# Ratio-dependent predation in a field experiment with wasps

Thierry Spataro, Sven Bacher, Louis-Félix Bersier, Roger Arditi

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## To cite this version:

Thierry Spataro, Sven Bacher, Louis-Félix Bersier, Roger Arditi. Ratio-dependent predation in a field experiment with wasps. Ecosphere, 2012, 3 (12), pp.1-12. 10.1890/ES12-00133.1. hal-01566981

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# Ratio-dependent predation in a field experiment with wasps

THIERRY SPATARO,<fnref n="1" />,<fnref n="2" />,<fnref n="3" />,<fnref n="4" />,† SVEN BACHER,<fnref n="5" /> LOUIS-FÉLIX BERSIER,<fnref n="5" /> AND ROGER ARDITI<fnref n="1" />,<fnref n="2" />,<fnref n="3" />,<fnref n="4" />

**Citation:** Spataro, T., S. Bacher, L.-F. Bersier, and R. Arditi. 2012. Ratio-dependent predation in a field experiment with wasps. Ecosphere 3(12):124. http://dx.doi.org/10.1890/ES12-00133.1

**Abstract.** The functional response is a key component of trophic interactions since it quantifies the per capita rate of prey consumption. Determining whether this rate depends on the prey density only (which is the standard assumption), on both prey and predator densities, or simply on their ratio is essential to understand interacting populations. Several experiments have convincingly demonstrated ratio dependence but, with very few exceptions, they were conducted in laboratory conditions. The difficulty of collecting the required data (initial prey density, prey consumption, predator density) probably explains the lack of evidence from functional responses observed in natural systems. A field experiment was previously conducted with a paper wasp and its prey, shield beetle larvae. Both densities were manipulated and the prey consumption was measured. A first analysis led to the conclusion that the functional response of the wasps depended on both prey and predator densities but could not be considered being ratio dependent.

Here, we perform an improved analysis of these data, making better justified assumptions and using more appropriate statistical methods. We fit several functional response models to the data and select the best one with information-theoretic criteria. We also estimate the model parameters and their confidence intervals. This more reliable analysis significantly modifies the original conclusion. Both model selection and parameter estimation indicate that ratio dependence governs the functional response of paper wasps preying on shield beetles in the field. Therefore, ratio dependence is not a laboratory artefact and should be more systematically considered as a potential model for describing functional responses.

**Key words:** *Cassida rubiginosa*; field experiment; functional response; model selection; mutual interference; *Polistes dominulus*; predation; predator dependence; ratio dependence.

**Received** 5 May 2012; revised 19 September 2012; accepted 11 October 2012; final version received 28 November 2012; published 28 December 2012. Corresponding Editor: J. Reeve.

**Copyright:** © 2012 Spataro et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits restricted use, distribution, and reproduction in any medium, provided the original author and sources are credited.

† **E-mail:** thierry.spataro@agroparistech.fr

## INTRODUCTION

The functional response *g* (the instantaneous rate of prey consumption by an average predator) is a key factor to understand the functioning of natural predator-prey systems. The functional

response had originally been defined as a function of the prey density only [*g* = *g*(*N*)] but many studies have shown that the predator density also influences the individual predation rate [*g* = *g*(*N*, *P*)]. This is known as “predator dependence” (Arditi and Ginzburg 1989). From

<footnote n="1">¹INRA, USC Écologie des populations et communautés, 7 quai Saint Bernard, case 237, 75252 Paris cedex 05 France</footnote>

<footnote n="2">²Université Pierre et Marie Curie, Sorbonne Universités, UMR Écologie et Évolution, case 237, 7 quai Saint Bernard, 75252 Paris cedex 05 France</footnote>

<footnote n="3">³CNRS, UMR Écologie et Évolution, 7 quai Saint Bernard, case 237, 75252 Paris cedex 05 France</footnote>

<footnote n="4">⁴AgroParisTech, 16 rue Claude Bernard, 75231 Paris cedex 05 France</footnote>

<footnote n="5">⁵Department of Biology, Unit of Ecology and Evolution, University of Fribourg, chemin du Musée 10, CH-1700 Fribourg, Switzerland</footnote>
an ecological point of view, the question of the
way in which the functional response depends on
the predator density is critical because this
dependence plays a key role for population
regulation, system stability, and community
structuring (e.g., Arditi and Ginzburg 1989,
Akçakaya et al. 1995, Hassell 2000, Rall et al.
2008). It has been proposed that the functional
response is likely to depend approximately on
both prey and predator densities simply via their
ratio [g(N, P) = g(N/P)] (Getz 1984, Arditi and
Ginzburg 1989). While the existence of predator
dependence is now widely accepted (e.g., Begon
et al. 2006, Gotelli 2008, Krebs 2009), a number of
authors consider ratio dependence as unlikely to
take place in natural systems or complain for a
lack of mechanistic explanations. In their view,
both prey density and predator density affect the
functional response independently [g = g(N, P)].
Mathematically, this implies one more variable
and a higher-dimensional parametric space. In
contrast, ratio dependence is of the same dimen-
sional complexity as the original prey-dependent
view. Some aspects of this debate were discussed
by Abrams and Ginzburg (2000) and the whole
theory of ratio dependence is treated compre-
hensively in a recent book by Arditi and
Ginzburg (2012), in which Chapter 5 reviews
the controversy. Moreover, many mechanisms
have been identified as leading to the emergence
of ratio dependence (see Chapter 4 of Arditi and
Ginzburg 2012).

The fact that the functional response is an
instantaneous concept prevents any immediate
observation of it. However, it can be calculated
from the direct observation of the number of prey
eaten over a finite period of time. Following
Holling (1959), hundreds of experiments have
measured the dependence of the functional
response on the prey density N. Several dozens
have measured the dependence on the predator
density P, a dependence known as 'interference'
(see, e.g., the reviews of Hassell 1978, 2000 and
DeLong and Vasseur 2011). However, surpris-
ingly few experimental studies have studied the
bivariate dependence on (N, P), which is clearly
what is needed in order to confirm or refute prey
dependence, predator dependence or ratio de-
pendence. The few existing studies deal with
various systems like microbial systems (Cabrera
F, 2011), aquatic invertebrates in microcosms

(Hansson et al. 2001, Kratina et al. 2009),
arthopod predators (Arditi and Akçakaya 1990,
Reeve 1997, Schenk et al. 2005, Hauzy et al. 2010),
insect parasitoids (Arditi and Akçakaya 1990,
Mills and Lacan 2004), big mammals (Owen-
Smith 2002, Vucetich et al. 2002, Jost et al. 2005).
Assessing the impact of prey and predator
densities on the functional response is not a
straightforward operation in natural systems.
Therefore, most experiments have been conduct-
ed in artificial or semi-natural settings. This is
really problematic because a classic explanation
of ratio dependence rests on spatial heterogeneity
and spatial behaviours of prey and predators
(Arditi and Isaiah 1992, Akçakaya et al. 1995,
Poggiale et al. 1998, Cosner et al. 1999, Arditi et
al. 2001, Tyutyunov et al. 2008). These features
are difficult to reproduce in non-natural settings.
Evidence from natural systems generally rests on
indirect observations used to infer the functional
response, for example from positive correlations
between the abundances of consecutive trophic
levels (Ginzburg and Akçakaya 1992) or from
time series (Jost and Arditi 2000, 2001).

In this article, we re-analyze a unique exper-
iment in which direct measurements of the
functional response were performed in a natural
setting by Schenk et al. (2005). They manipulated
in the field the densities of a predator, the paper
wasp *Polistes dominulus*, and its prey, shield
beetle larvae (*Cassida rubiginosa*), and monitored
the prey individually with video equipment. The
analysis of their observed data led these authors
to conclude that the predator functional response
was neither purely prey dependent nor purely
ratio dependent but intermediate (Schenk et al.
2005). We later realized that the methods used for
organizing the data, for parameter estimation,
and for model selection could all be improved in
significant ways, and we suspected that more
reliable results obtained from the same observa-
tions could lead to different conclusions.

A first improvement relates to the predator
densities used for the analyses. Schenk et al.
(2005) assumed a constant daily number for the
abundance of active predators, which was
calculated as the sum of the numbers of
predators present during each of the activity
hours. Three ways for refining the determination
of predator densities are explored in the present
study. In contrast with Schenk et al. (2005) who
determined the hourly numbers of predators in different ways in the different activity hours, we use a homogeneous approach that will be explained later. Then, we use the number of predators being present instead of the cumulated one as in Schenk et al. (2005), to be consistent with the definition of the functional response as a function of the “instantaneous” numbers of prey and predators present at a given time. Finally, we take account of the variations in the numbers of active predators in the course of the day rather than using a constant daily average, in order to make full use of the rich information that had been collected.

For computational reasons, Schenk et al. (2005) did not estimate all functional response parameters from the same data set: while the interference coefficient *m* was estimated with the 2005 data, the handling time and the searching efficiency were estimated from the previous experiment of Schenk and Bacher (2002) conducted at the same location. (This previous experiment observed the response to prey density only; the predator density was not varied.) However, the environmental conditions that prevailed during the 2002 experiments were quite different and it is very likely that the parameter values (particularly the searching efficiency) were not the same. In the present article, we estimate all parameters simultaneously from the same data set, avoiding this potential bias. This is the second improvement of the present paper.

A third major improvement is in the model selection procedure. Instead of relying entirely on the estimation of the mutual interference coefficient *m*, we fit a family of different functional response models, including the one used by Schenk et al. (2005). We then use information-theoretic criteria to compare the different models and select the best one (Burnham and Anderson 1998).

As will be seen, this improved analysis lead to the very clear conclusion that this predator-prey system follows a ratio-dependent functional response, rather than the previous conclusion of an “intermediate” degree of predator interference that was reached in 2005. In the context of the debate over the validity of the ratio-dependent hypothesis, we consider it necessary to publish the revised conclusion obtained from the same

data with improved analytical methods.

## MATERIAL AND METHODS

### Experimental data

Full details of the experimental setup are available in Schenk and Bacher (2002) and Schenk et al. (2005). Schenk and Bacher (2002) measured the predation rates of wasps on prey larvae presented at various densities. These prey densities *N* were set by removing all larvae occurring naturally in the experimental patch and distributing uniformly a given number of larvae reared in the lab. The naturally occurring predator density, i.e., the number of wasps observed foraging at the experimental site during one hour of the peak wasp activity, was approximately constant, uncorrelated to the prey density. Thus, this was a standard measurement of the response to prey density alone. This first experiment led the authors to the conclusion that the predator *P. dominulus* had a type-III functional response. Fitting the following classic type-III model,

$$g(N) = \frac{bN^2}{1 + bhN^2} \quad (1)$$

they estimated its two parameters *b* (the proportionality factor between predator density and searching efficiency) and *h* (the handling time).

Although the predator abundance was treated as a constant number across various prey densities, the authors monitored the variations of wasp activity during the day from the video recordings of predation events. We use the latter data and similar observations obtained in later years (Tschanz et al. 2007) to fit a gamma distribution in order to establish a predator activity profile during the day (Fig. 1). The consequences of these activity variations are studied in the present paper (see below).

Using basically the same experimental setup, Schenk et al. (2005) tested the influence of predator density in addition to the influence of prey density. For this, they conducted more complex experiments in which they manipulated not only the prey density, but also that of the predators by opening cages containing wasp nests in the immediate vicinity of the prey. They combined eight prey densities (*N* = 2, 4, 6, 8, 10, 20, 30, 40) with four numbers of predator cages (*c*
<figure><img src="image_3.png" /><figcaption>Fig. 1. Activity profile of predators during a day. This profile is obtained by fitting a gamma distribution (stars) to data (bars). The data correspond to the cumulated numbers of wasps preying on *C. rubiginosa* larvae at different day times in 2000, 2002, 2004, as recorded by video monitoring. The $p_i$ values correspond to the relative activity in comparison with the 13–14 h value, which is set to 1. The hourly predator numbers during the day ($P_i$) are obtained by multiplying the $p_i$ by $M_c$, the mean number of wasps caught during the 13–14 h slot ($M_0 = 1.75$, $M_{1,2,3} = 3.708$).</figcaption></figure>

= 0, 1, 2, 3) in a factorial design. Each of the 32 experimental bivariate combinations was conducted on a different day. The numbers of prey being consumed were measured from the counts of the prey remaining at the end of the day (Fig. 2).

The authors assessed the numbers of active predators foraging at the experimental site by sweep netting during one hour (13 h–14 h). By combining these observations with the gamma distribution describing the daily predator activity profile, effective hourly estimates of wasp numbers ($P_i$) can be reconstructed (Fig. 1). These hourly predator numbers differ from those used by Schenk et al. (2005) for three reasons. First, the activity profile in the current study calls on a gamma distribution fitted to data that include several years of observation (instead of a single year) and is extended upon ten time intervals (instead of eight). Second, according to the observations of Schenk et al. (2005) the number of predators caught during the 13 h–14 h hour ($M_c$) is not significantly different with $c = 1, 2$, or 3 open cages ($M_1 = 4.375, M_2 = 3.625, M_3 = 3.125$; Kruskal-Wallis test, $p = 0.89$). Therefore, these three cases are merged in the present study ($M_{1,2,3} = 3.708, n = 24$). Comparing with the

natural density (no cage open; $M_0 = 1.75, n = 8$), there is a significant difference between these two values (Mann-Whitney U-test, $p = 0.031$). Thus, only two predator densities can be distinguished (and not four). Third, unlike Schenk et al. (2005) who used the actual number of foraging wasps for the 13 h–14 h time interval but the average number $M_c$ for all other intervals, we reconstruct all hourly abundances from the activity profile and from $M_0$ and $M_{1,2,3}$, as explained in the legend to Fig. 1.

### Functional response models

Combining the standard type-II prey-dependent model of Holling (1959) with the interference model of Hassell and Varley (1969), Arditi and Akçakaya (1990) proposed the following expression for a type-II predator-dependent model of the functional response:

$$g(N, P) = \frac{\alpha N / P^m}{1 + \alpha h N / P^m} \quad (2)$$

where N and P are the prey and predator densities respectively, $\alpha$ is a measure of the searching efficiency, h the handling time, and m the mutual interference coefficient. According to
<figure><img src="image_4.png" /><figcaption>Fig. 2. Numbers of prey eaten per predator during a day when 0, 1, 2, or 3 cages were open as a function of (A) the initial density of prey and (B) the ratio between the initial number of prey and the average number of predators. Curves correspond to the best-fitting model (model 4, see Table 4). In (A), the curves at low predator density (grey line, no cage open) and at high predator density (black line, 1, 2, or 3 cages open) are distinct. In (B) the same curve stands for all predator densities.</figcaption></figure>

the classification proposed by Holling (1959), Eq. 2 simplifies to a type-I functional response if $h = 0$. This functional response is purely prey dependent for $m = 0$, purely ratio dependent for $m = 1$ and predator dependent for the other values of $m$.

A type-III (i.e., sigmoid) functional response

can be obtained by replacing the searching efficiency $\alpha$ by a function that increases linearly with $N/P^m$:

$$ \alpha \rightarrow \left( \alpha + \beta \frac{N}{P^m} \right). \qquad (3) $$

Hence:
$$g(N, P) = \frac{(\alpha + \beta \frac{N}{P_m}) \frac{N}{P_m}}{1 + (\alpha + \beta \frac{N}{P_m}) h \frac{N}{P_m}} \quad (4)$$

The original type-II Arditi-Akçakaya model (Eq. 2) is retrieved with $\beta = 0$. The type-III predator-dependent functional response used by Schenk et al. (2005) corresponds to the special case of Eq. 4 with $\alpha = 0$, but there is no a priori reason to make this restriction.

### Handling time estimate

Preliminary non-linear fits of the integrated form of Eq. 4 (needed to account for prey depletion; see below) yielded $h < 0$. This is incompatible with the biological meaning of this parameter. When constraining the parameter to be non-negative, the estimate of $h$ was consistently 0. Moreover, the graphical representation of the observed functional response (Fig. 2) shows that predator saturation did not occur within the range of prey densities used in the experiments. This makes estimation of the parameter $h$ mathematically impossible. Consequently, it is reasonable to set $h$ to 0. This does not imply that the handling time is really zero but only expresses the fact that predator saturation does not occur in this data set. This assumption simplifies Eq. 4 to:

$$g(N, P) = \left( \alpha + \beta \frac{N}{P_m} \right) \frac{N}{P_m} \quad (5)$$

In the rest of this work, we qualify as “type-III” responses all those with $\beta > 0$, even though Eq. 5 does not describe, strictly speaking, the type-III functional response as defined by Holling (1959) because of the absence of saturation. In the case that $\beta = 0$, Eq. 5 becomes a type-I model (and not a type-II because of the absence of the handling time).

### Prey depletion

To account for prey depletion during a finite time interval $[t_1, t_2]$, the instantaneous rate of prey consumption $g(N, P)$ must be integrated over the period $[t_1, t_2]$ in order to predict the actual number of prey eaten during this time. It is this integrated form that must be fitted to the data. The integrated form is obtained by solving the differential equation

$$\frac{dN}{dt} = -g(N, P)P.$$

By using the model of $g(N, P)$ defined by Eq. 5, this yields:

$$N_2 = \begin{cases} \frac{N_1 P_1^{2m-1}}{P_1^{2m-1} + \beta N_1 T} & \text{if } \alpha = 0 \\ \frac{\alpha N_1 P_1^m}{(\alpha P_1^m + \beta N_1) \exp\left(\frac{\alpha T}{P_1^{m-1}}\right) - \beta N_1} & \text{if } \alpha \neq 0 \\ \hfill \equiv f(N_1, P_1, T) & \text{if } \alpha = 0 \end{cases} \quad (6)$$

where $T = t_2 - t_1$ and $P_1$ is the predator density (assumed to remain constant between $t_1$ and $t_2$). The number of prey eaten during $[t_1, t_2]$ is thus $N_1 - N_2$.

### Predator densities

As mentioned previously, for each experimental day followed by Schenk et al. (2005), an estimate of the number of wasps can be reconstructed in each of the ten hours of the activity period. Consequently, the relation between the final and the initial numbers of prey (respectively $N_{11}$ and $N_1$) is given by the iterative relation:

$$N_{i+1} = f(N_i, P_i, 1) \quad \text{for } i = 1 \text{ to } 10 \quad (7)$$

with $f$ the function defined by Eq. 6, $N_i$ the number of prey alive at the beginning of the $i$th hour and $P_i$ the number of active predators during the $i$th activity hour. The latter is obtained by multiplying the relative predator activity (in comparison with the 13–14 h value which is set to 1) for this time interval by $M_c$, the mean number of wasps caught during the 13–14 h slot either with open cages ($M_{1,2,3} = 3.708$) or without open cage ($M_0 = 1.75$).

Schenk et al. (2005) ignored the hourly variations of the predator density and used a constant value $\tilde{P}$ over the day. If we make the same simplification, then the number of prey surviving at the end of an experimental day is given by:

$$N_{11} = f(N_1, \tilde{P}, 10) \quad (8)$$

with $\tilde{P}$ the mean number of active predators during the ten activity hours. Schenk et al. (2005) estimated $\tilde{P}$ as the sum of the densities in each hour:

$$\tilde{P} = \sum_{i=1}^{10} P_i \quad (9)$$
<table><caption>Table 1. Set of candidate models. All models assume no handling time ($h = 0$). Models with $m = 1$ are ratio dependent; they are nested into equivalent predator dependent models with free $m$. Models with $\beta = 0$ are of type I; those with $\beta > 0$ are of type III. The parameter $\alpha$ gives more generality but turns out to be unnecessary when $\beta > 0$ (see text).</caption><thead><tr><th>Model</th><th>Abundance of predators</th><th>No. free parameters</th><th>Functional response model</th></tr></thead><tbody><tr><td>1</td><td>constant (P̃, Eqs. 8 and 10)</td><td>3: m, β, α</td><td>g(N, P) = (α + β<sup>N</sup>/<sub>m</sub>)<sup>N</sup>/<sub>m</sub></td></tr><tr><td>7</td><td>variable (P̃<sub>i</sub>, Eq. 7)</td><td></td><td></td></tr><tr><td>2</td><td>constant (P̃, Eqs. 8 and 10)</td><td>2: β, α; (m = 1)</td><td>g(N, P) = (α + β<sup>N</sup>/<sub>P̃</sub>)<sup>N</sup>/<sub>P̃</sub></td></tr><tr><td>8</td><td>variable (P̃<sub>i</sub>, Eq. 7)</td><td></td><td></td></tr><tr><td>3</td><td>constant (P̃, Eqs. 8 and 10)</td><td>2: m, β, (α = 0)</td><td>g(N, P) = β<sup>(N/m)</sup></td></tr><tr><td>9</td><td>variable (P̃<sub>i</sub>, Eq. 7)</td><td></td><td></td></tr><tr><td>4</td><td>constant (P̃, Eqs. 8 and 10)</td><td>1: β; (m = 1, α = 0)</td><td>g(N, P) = β<sup>(N/m)</sup></td></tr><tr><td>10</td><td>variable (P̃<sub>i</sub>, Eq. 7)</td><td></td><td></td></tr><tr><td>5</td><td>constant (P̃, Eqs. 8 and 10)</td><td>2: m, α; (β = 0)</td><td>g(N, P) = α<sup>N</sup>/<sub>m</sub></td></tr><tr><td>11</td><td>variable (P̃<sub>i</sub>, Eq. 7)</td><td></td><td></td></tr><tr><td>6</td><td>constant (P̃, Eqs. 8 and 10)</td><td>1: α; (m = 1, β = 0)</td><td>g(N, P) = α<sup>N</sup>/<sub>P̃</sub></td></tr><tr><td>12</td><td>variable (P̃<sub>i</sub>, Eq. 7)</td><td></td><td></td></tr></tbody></table>

Obviously, this was an error and must be corrected to:

$$ \tilde{P} = \frac{1}{10} \sum_{i=1}^{10} P_i. \qquad (10) $$

This correction is of no consequence to the estimates of parameters that do not depend on the units of predator density. Particularly, it has no effect on $m$, which is the main parameter of interest.

In the present paper, we compare models using a constant predator density ($\tilde{P}$) with models accounting for the variations during the 10 hours ($P_{i}$, $i = 1$ to 10) the latter making a better use of the available information about the wasp daily activity pattern. The set of candidate models is given in Table 1. Among these models, model 3 is the most similar to that used by Schenk et al. (2005). Differences between these two models are summarized in Table 2.

## Parameter estimates

Non-linear regressions ($n = 32$) are performed with the *nls* function of the R software (R Development Core Team 2010). Graphs of residuals (not shown here) suggested that the

data do not fulfil the normality assumption. Consequently, a non-parametric bootstrap method is used to determine the confidence intervals of the parameter estimates. This method involves generation of new data sets by residual resampling. After fitting the model, fitted values and residuals are retained. New data sets are generated by adding a randomly chosen residual (with replacement) to each of the fitted values. For each of the 10,000 data sets generated in this way, we obtain new parameter estimates so that the distribution of model parameters can be reconstructed empirically. A confidence interval for each parameter can therefore be defined.

## Model selection

In order to compare the models, the second order Akaike criterion (AIC<sub>c</sub>) and the relative Akaike weight $w_i$ are calculated for each candidate model. It is generally considered that models for which $\Delta_i \le 2$ ($\Delta_i = \text{AIC}_i - \text{AIC}_{\text{min}}$) have substantial support, models having $\Delta_i$ of about 4 to 7 have considerably less support, while models with $\Delta_i \ge 10$ have essentially no support and can be omitted from further consideration (Burnham and Anderson 1998).

<table><caption>Table 2. Comparison between the model used by Schenk et al. (2005) and model 3, which is the most similar but corrects a few unsupported assumptions. See text for further explanations.</caption><thead><tr><th>Model</th><th>No. predators P̃</th><th>h</th><th>β</th><th>m</th></tr></thead><tbody><tr><td>Schenk et al. (2005)</td><td>Eq. 9</td><td>0.17‡</td><td>0.0039‡</td><td>free to adjust</td></tr><tr><td>This study, model 3</td><td>Eq. 10</td><td>0</td><td>free to adjust</td><td>free to adjust</td></tr></tbody></table>

<footnote>† From Schenk and Bacher (2002).</footnote>
<table><caption>Table 3. Parameter estimates after non-linear regression, and model ranking based on Akaike's criterion. Results for pure prey-dependent models ($m=0$) are not shown as they all fit very poorly ($\Delta$AIC$_c > 20$) in comparison with predator- and ratio-dependent models. Two models are considered in a tie in ranking if the difference between their AIC$_c$ is smaller than 1. Models 1, 3, 4 have the highest rank.</caption><thead><tr><th>Model</th><th>m</th><th>β</th><th>α</th><th>AIC<sub>c</sub></th><th>rank</th><th>w<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>1.59<br>[1.11; 2.33]</td><td>0.0081<br>[0.0012; 0.025]</td><td>0.041<br>[0; 0.15]</td><td>182.04</td><td>1(tie)</td><td>0.2456</td></tr><tr><td>2</td><td>1</td><td>0.0053<br>[0.0010; 0.0081]</td><td>0.0094<br>[0; 0.60]</td><td>184.31</td><td>4(tie)</td><td>0.0789</td></tr><tr><td>3</td><td>1.44<br>[0.98; 2.19]</td><td>0.012<br>[0.0062; 0.044]</td><td>0</td><td>181.28</td><td>1(tie)</td><td>0.3599</td></tr><tr><td>4</td><td>1</td><td>0.0062<br>[0.0052; 0.0094]</td><td>0</td><td>182.29</td><td>1(tie)</td><td>0.2177</td></tr><tr><td>5</td><td>2.21<br>[1.62; 2.94]</td><td>0</td><td>0.16<br>[0.097; 0.27]</td><td>183.90</td><td>4(tie)</td><td>0.0972</td></tr><tr><td>6</td><td>1</td><td>0</td><td>0.068<br>[0.051; 0.084]</td><td>193.87</td><td>6</td><td>0.0007</td></tr></tbody></table>

Note: Values in square brackets are 95% CIs.

The Akaike weight $w_i$ can be interpreted as the probability that model $i$ is the best model for the observed data, given the set of candidate models (Johnson and Omland 2004). A confidence set of models can be obtained by summing the $w_i$ from largest to smallest until that sum is just greater than a given probability that the best approximating model is contained within the confidence set (Burnham and Anderson 1998). As a complement, likelihood ratio tests (LRTs) are used to compare pairs of nested models. With respect to a “simple” model, the LRT provides a statistical criterion to assess the significance of the fitting improvement of a “richer” model that contains additional parameters. However, unlike AIC$_c$, LRTs cannot be used to test simultaneously all models in a candidate set and can only be applied to nested models (Burnham and Anderson 1998).

## RESULTS

The fits of purely prey-dependent models (with $m=0$) were clearly worse than fits of ratio-dependent and predator-dependent models ($\Delta$AIC$_c > 20$). Consequently, they are not presented and discussed thereafter.

Contrary to what was expected, using a constant number of predators over the whole activity period, thereby ignoring the variations of this number during the day is of no real consequence. The estimates of the parameter $m$ as well as the AIC$_c$ are identical when using either Eq. 7 or Eqs. 8 and 10 for the analyses.

Only the estimates of the other parameters are different. In fact, in the case in which the activity profile is the same on all experimental days, the estimate of the parameter $m$ depends only on the average number of predators present during the day and not on the hourly variations of this number. This numerical finding can be proved analytically in special cases. Consequently, we can reduce the set of candidate models to models 1 to 6 (see Table 1).

Fitting results and Akaike criteria for predator-dependent and ratio-dependent models are shown in Table 3. Responses of type I (i.e., with $\beta=0$; models 5 and 6) fit the data poorly. Model 6 is even the only model that fits clearly worse than the others ($\Delta$AIC$_c > 10$). Responses of type III ($\beta > 0$; models 1, 2, 3, and 4) fit better. This is also supported by the LRTs (Table 4): these tests show that the addition of the parameter $\beta$ always improves the fits significantly ($p < 0.05$). Among type-III models, those with $\alpha=0$ can be used with no significant loss of performance (LRT of model 3 vs. model 1: $p=0.19$; LRT of model 4 vs. model 2: $p=0.62$). According to the AIC$_c$ ranking, three nested models (models 1, 3 and 4) can be considered as the best ones. As explained above, the LRT indicates that model 3 can be preferred to model 1. The 95% confidence interval of $m$ in model 3 includes 1 (Table 3). Moreover the LRT of model 4 vs. model 3 (Table 4) indicates that the ratio-dependent model ($m=1$) performs as well as the equivalent model with free $m$ ($p=0.0699$). These results lead therefore to the conclusion that the ratio-dependent
<table><caption>Table 4. Results of likelihood-ratio tests for pairs of nested models. Within the set of three best models that was selected by Akaike's criterion (models 1, 3, 4, Table 3), LRTs show that model 1 is less parsimonious than both 3 and 4, and that model 3 is less parsimonious than model 4, which is therefore the best choice.</caption><thead><tr><th>Complex model</th><th>Simple model</th><th>LR</th><th>Additional parameters</th><th>df</th><th>p-value</th></tr></thead><tbody><tr><td rowspan="5">1</td><td>2</td><td>4.72</td><td>m</td><td>1</td><td>0.02978*</td></tr><tr><td>3</td><td>1.69</td><td>α</td><td>1</td><td>0.19401</td></tr><tr><td>4</td><td>4.97</td><td>m, α</td><td>2</td><td>0.08322</td></tr><tr><td>5</td><td>4.30</td><td>β</td><td>1</td><td>0.03804*</td></tr><tr><td>6</td><td>16.56</td><td>m, β</td><td>2</td><td>0.00025***</td></tr><tr><td rowspan="2">2</td><td>4</td><td>0.25</td><td>α</td><td>1</td><td>0.61666</td></tr><tr><td>6</td><td>11.83</td><td>β</td><td>1</td><td>0.00058***</td></tr><tr><td>3</td><td>4</td><td>3.29</td><td>m</td><td>1</td><td>0.06989</td></tr><tr><td>5</td><td>6</td><td>12.25</td><td>m</td><td>1</td><td>0.00046***</td></tr></tbody></table>

dent unsaturated type-III model (model 4) is the most parsimonious representation of the functional response of the wasp *P. dominulus* preying on *C. rubiginosa* (Fig. 2):

$$g(N, P) = \beta \left(\frac{N}{P}\right)^2. \qquad (11)$$

## DISCUSSION

It is generally difficult to monitor predation events in the field because of their secretive nature. However, the specific system that was set up by Schenk and Bacher (2002) has permitted to conduct in the field bivariate functional response experiments, i.e., to observe the number of prey eaten for various densities of both the prey and the predator populations. In their original analysis, Schenk et al. (2005) showed that predator interference occurred in *P. dominulus* and suggested that both direct (behavioural interactions) and indirect (non-random search for prey) mechanisms could be implicated. Moreover, they concluded that the functional response of *P. dominulus* was neither prey dependent nor ratio dependent, with both extremes clearly rejected. By re-analyzing the same data incorporating more realistic assumptions, we have shown here that the functional response of this predatory wasp can actually be considered as ratio dependent. Not only is the possibility of ratio dependence included in the confidence interval of the best-fitting model, but a ratio-dependent model fits as well as this best-fitting model in spite of a smaller number of parameters.

In artificial settings, the predator density can be controlled and is generally held constant. In

natural settings, the predator density may fluctuate during the interaction sessions. This can have consequences on the parameter estimates. However, we have shown here that this has no effect on the estimate of the mutual interference parameter *m*. It is sufficient to know the daily average of the predator density. Since *m* is the major parameter of interest, this property has a great practical interest.

In their original analysis, Schenk et al. (2005) did not estimate all functional response parameters from the same data set. The handling time *h* and the parameter *β* were taken from the observations of Schenk and Bacher (2002) ($h = 0.17 \pm 0.02$ h; $\beta = 0.0039 \pm 0.0005$). When conducted at another site (Schenk and Bacher 2002), experiments led again to quite different estimates of *h* (0.07 ± 0.01 h) and of *β* (0.0074 ± 0.001). Thus, these two parameters can change according to environmental conditions. Let us consider modifications of models 3 and 9 so as to re-incorporate the effect of a handling time (i.e., use Eq. 4 instead of Eq. 5). If we follow Schenk et al. (2005) by setting $h = 0.17$ and $\beta = 0.0039$, we find $m = 0.698$ with the modified model 3 and $m = 0.663$ with the modified model 9. These values are closer to the value obtained by Schenk et al. (2005) than to those obtained in the present study by estimating all parameters simultaneously. We can conclude that the “intermediate” value of *m* obtained by Schenk et al. (2005) is related to the fact that the various parameters had been estimated in separate experiments. While such procedure is sometimes unavoidable, the present data set is of sufficient quality to estimate all parameters simultaneously, thereby providing more reliable values.

Much theoretical work focusing on the impli-
cations of mutual interference for population
dynamics has demonstrated its stabilizing effect
in predator-prey systems. Although top-down
effects are reduced in comparison with prey-
dependent models, predators can still regulate
the prey and check their outbreaks whether
predator dependence is moderate (m < 1), ratio
dependent (m = 1) or strong (m > 1) (Arditi and
Berryman 1991, Berezovskaya et al. 2001, Arditi
et al. 2004). Consequently, our re-evaluation of
the mutual interference strength in *P. dominulus*
does not mean that this predator cannot play a
key role in the regulation of its prey *C. rubiginosa*.
However, the study of the dynamics of this
system is not the aim of the present work because
the available data do not allow it: time series of
freely reproducing populations would be neces-
sary.

Our revised analysis of the 2005 experiment
has shown that the handling time could be
neglected in these data. This is due to the low
prey densities used in this experiment. Schenk
and Bacher (2002) had shown that the inflexion of
the functional response due to the saturation
effect of the handling time occurred at a prey
density higher than 40, which is the highest value
considered in the 2005 experiment. As we have
already said, there is absolutely no sign of
saturation in these data (Fig. 2), a situation that
might also be amplified by the fact that wasp
densities were much higher in the 2005 experi-
ment (median wasp densities = 3 counts per
hour) than in the 2002 experiment (median
densities = 1 count per hour).

The main feature of the type-III functional
response is the acceleration (convexity) at low
prey densities. Our statistical analysis confirms
this convexity, which was visually evident in Fig.
2, and which had previously been demonstrated
by Schenk and Bacher (2002). This can be
modelled with the searching efficiency being an
increasing function of prey density. Schenk and
Bacher (2002) had found a proportional relation-
ship between the searching efficiency and the
initial number of prey. We have considered here
the actual number of available prey (our Eq. 3)
rather than the initial number, which is more in
line with the instantaneous concept of the
functional response. As noted by Schenk and
Bacher (2002), the acceleration of the predation
rate at low prey densities could be due to the fact

that *P. dominulus* is a generalist predator. Tschanz
et al. (2007) showed that prey switching between
*C. rubinigosa* and larvae of flour beetles *Tenebrio
molitor*, offered in an experiment as an alternative
prey of *P. dominulus*, could not explain the type-
III functional response, but they proposed
alternative mechanisms, such as learning or the
presence of still another prey than *C. rubiginosa*
and *T. molitor*.

In the context of the debate over the ratio-
dependent hypothesis, a number of experimental
studies could be analyzed to support or reject
this hypothesis (see Chapter 2 of Arditi and
Ginzburg 2012). They all demonstrate predator
dependence to various degrees and many of
them provide support for ratio dependence as an
acceptable approximation (DeLong and Vasseur
2011). However, the majority of these empirical
data come from laboratory experiments, with
spatial and temporal scales that pertain more to
behaviour than to population dynamics. To the
best of our knowledge, the present experiment is
only the second that demonstrates so clearly a
ratio-dependent functional response by direct
measurements in a natural setting. The only
other field study that we are aware of is the Isle
Royale wolf-moose system studied by Vucetich et
al. (2002) and Jost et al. (2005). Together, they
provide evidence that ratio dependence is not a
laboratory artefact and can arise in natural
conditions. By contrast, as reviewed by Arditi
and Ginzburg (2012), there does not exist a single
study, whether performed in the lab or in the
field, that has ever demonstrated pure prey
dependence.

All the empirical evidence thus strongly
suggests that it is time to abandon models with
pure prey dependence for understanding preda-
tor-prey population dynamics. Instead, models
incorporating predator dependence must always
be considered. The only two existing field studies
have demonstrated ratio dependence, which is a
dimensionally and parametrically simpler form
of predator dependence. Clearly, there is a need
for more field studies.

ACKNOWLEDGMENTS

We thank Sébastien Ballesteros and Cyril Depoudent
for conducting a preliminary analysis as part of a
student project, Rudolf Rohr for interesting discus-
sions, John D. Reeve and two anonymous referees for
commenting on a previous version of the manuscript.
External support was provided by the French National
Research Agency (ANR), by the French Ministry of
Agriculture (CNECA2), and by the Swiss National
Science Foundation through the National Centre of
Competence in Research "Plant Survival".

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