $$

(29)

We analyzed the resulting dynamics of the FTD model and the aggregate model as described in the methods of the main text. For all selection regimes, Fig. 8 shows very similar patterns for the FTD model with mutation compared to the FTD model with immigration (cf. Fig. 3). In contrast, the aggregate model no longer displayed large oscillations in its mean trait values for intermediate values of β, but high trait variances instead, indicating that even the type of dynamics it generates is not robust to such structural modification. Importantly, immigration influenced the mean trait values directly, pulling toward intermediate values and facilitating recurrent changes in trait values. Even for stabilizing selection, the agreement between the two types of models in the temporal averages of the total biomasses and mean trait values is worse than in the scenario with immigration, since variance levels of the FTD model are higher in the mutation variant (4×10⁻³ compared to 2×10⁻⁵, when β=0.2), leading to worse performance of the aggregate model due to Jensen’s inequality.

Fig. 8

Temporal averages (lines) and variation (± 1 standard deviation; shades) of biomass, mean trait, and standing trait variance of the solutions of the FTD (red) and aggregate (green) models with mutation instead of immigration (25, 29), as the shape of the trade-offs (β) is varied. Mutation rates were set to μ _A =10⁻⁴ and μ _C =2×10⁻⁵, with all other parameters as in Table 1. The dotted lines correspond to alternative long-term solutions, obtained by exploring different initial conditions, leading to distinct basins of attraction

In the absence of immigration, the FTD model no longer exhibited bistability. Also, the transition between states of very different dominant trait values occurred more slowly, leading to longer cycles. Nonetheless, it still presented bimodal trait distributions for a wide range of parameters. For instance, P was above 1.2 (see “Numerical simulations and data analysis” section) during 12 and 68 % of the time for prey and predators, respectively, with β=3, and during 5 and 45 % for β=5. This confirms that bimodal trait distributions were stabilized but not caused by immigration, which is in accordance with findings from Pigolotti et al. (2010).