# Spatial heterogeneity and functional response: an experiment in microcosms with varying obstacle densities

**ABSTRACT** Spatial heterogeneity of the environment has long been recognized as a major factor in ecological dynamics. Its role in predator-prey systems has been of particular interest, where it can affect interactions in two qualitatively different ways: by providing (1) refuges for the prey or (2) obstacles that interfere with the movements of both prey and predators. There have been relatively fewer studies of obstacles than refuges, especially studies on their effect on functional responses. By analogy with reaction-diffusion models for chemical systems in heterogeneous environments, we predict that obstacles are likely to reduce the encounter rate between individuals, leading to a lower attack rate (predator-prey encounters) and a lower interference rate (predator-predator encounters). Here, we test these predictions under controlled conditions using collembolans (springtails) as prey and mites as predators in microcosms. The effect of obstacle density on the functional response was investigated at the scales of individual behavior and of the population. As expected, we found that increasing obstacle density reduces the attack rate and predator interference. Our results show that obstacles, like refuges, can reduce the predation rate because obstacles decrease the attack rate. However, while refuges can increase predator dependence, we suggest that obstacles can decrease it by reducing the rate of encounters between predators. Because of their opposite effect on predator dependence, obstacles and refuges could modify in different ways the stability of predator-prey communities.

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**ABSTRACT:**Predator-prey interactions are mediated by the structural complexity of habitats, but disentangling the many facets of structure that contribute to this mediation remains elusive. In a world replete with altered landscapes and biological invasions, determining how structure mediates the interactions between predators and novel prey will contribute to our understanding of invasions and predator-prey dynamics in general.Here, using simplified experimental arenas, we manipulate predator-free space, whilst holding surface area and volume constant, to quantify the effects on predator-prey interactions between two resident gammarid predators and an invasive prey, the Ponto-Caspian corophiid Chelicorophium curvispinum.Systematically increasing predator-free space alters the functional responses (the relationship between prey density and consumption rate) of the amphipod predators by reducing attack rates and lengthening handling times. Crucially, functional response shape also changes subtly from destabilising Type II towards stabilising Type III, such that small increases in predator-free space to result in significant reductions in prey consumption at low prey densities.Habitats with superficially similar structural complexity can have considerably divergent consequences for prey population stability in general and, particularly, for invasive prey establishing at low densities in novel habitats.This article is protected by copyright. All rights reserved.Functional Ecology 09/2014; · 4.86 Impact Factor -
##### Conference Paper: The paradox of enrichment in metaecosystems

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**ABSTRACT:**Background/Question/Methods At the convergence of community, ecosystem and spatial ecology, the concept of metaecosystems considers a spatially structured environment in which species as well as inorganic nutrients and detritus can diffuse between localities. This new framework promises to be a powerful tool to address questions at large spatial scales for which feedbacks between nutrient dynamics and species communities are crucial for ecosystem functioning. Here we revisited the renowned paradox of enrichment within the metaecosystem framework. This paradox states that ecosystem enrichment could destabilize consumer-resource interactions instead of benefiting their demography. Species dispersal has been proved to dampen this enrichment destabilization in metacommunities. However, spatial flows of nutrient, detritus and local recycling could change local enrichment and potentially modify this outcome. We studied the effect of either nutrient, detritus, producer or consumer spatial flows, combined with changes of regional enrichment, on the stability of a two-patch metaecosystem model. Results/Conclusions We found first that spatial flows of nutrient and detritus are destabilizing whereas spatial flows of producers and consumers are either neutral or stabilizing. These opposite effects on stability are linked to opposite effects on spatial synchrony. We also found unexpected additional stabilizing effects of consumer spatial flows at intermediate diffusion rates, for higher enrichment. Furthermore, we found that consumer spatial flows could lead to different equilibriums in strongly enriched metaecosystems. Then, initial small changes in densities could result in radically divergent distributions of biomasses. Our metaecosystems' study reveals that nutrient spatial dynamics can produce complex interactions with species dynamics. Source-sink dynamics do not inevitably produce stabilizing compensatory effects, for instance when only nutrients can diffuse. However, the paradox of enrichment is unlikely to occur in well connected localities because it can be dampen by a global dispersal, which redistributes regionally the local excess of fertility. Stabilize the effects of a more global enrichment (regional scale) requires consumer spatial flows at intermediate dispersal rates. This points out the specific importance of consumers' dispersal to neutralize this paradox in nature.97th ESA Annual Convention 2012; 08/2012 - SourceAvailable from: Roger Arditi[Show abstract] [Hide abstract]

**ABSTRACT:**dependent predation in a field experiment with wasps. Ecosphere 3(12): 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 depen-dence 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.Ecosphere 12/2012; art124. · 2.60 Impact Factor

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POPULATION ECOLOGY - ORIGINAL PAPER

Spatial heterogeneity and functional response: an experiment

in microcosms with varying obstacle densities

Ce ´line Hauzy•Thomas Tully•Thierry Spataro•

Gre ´gory Paul•Roger Arditi

Received: 9 July 2009/Accepted: 3 February 2010/Published online: 9 March 2010

? Springer-Verlag 2010

Abstract

long been recognized as a major factor in ecological

dynamics. Its role in predator–prey systems has been of

particular interest, where it can affect interactions in two

qualitatively different ways: by providing (1) refuges for

the prey or (2) obstacles that interfere with the movements

of both prey and predators. There have been relatively

fewer studies of obstacles than refuges, especially studies

on their effect on functional responses. By analogy with

reaction–diffusion models for chemical systems in hetero-

geneous environments, we predict that obstacles are likely

to reduce the encounter rate between individuals, leading to

a lower attack rate (predator–prey encounters) and a lower

interference rate (predator–predator encounters). Here, we

test these predictions under controlled conditions using

Spatial heterogeneity of the environment has

collembolans (springtails) as prey and mites as predators in

microcosms. The effect of obstacle density on the func-

tional response was investigated at the scales of individual

behavior and of the population. As expected, we found that

increasing obstacle density reduces the attack rate and

predator interference. Our results show that obstacles, like

refuges, can reduce the predation rate because obstacles

decrease the attack rate. However, while refuges can

increase predator dependence, we suggest that obstacles

can decrease it by reducing the rate of encounters between

predators. Because of their opposite effect on predator

dependence, obstacles and refuges could modify in dif-

ferent ways the stability of predator–prey communities.

Keywords

Mutual interference ? Predator–prey interaction

Encounter rates ? Habitat complexity ?

Introduction

The functional response of the predator describes the pre-

dation rate as a function of prey density (Solomon 1949;

Holling 1959a; Holling 1959b) and of predator density

(Hassell and Varley 1969; Beddington 1975; DeAngelis

et al. 1975; Arditi and Ginzburg 1989; Arditi and Akc ¸akaya

1990). It is a key component of dynamic population models

that include trophic interactions. The dynamics and stability

of predator–prey communities, and also their response to

enrichment, are sensitive to the shape of prey and predator

dependences (Rosenzweig 1971; Oksanen et al. 1981; Ar-

diti and Ginzburg 1989; Arditi et al. 1991; Hulot et al. 2000;

Gross et al. 2004), making it very important to understand

the mechanisms that determine the functional response.

The functional response is defined at the level of the

population. Models must express the way in which

Communicated by Stefan Scheu.

C. Hauzy ? T. Tully ? T. Spataro ? R. Arditi

UMR7625 E´cologie et E´volution,

Universite ´ Pierre et Marie Curie, 75005 Paris, France

T. Tully

IUFM de Paris, Universite ´ Paris 4 - Sorbonne, 75016 Paris,

France

C. Hauzy ? T. Spataro ? R. Arditi

USC2031-INRA E´cologie des populations et communaute ´s,

AgroParisTech, 75005 Paris, France

C. Hauzy (&)

IFM Theory and Modelling, Linko ¨ping University,

581 83 Linko ¨ping, Sweden

e-mail: celha@ifm.liu.se

G. Paul

Institute of Theoretical Computer Science and Swiss Institute of

Bioinformatics, ETH Zu ¨rich, 8092 Zurich, Switzerland

123

Oecologia (2010) 163:625–636

DOI 10.1007/s00442-010-1585-5

Page 2

behavioral mechanisms of prey and predator individuals

together with environmental structures (e. g. heterogene-

ities) translate into variations of the prey abundances.

The shape of the functional response is linked to the

behavior of both prey and predators. For most species, the

predation rate initially increases with prey density, and

then gradually saturates. This saturation can be explained

by the handling time, i.e., the time the predator takes to

handle its prey (Holling 1959b). At low prey densities,

different shapes have been described, resulting in either

Holling type-II or type-III dependence (Holling 1959a).

The type-III (sigmoidal) functional response differs from

that of type II by an acceleration phase of the predation rate

at low prey density preceding the deceleration phase. This

acceleration can be explained, for example, by an increase

in the predator’s ability to detect its prey as prey density

increases (e.g., Pietrewicz and Kamil 1979). Holling’s

mechanistic model (1959b) is a well-known model of type-

II functional response. It describes the increase in the

number of prey captured with increasing prey density due

to a constant attack rate, followed by a saturation due to the

handling time.

Predator density can also affect the predation rate: an

increase in predator density can reduce the predation rate

(Hassell and Varley 1969). Such ‘‘predator dependence’’

has been reported for several species (e.g., Fussmann et al.

2005; Rickers and Scheu 2005; Schenk et al. 2005; Kratina

et al. 2009). In his mechanistic model, Beddington (1975)

further developed Holling’s model to explain predator

dependence by the time predators spend interfering with

each other when they encounter. Many other behaviors,

such as aggregation of prey (Cosner et al. 1999; Arditi et al.

2001), social interactions between predators (Abrams and

Ginzburg 2000), or prey-taxis of predators (Tyutyunov

et al. 2008), have also been proposed as additional mech-

anisms that can generate predator dependence in the

functional response. These mechanisms can be modeled by

the Beddington model or by phenomenological models by

considering that interference between predators directly

affects their attack rate through an interference coefficient

(Hassell and Varley 1969; Arditi and Ginzburg 1989; Ar-

diti and Akc ¸akaya 1990). The Arditi–Akc ¸akaya model

consists of the inclusion of the Hassell–Varley approach

into Holling’s model.

In almost all ecological systems, predators search for

prey in heterogeneous environments (Li and Reynolds

1994). Small-scale environmental heterogeneities, which

are relevant to individual mobility, may not split popula-

tions into distinct subpopulations, as would larger discon-

tinuities in the manner suggested by metacommunity

theory (e.g., Hanski and Gilpin 1991; Holyoak et al. 2005).

At a sufficiently small scale, heterogeneities in the envi-

ronment can decrease the ability of predators to catch prey

by modifying the movement of individuals. Heterogene-

ities can prevent predators from reaching parts of their

environment and provide prey with absolute refuges from

predators. Such refuges can reduce the predation rate

(Hildrew and Townsend 1977, 1982; Folsom and Collins

1984). For example, the predation rate of caddisfly larvae

on stonefly larvae and Chironomids has been found to be

higher on homogeneous substrates than on more hetero-

geneous substrates that allow prey to hide (Hildrew and

Townsend 1982). Moreover, environmental heterogeneities

can constitute obstacles to the movements of both predators

and prey without providing refuges, such as dead leaves or

needles in the soil litter, which can provide a complex

arrangement of fine-scale obstacles to soil fauna. Such

obstacles can also reduce the predation rate (Savino and

Stein 1982; Kaiser 1983). Kaiser (1983) showed that the

predation rate falls when the complexity of labyrinth bar-

riers to the movements of both prey (phytophagous mite)

and predators (predatory mite) increases. However, none of

these studies has truly characterized the effect of environ-

mental heterogeneities on the shape of the functional

response.

Several other studies have investigated the effect of

refuges on the functional response. The results of various

theoretical studies indicate that refuges are able to modify

the dependence of the functional response on prey density.

When refuges are limited in both size and number, the

predation rate is reduced by refuges only at a low prey

density. This effect vanishes at higher prey density because

the refuges become saturated, resulting in a switch of the

functional response from type II to type III (Murdoch and

Oaten 1975; Hassell 2000). The findings of several experi-

mental studies support these theoretical predictions (e.g.,

Lipcius and Hines 1986). Spatial heterogeneity in terms of

the presence of refuges can affect not only the prey depen-

dence but also the predator dependence of the functional

response. Theoretical models show that a predator exhibiting

a type-II prey-dependent functional response in a homoge-

neous environment could exhibit predator dependence when

refuges are introduced (Cosner et al. 1999).

To the best of our knowledge, no experimental or the-

oretical studies have specifically investigated the effect of

obstacles on the shape of the functional response. Because

predator–prey systems are analogous to enzyme–substrate

systems (e.g., Real 1977), we have looked at predictions

stemming from a non-ecological context: the effect of

obstacles on chemical reactions in a heterogeneous envi-

ronment. The process of chemical reactions accounts for

the spatial spread of the chemical species and their reac-

tions. Reaction–diffusion models show that the density of

obstacles affects only the diffusion process, not the reaction

between particles (Bouchaud and Georges 1990; Berry

2002): i.e., the encounter rate monotonously falls, whereas

626Oecologia (2010) 163:625–636

123

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the reaction rate remains unchanged. The decrease in the

encounter rate arises because the area explored by particles

is decreased by obstacle density. This result is quite robust

and has been reported for various different shapes of

obstacles (Saxton 1994). For predator–prey systems, it is

predicted that obstacle density will decrease predator

encounter rate with prey or other predators—without

affecting the interaction time. More specifically, this means

that the attack rate should decrease with the density of

obstacles, whereas the handling time should not be affec-

ted. As a consequence, obstacles should not change a

functional response of type II to that of type III (as refuges

would), but they would reduce the ascending part of the

type-II functional response. Similarly, the rate of encoun-

ters between predators can be expected to decrease with

obstacle density—but not their interaction times. Conse-

quently, obstacle density should decrease the interference

between predators, and the predator dependence of the

functional response should become weaker.

Here, we report our investigation of the hitherto un-

derexplored effect of obstacles on the functional response

using experiments conducted under controlled conditions.

We used the soil litter mite Pergamasus crassipes as the

predator and the springtail Folsomia candida as its prey. In

a homogeneous environment, this predator–prey system is

known to display a type-II functional response when a

single predator is used (Tully et al. 2005). We studied the

effect of obstacle density on the parameters of the func-

tional response combining two approaches: (1) through a

close behavioral follow-up, we directly measured some key

behavioral parameters involved in the functional response

(later called ‘‘behavioral approach’’); (2) by measuring the

number of prey killed at the end of each trial, we were able

to directly measure the shape of the functional response

(‘‘population approach’’). We address the following ques-

tions regarding the effect of obstacle density on the func-

tional response: is the predator attack rate reduced by

increasing density of obstacles? Is the type of functional

response (type II or III) modified by obstacle density? Does

obstacle density reduce the encounter rate between preda-

tors? Is there predator dependence in the functional

response; if so, is it reduced for higher levels of obstacle

density? Finally, are the handling and interference times

modified when the density of obstacles varies?

Materials and methods

Functional response models and theoretical predictions

We used a set of four classical models, one without any

predator dependence, and the other three with different

forms of predator dependence (Tables 1, 2). The first

model is the classical Holling type-II functional response,

also called the Holling disk equation (1959b). In this

model, the number of prey eaten increases when prey

density N increases according to the attack rate of the

predator a, and it is limited by handling time th, which is

the time spent by the predator handling the captured prey.

The second model is the Beddington–DeAngelis model

(Beddington 1975; DeAngelis et al. 1975). This model

takes into account the interference between predators by

incorporating the time, tI, spent by predators on each

interaction with conspecifics, which they encounter at rate

b. The Beddington–DeAngelis model reverts to the Holling

model when b or tIis equal to 0. The third model (Arditi

and Akc ¸akaya 1990) combines the Holling disc equation

and the Hassell and Varley expression (1969) for the effect

of predator density P on the attack rate: the parameter m

indicatesthestrengthofpredatordependenceand

Table 1 Model

dimension

variablesandparameters: names,definitions,

Parameter Description

Ne

N

Number of prey captured per predator and per unit of time

Prey density expressed as the number of prey per unit area

P

Predator density expressed as the number of predators per

unit area

X

Obstacle density expressed as the number of obstacles per

unit area

a

Attack rate or searching efficiency: rate of capturing an

individual prey per unit of prey density and per unit of

time spent searching for prey. It can be expressed as the

surface area cleared of prey per unit searching time

th

Handling time: the time spent handling and consuming a

captured prey. During this time the predator is

considered to be unable to search for another prey

b

Encounter rate with other predators: rate of encountering

a predator per unit predator density and per unit of time

spent searching; can also be expressed as the surface

area per unit of time

tc

Interference time: time spent interacting directly with

another predator

m

Interference coefficient: dimensionless

Table 2 Functional response models

Model nameMathematical expression

Holling

Ne¼

aðXÞN

1 þ aðXÞthN

Beddington–DeAngelis

Ne¼

aðXÞN

1 þ aðXÞthN þ bðXÞtcP

aðXÞNP?mðXÞ

1 þ aðXÞthNP?mðXÞ

aðXÞN=P

1 þ aðXÞthN=P

Arditi–Akc ¸akaya

Ne¼

Arditi-Ginzburg

Ne¼

See Table 1 for definitions of parameters and variables

Oecologia (2010) 163:625–636 627

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interference. By setting m to 0, the model reduces to the

purely prey-dependent model of Holling. If m = 1, the

functional response depends simply on the N:P ratio, and

the Arditi–Akc ¸akaya model reduces to a fourth model: the

ratio-dependent model of Arditi and Ginzburg (1989).

Reaction–diffusion models from chemistry indicate that

obstacles reduce the encounter rate between individuals

without affecting their interaction times. This implies the

following predictions. First, the attack rate a would be a

decreasing function of obstacle density X [noted a(X) in the

four models]. Second, the encounter rate between predators

b and the interference coefficient m would be decreasing

functions of obstacle density [noted b(X) in the Bedding-

ton–DeAngelis model and m(X) in the Arditi–Akc ¸akaya

model, Table 2]. Third, the handling time th and the

interference time tcwould not depend on obstacle density.

Experimental methods

Organisms

In our experiments, we used two small species that are

found in the soil litter: a predatory mite, Pergamasus

crassipes, and one of its natural prey species, the parthe-

nogenetic springtail (collembola), Folsomia candida. Both

species are blind. Adult mites were collected in an oak and

beech litter in the deciduous forest surrounding the field

station where the experiments took place (CEREEP Foljuif

field station, Saint-Pierre-le `s-Nemours, Seine et Marne,

France). We only used females to avoid mating behavior

during the experiments. Mites were kept isolated in small

rearing boxes (2 9 2 cm) overnight before their use in the

assay so as to increase and standardize their hunger level.

For the collembolan prey, we used young adult females of

similar size and age (approx. 1 month old, 1.2–1.5 mm in

length), all of which belonged to the same clonal popula-

tion originating from the south of France (‘‘TO’’ strain;

Tully et al. 2005, 2006). Prey individuals were randomly

taken from several replicated populations that had been

reared under similar conditions (21?C, 100% relative

humidity) for several months. In order to avoid pseudo-

replication, each individual predator or prey was used only

once during the experiments.

Experimental arena

The experimental arenas consisted of circular plastic boxes

(280.8 cm2) filled with a 2-cm-thick layer of plaster of

Paris. Moist plaster was used because it provides a smooth

surface that maintains the high hygrometry needed by both

species. Obstacles were set over the arena in such way that

they did not provide any refuges for the prey (Fig. 1a). Our

obstacles were designed to block the movements of both

prey and predators without substantially reducing the

unencumbered surface area of the arenas: each obstacle

occupies about 0.04 cm2, meaning that the area covered by

the obstacles at the highest obstacle density was less than

1% of the whole surface of the arena. The obstacles and the

plastic borders of the arenas were both painted with Fluon

to prevent the collembolans and mites from climbing on

them. The accessible surface was therefore very similar for

all the obstacle densities. For the sake of simplicity,

obstacles were distributed regularly over the surface of the

arenas (Fig. 1b–d).

Experimental design

We manipulated three variables: obstacle density X, prey

density N, and predator density P (Table 3). We used three

levels of obstacle density: 1, 23, and 52 obstacles per arena.

For each of these levels, prey and predator densities were

manipulated. We used four levels of prey density (4, 8, 16,

and 32 prey individuals per arena) and three levels of

predator density (1, 4, and 8 predators per arena). We

added a further two combinations (N:P combinations of 2:1

and 64:8) in order to provide three N:P ratios (2, 4, and 8)

(Table 3). This design enabled us to test both the Bedd-

ington–DeAngelis and Arditi–Akc ¸akaya predator-depen-

dent models. Each set of conditions (obstacle density 9

prey density 9 predator density) was replicated at least

seven times, with slightly more replicates (9 or 10) for the

lowest prey density (Table 3), yielding a total of 301 trials.

90°

2 cm

1 cm

abcd

Fig. 1 Structure and organization of the obstacles. a The obstacles

comprise small transparent plastic crosses with four branches of equal

size (branch length 1 cm, branch height 2 cm, thickness 0.01 cm). b–

d Distribution of the obstacles (crosses) in circular arenas with one

(b), 23 (c), or 52 (d) obstacles

628 Oecologia (2010) 163:625–636

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The experiments took place at room temperature

(approx. 18?C) and under constant light. For each trial, we

first randomly deposited the N prey in an arena and after

about 1 min we introduced P predator(s) that had been

randomly selected. The last predator to be introduced was

chosen as the focal predator, and its behavior was carefully

monitored by eye for 1 h. During our 1-h follow-up, we

kept prey density constant by introducing a new springtail

in the arena each time one was captured and killed by a

predator. This enabled us fulfill the hypothesis of constant

prey density, which is a common condition in the different

models of functional response used here.

Measurements

With the help of EthoLog software ver. 2.2.5 (Ottoni

2000), we recorded the times at which the following events

occurred: (1) the capture of a prey by the focal predator, (2)

the release of the dead prey by the focal predator, and (3)

the beginning and (4) the end of contacts between the focal

predator and other predators. These data provided us with

direct measurements on the four functional response

parameters (behavioral approach): the attack rate a, the

encounter rate between predators b, the handling time th,

and the interference time tcbetween predators. We also

counted the total number of prey captured and eaten per

predator Neat the end of each trial in order to determine the

average effect of a predator on the prey population and

obtain fitted estimates of the functional response parame-

ters (population approach).

Statistical analyses

All statistical analyses were performed using R software v.

2.8 (Ihaka and Gentleman 1996). Prey density (six levels)

was treated as a continuous variable. Predator and obstacle

densities (three levels each) were treated as non-ordered

categorical variables. In tables, figures and in the text, we

provide parameter estimates with their associated 95%

confidence intervals (95% CI) in parenthesis.

Behavioral approach: attack and encounter rates

We estimated the attack rate a and the encounter rate

between predators b using failure–time analysis (also

known as survival analysis). This approach consists of

analyzing the time the focal predator takes to capture its

first prey (for a) or to encounter its first predator (for b). For

each level of prey, predator, and obstacle density, the rates

of these two events, also known as ‘‘hazard rates’’—

h(N,P,X) in statistical jargon—were estimated directly by

fitting a parametric survival model to the time of the first

capture or first encounter. We used the exponentially dis-

tributed ‘‘survreg function’’ from the ‘‘survival library’’

(see Fox 2001 and Tully et al. 2005 for more details on this

estimation method). This model assumes that these two

hazard rates remain constant over time. From these hazard

rates, the attack and encounter rates can be estimated for

each level of prey, predator, and obstacle density as

a(N,P,X) = ha(N,P,X)/N

and

with haand hbbeing the hazard rates estimated for captures

of prey and encounters between predators, respectively.

Finally, the relationship between a (or b) with the densities

of prey, predators, and obstacles was studied by fitting a

linear model with generalized least squares between these

variables (‘‘gls’’ function of the ‘‘nlme’’ library).

After model selection based on Akaike Information

Criteria (AIC) (the ‘‘stepAIC’’ function of the ‘‘MASS’’

library), multiple comparisons were performed using Tu-

key’s post hoc tests. Because the distribution of residuals

was not different from a normal distribution, confidence

intervals were calculated using a parametric method (the

‘‘confint’’ function of the ‘‘stats’’ library).

b(N,P,X) = hb(N,P,X)/P,

Behavioral approach: interaction times

Interaction times th and tc were measured as the time

intervals between the beginning and the end of interactions.

The prey handling time, th, associated with each capture

was calculated as the time between the capture of a prey

and the release of its remains by the focal predator.

However, in 25% of the measurements, the focal predator

was still handling its prey at the end of the hour-long assay

(censored data). We analyzed these data using failure-time

analysis because this method enabled us to take these

censored measurements into account. More specifically, we

used the cox-proportional hazard model (non-parametric

failure-time models, the ‘‘coxph’’ function of the ‘‘sur-

vival’’ library) to quantify the effects of prey, predator, and

obstacle densities on the first handling time, th (244

measurements).

The interference time, tc, is the duration of interactions

between the focal predator and other predators. These

interference times were short enough to have no censored

Table 3 Prey:predator density ratio for each combination of prey and

predator densities

Predator density

(no. per arena)

Prey density (no. per arena)

248 1632 64

12a

481632

41248

80.51248

All of these combinations were studied at each obstacle density. Each

condition was replicated at least seven times

aIndicates a condition with nine (for 23 and 52 obstacles per arena)

or ten (for one obstacle per arena) replicates

Oecologia (2010) 163:625–636629

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measurements. After log transformation, these times were

analyzed using a linear mixed effect model (the ‘‘lme’’

function of the ‘‘nlme’’ library) to investigate how they

were influenced by prey, predator, and obstacle densities.

The experiment code was included as a random effect in

order to take into account the repeated measurements

performed during each assay (2,492 times measured for

301 assays).

The confidence intervals of the handling time th,and

interference time tc, were calculated using a parametric

method (the ‘‘confint’’ function of the ‘‘stats’’ library). We

calculated parametric confidence interval for the combined

parameter btc (see the section ‘‘Population approach:

predator dependence and the impact of obstacle density’’)

where the encounter rate between predators, b, and

the interference time, tc, were estimated directly. The

standard deviation of btc used for its confidence inter-

val was calculated using the following expression:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rbtc; rband rtcare the respective standard deviations.

rbtc¼ btc

r2

b=b2þ r2

tc=t2

cþ 2covðb;tcÞ=btc

q

;

where

Population approach: prey dependence

An effective and powerful approach to distinguish between

type-II and type-III functional responses is to analyze the

relationship between the proportion of prey captured (Ne/

N), and the number of prey available N (described and

reviewed in Juliano 2001). This relationship always

decreasesmonotonicallyfor

response, whereas for the type-III functional response, it

first increases at low prey densities and then decreases with

increasing prey density. By studying the sign of the slope at

the origin, it is possible to determine whether the functional

response is type II (negative slope) or type III (positive

slope). To this end, we used a generalized linear model (the

‘‘glm’’ function, ‘‘logit link’’) to study the relationship

between the proportion of prey captured (binomial data)

and prey density. We used a third order polynomial in the

prey density as the linear regressor to let enough freedom

in the relationship to capture a potential positive slope at

the origin (Juliano 2001). We tested the effect of predator

and obstacle densities on each term of the polynomial and

simplified the full model using AIC.

thetype-IIfunctional

Population approach: predator dependence and the impact

of obstacle density

We used non-linear regression analysis to determine (1) the

existence and shape of predator dependence and (2) the

effect of obstacle density on prey and predator depen-

dences in the functional response. To do this, we compared

a set of functional response models (Tables 1, 2) fitted to

Ne, the number of prey captured per assay.

To investigate the effect of obstacle density, X, on the

functional response, we modeled its effect on parameters of

the four models considered: the attack rates a, the

encounter rate between predator b, and the interference

strength m (Tables 1, 2). We made no assumption regard-

ing the shape of the relationship between obstacle density

and each parameter: we used categorical coding, resulting

in five possible functions depending on the grouping of the

levels of obstacle density. For example, a(X) = (a1, a23,

a52) refers to the case where the attack rate takes three

values, one for each of the three obstacle density levels (1,

23, and 52 obstacles, respectively). a(X) = (a1, a23–52)

refers to a models where a take two values, one for

the lowest density of obstacles and another one for the

two highest levels of obstacles densities. Functions

a(X) = (a1–23, a52) and a(X) = (a1–52, a23) are constructed

in the same way. Finally, a(X) = a means that a does not

vary with X. In this way, we obtained five different models

for the Holling and Arditi–Ginzburg models and 25 models

forthe Beddington–DeAngelis

models.

In the Beddington–DeAngelis model, b and tccannot be

estimated separately: we can only estimate the combined

parameter btc. The different models were fitted to the data

using non-linear least-squares regression (combination of

the ‘‘genoud’’ function from the ‘‘rgenoud’’ library and the

‘‘nls’’ function from the ‘‘nlme’’ library). The ‘‘genoud’’

function uses a genetic algorithm to explore the whole

parameter space in order to give good initial parameter

estimates for the ‘‘nls’’ function. This reduces the risk of

being trapped in local minima. The models were compared

in two successive steps using information theory criteria

(Burnham and Anderson 2002). For each of the four

functional response models, we first selected the best

function for the dependence of the parameters on the

obstacle density using a stepwise AIC procedure. In the

second step, we selected the best model for predator

dependence, once again using AIC because some models

are not nested (for instance, Arditi–Ginzburg and Bedd-

ington–DeAngelis). We computed the difference, DAICi,

between the AIC of each model, i, and the AIC of the

‘‘best’’ model (i.e., the one with the lowest AIC). From

these DAICi, we computed the Akaike weights (the ‘‘ak-

aike.weights’’ procedure, ‘‘qpcR’’ library), interpreted as

‘‘the probability that model i is the best model for the

observed data, given the candidate set of models’’ (Burn-

ham and Anderson 2002). Likelihood ratio tests were used

to test particular hypotheses on nested models.

Confidence intervals were estimated with a bootstrap

procedure: the BCa (bias-corrected and accelerated) con-

fidence interval (the ‘‘boot.ci function’’, ‘‘boot’’ library)

andArditi–Akc ¸akaya

630Oecologia (2010) 163:625–636

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was computed from 1,000 bootstrap samples (the ‘‘boot.ci

function’’, ‘‘boot’’ library).

Results

Behavioral approach: attack and encounter rates

The attack rate was not affected by either prey density

(F1= 1.83, P = 0.18) or predator density (F2= 0.33,

P = 0.72). The fact that the attack rate did not vary with

prey density means that we were dealing with a type-II

functional response (Hassell 1978). The absence of effect

of predator density on the attack rate suggests the absence

of predator dependence modeled in the way of the Arditi–

Akac ¸akaya or Arditi–Ginzburg models. Moreover, the

attack rate was significantly modified by obstacle density

(F2= 8.20, P = 0.001; Fig. 2a). Tukey’s post hoc tests

indicate that the attack rate for the lowest level of obstacle

density (1 obstacle: a1) is significantly higher (a1vs. a23:

P = 0.003; a1vs. a52: P = 0.003) than that for either of the

higher obstacle densities (23 and 52 obstacles: a23and a52,

respectively). The attack rate did not differ significantly

between the two highest obstacle densities (a23vs. a52:

P = 0.99; Fig. 2a).

The encounter rate between predators was affected by

neither prey density (F1= 0.15, P = 0.70) nor predator

density (F1= 2.41, P = 0.13), whereas it was significantly

10 10

15 15

20

20

0.50.5

1.01.0

1.5

1.5

2.0

2.0

2.5

2.5

3.0

3.0

3.5

3.5

-0.2-0.2

0.00.0

0.20.2

0.40.4

0.60.6

0.80.8

20 20

2525

3030

3535

§§

****

§*

**

§§

123 52

12352

DirectFitting

DirectFitting

a

§*

b

d

Attack rate a (cm2.min-1)

Handling time th(min)

btc(cm2)

Encounter rate between

predators b (cm2.min-1)

c

Obstacle density X (no.arena-1)

Method

Fig. 2 Estimates of functional

response parameters. The effect

of obstacle density on the attack

rate (a) and on the encounter

rate between predators (c). The

handling time (b) and the

combined parameter btc(d).

The parameters (mean ± 95%

CI) are estimated either directly

(behavioral approach: black,

filled circles) or indirectly using

non-linear fitting of functional

response models (population

approach: gray, open circles).

For the latter method, the attack

rate and the handling time are

derived from the selected

Holling model (Table 4, model

2), and the btcis estimated by

fitting the Beddington–

DeAngelis model (Table 4,

model 4). § and * indicate

significant differences in the

direct estimation of the attack

rate (a) or of the encounter rate

between predators (c) at

different obstacle densities

(Tukey’s post hoc tests)

Oecologia (2010) 163:625–636 631

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decreased by increasing obstacle density (F2= 4.06,

P = 0.03; Fig. 2c). Tukey’s post hoc tests indicate that the

encounter rate was significantly higher (b1 vs. b52:

P = 0.03) for the lowest obstacle density than for the

highest obstacle density, with the intermediate obstacle

density not differing significantly (b1vs. b23and b23vs b52:

P[0.12) from the other two (Fig. 2c).

Behavioral approach: interaction times

Neither obstacle density (v22= 1.48, P = 0.48), prey

(v21= 0.61,

P = 0.44) density, or predator

(v22= 2.22, P = 0.33) significantly affected the handling

time th. By means of direct estimation, the average duration

of prey handling was found to be 24.9 (23.0, 26.8) min

(Fig. 2b).

The interference time between the focal predator and

other predators was also found to be independent of prey

density(F1= 0.29,

P = 0.59),

(F1= 1.44, P = 0.23), and obstacle density (F2= 0.17,

P = 0.84). Interactions between predators were quite brief,

lasting, on average, slightly less than 1 s [mean 0.9 (0.8,

1.0) s].

density

predator density

Population approach: prey dependence

The proportion of prey captured decreased with prey den-

sity (v21= 145, P\0.0001) and with prey density

squared (v21= 13, P = 0.0003), but did not vary with the

cubic term (v21= 1.37, P = 0.24). Neither the density of

predators (predator density 9 prey density: v22= 2.28,

P = 0.32) nor that of obstacles (obstacle density 9 prey

density: v22= 0.90, P = 0.64) modified the effect of prey

density. Thus, the proportion of prey captured decreased

monotonously with increasing prey density (Fig. 3a)

whatever the levels of predator or obstacle densities. This

means that the shape of the functional response remained

type II under all of our conditions. This result was quali-

tatively robust to the withdrawal of the highest (2) and

lowest (64) prey densities.

Population approach: predator dependence

and the effect of obstacle density

The ‘‘best’’ model, i.e., the one with the lowest AIC,, was

the Holling model in which the attack rate is reduced for

the highest levels of obstacle density (model 2 in Table 4).

Three other models also fitted the data quite well (small

DAIC): the Holling model with no obstacle density effect

(model 1 in Table 4) and the Beddington–DeAngelis and

the Arditi–Akac ¸akaya models with a negative effect of

obstacle density on the attack rate (models 4 and 6,

respectively, Table 4).

A comparison of the models indicates that the Holling

model is the most likely one for our springtail–mite system,

with a 55.4% cumulative probability of being the best

model. This is supported by likelihood ratio tests (LRTs)

indicating that the interference parameters, btcand m, are

not significantly different from 0 (Table 4). Note that the

likelihood that the Arditi–Ginzburg model is the best one is

0.50.5

1.01.0

1.5

1.5

2.0

2.0

0.00.0

0.10.1

0.20.2

0.30.3

0.40.4

0.50.5

0.60.6

0.70.7

0102030405060

0102030405060

ba

Prey density N (no.arena-1)Prey density N (no.arena-1)

Number of prey captured Ne

Proportion of prey captured

Fig. 3 Effect of obstacles on prey dependence. a Proportion of prey

captured per predator per hour as a function of prey density. Data

(mean ± 95% CI) and fitted model [ln(p/(1-p)) * -1.1765-

0.0725 N ? 0.0006 N2], where N is the prey density, and p is the

proportionofpreycaptured.

(mean ± 95% CI) per predator per hour as a function of prey density

for each level of obstacle density. Light gray One obstacle, medium

gray 23 obstacles, black 52 obstacles. 19% of predators did not

b

Numberofpreycaptured

capture any prey, 45% captured one prey, and 32% captured more

than one prey. The three curves represent the fitting of the Holling

model where the attack rate differs for each level of obstacle density,

a1 (solid light-gray line), a23 (solid medium-gray line), and a52

(dashed black line) corresponding, respectively, to a density of one,

23, and 52 obstacles. Parameter estimates are the following:

a1= 2.20 (1.54, 3.24) cm2min-1, a23= 1.31 (0.86, 2.14) cm2

min-1, a52= 1.29 (0.86, 2.01) cm2min-1, th= 29.0 (24.9, 33.7) min

632Oecologia (2010) 163:625–636

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almost zero (Table 4), which means that this model is not

supported by the results of this experiment. In addition, the

LRTs indicate that the interference parameter, m, is sig-

nificantly different from 1 (Table 4). Thus, our results

suggest that predator dependence is not an important factor

in the functional response of our springtail–mite system. A

purely prey-dependent functional response is here more

relevant.

Models that take obstacle density into account in the

attack rate only always had a lower AIC than equivalent

models without obstacle density (odd vs. even models,

Table 4). Of the four types of functional response model,

the best model with an obstacle density effect had the

attack rate as a(X) = (a1, a23–52) (Table 4; Fig. 3b). The

two attack rate estimates, a1and a23–52, were fairly similar

between the three best models (Holling, Beddington–

DeAngelis, and Arditi–Akc ¸akaya; Table 4). The null

hypothesis test, a1= a23–52, using LRT was marginally

significant for the Holling, Beddington–DeAngelis, and

Arditi–Akc ¸akaya models (Table 4), which suggests that it

is quite likely that the attack rate is reduced when obstacle

densities are high. For both the Beddington–DeAngelis and

the Arditi–Akc ¸akaya models, the best model with an

obstacle density effect had btcand m parameters that did

not vary with obstacle density (Table 4). To conclude,

these results suggest that obstacle density can reduce the

Table 4 Comparison of the fits of the functional response models, parameters estimated, and likelihood ratio tests

Model nameModel no.a

kb

DAICic

wid

Parameters estimatede

Likelihood ratio tests

(hypothesis tested)

Holling12 1.660.168

a = 1.58 (1.20, 2.16)

th= 29.5 (25.2, 34.1)

a1= 2.20 (1.54, 3.30)

a23–52= 1.30 (0.95, 1.86)

th= 29.0 (24.9, 33.7)

a = 1.70 (1.14, 2.50)

2300.386

model 2 vs. 1 (a1= a23–52):

F1= 3.65, P = 0.057

Beddington–DeAngelis333.410.070

btc= 0.10 (-0.19, 0.58)

th= 29.3 (24.9, 34.0)

a1= 2.27 (1.48, 3.73)

a23–52= 1.41 (0.95, 2.09)

btc= 0.11 (-0.17, 0.58)

th= 28.8 (24.7, 33.2)

a = 1.63 (1.19, 2.31)

44 1.700.165

model 4 vs. 3 (a1= a23–52):

F1= 3.69, P = 0.055

model 4 vs. 2 (btc= 0):

F1= 0.30, P = 0.586

Arditi–Akc ¸akaya533.610.063

m = 0.03 (-0.19, 0.25)

th= 29.4 (25.4, 34.3)

a1= 2.17 (1.44, 3.56)

a23–52= 1.35 (0.95, 2.07)

m = 0.04 (-0.19, 0.26)

64 1.920.147

model 6 vs. 5 (a1= a23–52):

F1= 3.67, P = 0.056

model 6 vs. 2 (m = 0):

F1= 0.08, P = 0.782

th= 28.9 (24.9, 33.7)

a = 8.54 (5.44, 14.46)Arditi–Ginzburg7240.03

\0.001

th= 35.8 (30.9, 41.0)

a1= 11.78 (6.53, 28.24)

a23–52= 6.82 (4.08, 12.68)

th= 35.3 (52.4,162.5)

8340.02

\0.001

model 8 vs. 7 (a1= a23–52):

F1= 0.30, P = 0.586

model 8 vs. 6 (m = 1):

F1= 42.32, P\0.001

See Table 1 for parameter definitions

aX, Attack rate for obstacle density X (cm2min-1); th(min); btc(cm2); m (dimensionless)

aModels with an odd number are simple models with no effect of obstacle density on the attack rate or interference parameter. Models with an

even number are those selected using the Akaike Information Criteria (AIC) within all possible models, including obstacle effect on attack rate

and/or interference parameter

bk, Number of parameters

cDAICiDifference in AIC between the ith model and the best model

dwiAkaike weights of the ith model

eValues are given as the mean with the 95% CI

Oecologia (2010) 163:625–636633

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attack rate, but it does not seem to affect the interference

parameters. In a nutshell, the prey dependence of the

functional response seems to be sensitive to obstacle den-

sity, whereas the predator dependence is not.

Discussion

Predators usually search for prey in heterogeneous envi-

ronments. By modifying the manner in which individuals

interact, small-scale heterogeneity, such as refuges or

obstacles, can modify the predator’s functional response.

While the consequences of refuges on the functional

response have been well documented, the impact of

obstacles has been poorly investigated. Here, we show that

obstacles do not affect the functional response in the same

way as refuges.

The effect of obstacles on the functional response

Based on predictions from reaction–diffusion models, we

expected that the encounter rates (predator–prey and

predator–predator) would decrease with an increasing

density of obstacles but that interaction times (handling and

interference time) would remain unaffected by obstacles.

Tracking individual behavior allowed us to test these pre-

dictions directly (behavioral approach). As expected, we

found that both the attack rate and the encounter rate

between predators were lower when obstacle density is

high (Fig. 2a, c). Our analyses suggest that the effect of

obstacle density is not linear and tend to saturate at high

densities. Our second prediction was also fulfilled: neither

the handling time nor the interference time were affected

by obstacle density. Thus, our results for the behavioral

approach were consistent with our theoretical predictions.

As expected in the light of the behavioral approach,

obstacles did not modify the type of prey dependence: the

functional response followed a type-II pattern, whatever

the obstacle density. However, the degree of type-II prey

dependence was slightly affected by obstacle density.

Increasing obstacle density marginally reduced the attack

rate (LRT in Table 4, Fig. 3b). This analysis also shows

that the effect of obstacles saturates at high densities. The

results obtained with the population approach are thus

pretty similar to those derived independently from the

behavioral approach (Fig. 2a).

We did not find any significant effect of predator density

on the functional response. The interference parameters of

the Beddington–DeAngelis and Arditi–Akc ¸akaya models

(btIand m) were not significantly different from 0 (LRT,

Table 4). We can relate this finding to the fact that inter-

ference time between predators (directly estimated in the

behavioral approach) was found to be very short, less than

1 s on average. This was much shorter than the 25 min of

handling time. Even if the predators encountered each

other fairly frequently (an order of magnitude more often

than each of them encounters a prey), the total time spent

by predators interacting with one another was short and

basically negligible. Thus, once again, both approaches

gave a consistent picture of predator–prey interaction in

our experimental system. This absence of any effect of

predator density (Table 4) on the functional response

explains why our system could not reveal any effect of

obstacle density on predator dependence.

Behavioral basis of the functional response

As well as providing a qualitatively consistent picture of

the predator–prey interaction, our two approaches also led

to quantitative agreement in the parameter values estimated

either directly (behavioral approach) or indirectly (popu-

lation approach). Indeed, the mean values and confidence

intervals for the attack rate, a, (Fig. 2a), the handling time,

th, (Fig. 2b), and the combined interference parameter, btc

(Fig. 2d), estimated from the direct observation of indi-

vidual behavior were very close to those obtained by fitting

the functional response models. The quantitative agreement

in the handling time could be due to the fact that in our

experimental conditions (i.e., experiment duration), pre-

dators did not get satiated during trials. The estimated

handling time from functional response models (the pop-

ulation approach), which can include time for eating and

digesting the prey, is reduced to the time needed for han-

dling and sucking up a prey, which corresponds to the

handling time that we measured directly through the

behavioral approach. The similarity between the results of

both approaches supports the behavioral interpretation of

the parameters of the functional response model proposed

by Holling (1959b) and Beddington (1975) and summa-

rized in Table 1.

Mechanisms of obstacle effect

We observed that obstacle density decreased the encounter

rates. Reaction–diffusion models show that the density of

obstacles decreases the spatial spread of particles (in our

case, predators). By modifying the trajectory of predators,

obstacles increase the probability that a predator explores a

place he already went through and that he had previously

clearedofprey.Therefore,

encounter rate of the predator with prey and other preda-

tors. Another mechanism can reduce the encounter rates

when obstacle density increases in our experiment: preda-

tors could lose time to detect and handle obstacles they

encounter during their search for prey. The handling of

obstacles could decrease the attack rate in the same manner

obstacles decreasethe

634Oecologia (2010) 163:625–636

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as the handling of inedible prey can decrease the attack rate

of predators foraging in a plurispecific prey community

(Wei and Walde 1997; Kratina et al. 2007). The duration of

this obstacle handling was not measured in our experiment.

Although this obstacle handling is probably very short,

similar to the interference time (\1 s), the accumulation of

a high number of short handling of obstacles could

decrease the time available to search for prey and, ulti-

mately, also participate in decreasing the attack rate and the

encounter rate between predators.

Refuges versus obstacles and predator dependence

Previous experimental studies (Hildrew and Townsend

1977; Folsom and Collins 1984; Lipcius and Hines 1986)

have shown that spatial heterogeneity produced by refuges

can reduce the predation rate. Here, we demonstrate that

refuges are not required for a decrease in the predation rate

to be observed in response to environmental heterogeneity.

We proposed and tested an additional dimension of envi-

ronmental heterogeneity, namely that of obstacles to indi-

vidual movements, and show that this too can reduce the

predation rate by reducing the attack rate. However, ref-

uges and obstacles could have opposite effects on the

interference between predators. Theory suggests that ref-

uges could increase interference between predators, thus

enhancing the predator dependence of the functional

response (Cosner et al. 1999). In contrast, we demonstrate

that obstacles can reduce the encounter rate between pre-

dators and therefore reduce their interference. As far as we

are aware, our study is the first characterization of the

effect of spatial heterogeneity on the dependence of the

functional response on predator density.

Because small-scale spatial heterogeneities, such as

refuges or obstacles, can have opposite effects on predator

dependence, it is important to clarify the nature of spatial

heterogeneities when their consequences on trophic inter-

actions are to be studied.

Refuge versus obstacle and stability of predator–prey

systems

Small-scale heterogeneities, such as refuges or obstacles,

are likely to modify the stability of predator–prey com-

munities. Because refuges and obstacles can have opposite

effects on predator dependence, they could have different

effects on the stability of predator–prey systems. Refuges

are considered to increase the stability of predator–prey

systems. Indeed, refuges can switch the functional response

from the type-II function response to that of type III, which

in turn can increase the stability of predator–prey systems

(e.g., Murdoch and Oaten 1975). Moreover, refuges can

increase interference, which can also increase the stability

(e.g., DeAngelis et al. 1975). Thus, the effects of refuges

on prey and predator dependence can promote the stability

of predator–prey systems.

Our study shows that obstacles can reduce the attack

rate, which is also a factor of stability. A reduced attack

rate can stabilize predator–prey dynamics by damping the

oscillations of population densities and by shifting them

away from extinction thresholds (Crowley 1978). Thus, in

predator–prey systems having low levels of interference

between predators (as in our collembola–mite system),

obstacles, through their effect on attack rate, are expected

to increase the stability. However, if the interference is

strong, the consequences of the presence of obstacles are

not known. Indeed, our study shows that obstacles can

reduce the interference between predators that is known to

promote the stability of predator–prey systems (e.g.,

DeAngelis et al. 1975). Thus, the stability of predator–prey

communities will result from the balance between these

two antagonistic effects.

Since small-scale spatial heterogeneities such as refuges

or obstacles can have opposite effects on predator depen-

dence, we encourage investigators to clarify the nature of

the spatial heterogeneities and the strength of the predator

interference in their future experiments in order to be able

to understand and predict the effect of these heterogeneities

on the stability of communities.

Acknowledgments

Renault for their technical assistance. We are grateful to CEREEP for

providing an excellent working environment at the field station where

this work was carried out. We thank M. van Baalen for stimulating

discussions and Monika Ghosh for assistance with the English. C.H.

thanks R.A. for funding this research during a short postdoc at Ag-

roParisTech. This research complies with all applicable regulatory

requirements.

We thank A.-S. Barbero, K. Jaouannet and Q.

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