1

From individual decisions to population spatial distribution:

Model and field experiments on the parasitoid Aphelinus asychis

Xavier Fauvergue1, Keith R. Hopper

2 and Eric Wajnberg

1

1 Institut national de la recherche agronomique

37 boulevard du Cap

06600 Antibes

France

mail: fauverg@antibes.inra.fr

tel. 33 4 93 67 89 04

fax 33 4 93 67 88 97

2 Beneficial Insect Introduction Research Unit

Agricultural Research Service

United States Department of Agriculture

University of Delaware

501 South Chapel Street

Newark, Delaware 19713

USA

Corresponding author: Xavier Fauvergue

Running headline: from individual decisions to population spatial distribution

Submitted to Behavioural Ecology (21 Feb. 02)

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From individual decisions to population spatial distribution:

Model and field experiments on the parasitoid Aphelinus asychis

Abstract. To predict the effect of density of foraging parasitoids on their dispersal, we

developed a spatially explicit individual-based model in which female parasitoids visit

host patches at random but use optimal decision rules to leave these patches. We used a

patch-leaving decision-rule inspired by the marginal value theorem where each forager

compares its instantaneous attack rate with its learned estimate of average attack rate.

Simulations with different combinations of host patch distribution, host density, mutual

interference between parasitoids, and parasitoid density suggested the average dispersal

distance of parasitoids should increase with parasitoid density. To test this prediction,

we used data from two large-scale field experiments where we had released either few

or many adults of the parasitoid Aphelinus asychis (Hymenoptera: Aphelinidae) and

subsequently assessed their dispersal distances via the spatial distribution of parasitism

on their hosts Diuraphis noxia (Homoptera: Aphididae). Dispersal distances increased

with the number released as predicted by the model in one experiment but not in the

other. We propose three hypotheses to explain this inconsistency among experiments.

Two hypotheses involve differences in the number released, and one involves

environmental variability among experimental sites. We discuss potential limits to field

research linking individual decisions and population-level phenomena.

Key words (different from words in the title): dispersal, patch time allocation,

marginal value theorem, diffusion, biological control, individual-based modelling.

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Introduction

The link between individual decisions and population-level phenomena is a

rising issue in animal ecology (reviews in Hassell and May, 1985; Koehl, 1989; Lima

and Zolner, 1996; Lomnicki, 1999; Smith and Sibly, 1985; Sutherland, 1996). From the

perspective of population dynamics, founding population models on individual

behaviour fulfils the need for a general a priori theory of population phenomena, the

theory to evolution by natural selection (Lomnicki, 1999; Sutherland, 1996). From the

perspective of behavioural ecology, integrating individual decisions into population

models is a means to derive individual-based predictions at the population level.

Discrepancies between such predictions and field-collected data serve to evaluate the

importance of finely tuned behaviour under naturally occurring environmental

variability. Optimal foraging theory is a cornerstone of this individual-population

approach. First, it offers an evolutionary framework to analyse behaviour because it

assumes that, given a number of constraints, individual decisions contribute to

maximise fitness (Stephens and Krebs, 1986). Second, it decomposes behaviours into

processes that are sufficiently simplified to be incorporated in population models.

One of the most productive attempts to integrate individual behaviour and

population phenomena has focussed on the interplay between foraging decisions in a

patchy environment, spatial distribution, and long-term population dynamics. The ideal

free distribution (Fretwell and Lucas, 1970) has been central to this approach

(Sutherland, 1983, for the seminal work); given the assumptions that foragers are

omniscient and have equal competitive abilities, their distribution should result in equal

fitness gains on every patch, whatever the amount of resources or the number of

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competing foragers (Kacelnik et al., 1992, for a clear discussion of underlying

assumptions and predictions). As an evolutionary stable strategy, the ideal free

distribution model is rooted in evolutionary theory but proposes no behavioural

mechanism that could lead to such a population distribution. Nonetheless, a family of

individual-based models, that we will henceforward refer to as BKK models (sensu

Ward et al., 2000), suggest that for some combinations of individual behaviour

involving movement between patches, patch residence times, and learning (see model

section for details), the population spatial distribution should converge toward the ideal

free distribution (BKK parents: Regelmann, 1984; Bernstein et al., 1988; 1991; BKK

offspring: Beauchamp et al., 1997; Tyler and Rose, 1997; Ward et al., 2000). In turn,

the distribution of foragers in a patchy environment can affect population dynamics

(Bernstein et al., 1999; Comins and Hassell, 1979; Krivan, 1997; Sutherland, 1983; van

Baalen and Sabelis, 1999; 1993). This is particularly significant in host-parasitoid

systems because the distribution of parasitoids has immediate consequences for their

reproduction and thus, for growth of their population and mortality in the population of

their hosts. Whether or not spatial aggregation stabilises otherwise unstable population

models has been the focus of much controversy (Bernstein, 2000, for a lucid and recent

review).

Theoretical models linking individual behaviour and population dynamics have

rarely been tested (but see Cronin and Strong, 1999; Goss-Custard et al., 1995; Levin et

al., 2000; Pulido and Diaz, 1997; Shimada, 1999; Visser et al., 1999), and BKK models

have not been tested at all. One reason is that the primary objective of BKK models was

to study the consequences of relaxing some unrealistic assumptions of the ideal free

distribution. BKK models incorporated assumptions such as imperfect knowledge of the

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environment (Bernstein et al., 1988), non-negligible travel times (Bernstein et al.,

1991), conspecific attraction among foragers (Beauchamp et al., 1997) or the presence

of predators (Tyler and Rose, 1997; Ward et al., 2000). In some cases, model

predictions were compared with those from the ideal free distribution rather than with

data (Beauchamp et al., 1997; Bernstein et al., 1988; 1991; Ward et al., 2000). Another

reason for the unbalance between theory and data may be the difficulty of carrying

experiments to test theoretical predictions. Insect parasitoids are excellent organisms to

test BKK models because their foraging decisions are tightly linked to fitness (Godfray,

1994; Godfray and Shimada, 1999), but their small size represent a serious

methodological challenge for field experiments (Casas, 2000). The difficulty of

obtaining experimental field data has sometimes been circumvented with the use of

multi-patch laboratory microcosms as experimental environments (Cook and Hubbard,

1977; Hubbard and Cook, 1978; Tregenza et al., 1996; Visser et al., 1999), but

inferences about population distribution from such small-scale experiments are

questionable.

Here we develop an individual-based model for population spatial distribution,

with realism in assumptions about individual behaviour comparable to that first used by

Bernstein et al. (1988), but with better insight concerning which predictions can be

tested with parasitoids in the field. For this, we developed predictions testable with

time-integrated mass mark-recapture experiments, where all individuals are released at a

particular point in space and recaptured later with traps cumulating their progeny over a

non-negligible trapping period (Turchin, 1998). We focused on the effect of the number

of individuals released because density is expected to affect patch use via competition

(Milinski and Parker, 1991), and release size is a variable relatively easy to manipulate

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in the field. The second aim of this work is to compare our predictions with results from

two large-scale field experiments where the number released was manipulated, and the

post-release population distribution measured.

The model

Environment. The model environment consisted of 1000 host patches positioned within

a regular array of 100×100 spatial coordinates. Such an environment is about 200 times

larger than the one modelled by Bernstein et al. (1988; 1991; see also Ward et al., 2000)

because our aim is to simulate the movement of foragers in field conditions. For the

same reason, we also used three different spatial distributions of host patches: regular,

random and aggregated. In the regular pattern, patches with hosts were regularly spaced.

In the random pattern, the 1000 patches were placed at coordinates selected randomly,

but with the condition that no more than one patch occurred per coordinate. In the

aggregated pattern, the 100×100 grid was divided into 100 10×10 plots and the number

of patches in each plot was drawn from a negative binomial distribution with a mean of

10 patches per plot and k=0.5 (which gave moderate levels of aggregation).

Migration rule. Movement of parasitoids between spatial coordinates was random but

constrained to one of the eight neighbouring coordinates between consecutive time

steps. The time individuals remained at a coordinate varied according to the departure

decision rule. Following McNamara & Houston (1985; but see also Ollason, 1980, and

other BKK models), we used a version of Charnov’s (1976) marginal value theorem in

which each individual has no a priori information about patch quality, but learns its

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“personal” estimate of maximum average attack rate in the environment. In our model,

each individual left each encountered patch when its instantaneous attack rate fell below

its estimate of average attack rate, a rule known to converge asymptotically to a

maximisation of average attack rate in different environmental situations (McNamara

and Houston, 1985). Coordinates without patches were considered similar to

coordinates with completely depleted patches. On such coordinates, parasitoids stayed a

single time step. For computations of instantaneous attack rate, we used Hassell &

May’s (1973) model that assumes a type II functional response (Holling, 1959) and

varying amounts of direct mutual interference between foraging females (Hassell and

Varley, 1969):

NaQ N P

Q th N Pi

i i

m

i i

m=+

−

−1 (Equation 1)

where Nai is the number of hosts attacked per female at each time step on patch i (Nai is

thus the instantaneous attack rate), Ni and Pi the number of healthy hosts and the

number of parasitoids on patch i, Q the quest constant, m the coefficient of interference,

and th the handling time. On a given patch, the instantaneous attack rate Nai decreased

with time because of host depletion (Nai Pi hosts were removed from each patch at each

time step). For computations of average attack rate, we used a linear operator weighing

past and present experience with a memory factor (Bush and Mosteller, 1955, and other

BKK models):

γt = αNai + (1-α)γt-1 (Equation 2)

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where γt is the average attack rate learned at time t and α the memory factor. The initial

condition for γt was γ0=0. Given these formulations for attack rates, the effect of forager

density could appear both instantaneously, as a consequence of direct mutual

interference when m>0, and through time as a consequence of patch depletion.

Simulations. In each run, parasitoids were released at the centre of the grid and allowed

to move among coordinates during 200 time steps. The resulting population spatial

distribution was evaluated after 50, 100, 150 or 200 time steps by relating the

proportion of progeny produced by the released parasitoids to the distance from the

release point. We used proportions rather than counts to allow the comparison of spatial

distributions among populations of different sizes. To predict the effect of number

released, the model was run with different values for the number of parasitoids released

(1, 10, 100 and 1000). Furthermore, because we suspected parasitoid dispersal to be

influenced by the number of hosts per patch, the spatial distribution of host patches, and

the amount of interference between wasps, each number released was tested in the three

different spatial distributions of patches described above, in combination with three

different initial numbers of hosts per patch (10, 100 and 1000) and three values of

interference (0.0, 0.5 and 1.0). One hundred replicated simulations were performed for

each combination of initial conditions.

Predictions. The simulations showed an effect of number of parasitoids released on the

spatial distribution of their progeny. Higher number of parasitoids released resulted in

higher proportions of their progeny further from the release point (Fig. 1). This pattern

appeared soon after introduction and perpetuated through time. Such a shift in the

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distribution of distances between the release point of parasitoids and the capture of

progeny suggests that the average distance between emergence and reproduction

increases with the number released. Our model thus predicts a form of density-

dependent dispersal. This prediction was robust; the quantitative effect of number

released on post-dispersal population distribution was unaffected by host density, host

spatial distribution or mutual interference between parasitoids. Only host density had a

significant main effect on the spatial distribution of progenies: populations distributed

globally closer to the release point in environment with richer host patches. Host spatial

distribution or mutual interference between parasitoids had no such main effects.

The field experiments

Insects. To test the main prediction of the model, we studied the post-release spatial

distribution of populations of the parasitoid Aphelinus asychis Walker (Hymenoptera:

Aphelinidae). A. asychis is a solitary endoparasitoid of a number of aphid species

(Jackson and Eikenbary, 1971; Thompson, 1953) including the Russian wheat aphid

Diuraphis noxia Mordvilko (Homoptera: Aphididae) (Chen and Hopper, 1997).

Individual D. noxia have a characteristic colonial distribution on wheat leaves, which

can be modelled as a patchy distribution. Adult A. asychis are small (about 1 mm long)

and although they have wings that can aid in long-distance air-borne dispersal, field and

laboratory observations suggest that they forage mostly on foot (Fauvergue et al., 1995;

Rao et al., 1999). Females discriminate between unparasitised and parasitised hosts, and

except in some crowded laboratory conditions, they avoid ovipositing in the latter (Bai

and Mackauer, 1990a; but see also Bai and Mackauer, 1990b). Hence, parasitised hosts

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“disappear” from the population of available hosts as assumed in the model. Female A.

asychis are synovogenic (i.e., eggs are slowly matured in the course of adult life) and

live an average of one month when fed with honey in the laboratory. With such a life

span, females probably encounter many patches in the wild, and this should favour

plasticity in patch residence times. As predicted by the marginal value theorem, patch

residence time of A. asychis increases with increasing number of hosts (Li et al., 1993),

but we do not know whether patch leaving decisions rely on comparisons between

instantaneous and average attack rates.

Releases. To test for the effect of the number of parasitoids released on population

distribution, we put either few or many mummified aphids parasitised by Aphelinus

asychis at ten release points (1994; five replicated releases per release size) or eight

release points (1996; four replicated releases per release size) and studied the

distribution of the resulting populations. In 1994, A. asychis were from Italy and mass-

reared on either Diuraphis noxia or Schizaphis graminum (Homoptera: Aphididae) on

wheat. In 1996, A. asychis were from Montpellier, France, and mass-reared on S.

graminum only. For the releases, mummified aphids naturally stuck on dried tillers were

put in 0.5 litre canisters with open mesh tops. To manipulate release size, we put

various numbers of these canisters in wire mesh baskets covered with wooden lids (for

protection from rain) at each release point. Two weeks after the releases, we estimated

the actual number of parasitoids released by counting the number of mummies with

emergence holes in a subsample of each canister. In 1994, we released 334, 356, 366,

442, 628 wasps in small releases and 2506, 2633, 2712, 2731, 3021 wasps in large

releases. In 1996, we released 687, 731, 781, 1050 wasps in small releases and 9187,

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9201, 9493, 9784 wasps in large releases. Subsamples kept in the laboratory to assess

sex ratio showed that males and females were released in equal proportions.

Experimental sites. In 1994, each release point was in a different field planted with

dryland winter wheat Triticum aestivum. The ten fields were scattered along the western

edge of the Great Plains, Weld County, Colorado, USA (40°42’–43’N, 104°48’–54’ W,

elevation 1450–1740 m). In 1996, all the releases were done in a single irrigated wheat

field located on the western slope of the Rocky Mountains (South-western Colorado

Research Centre, Colorado State University) at Yellow Jacket, Colorado, USA

(37°32’N, 108°44’W, elevation 2085 m). In both years, release points were separated

from one another by more than 100 m and were at least 30 m inside the border of the

fields. In 1994, we estimated natural aphid infestation in each field at the beginning of

the experiment to be greater than 10% of wheat tillers infested with D. noxia (i.e., >40

infested tillers per m²). In 1996, the field had an initial natural aphid infestation of 252 ±

105 (mean ± standard deviation, n=1176) infested tillers per m². Naturally occurring D.

noxia were parasitised by the braconid wasp Diaeretiella rapae but at a very low rate. In

contrast, we assumed the initial density of A. asychis to be null because A. asychis

originates from the Old World and no population had been introduced in Colorado

before the experiments.

Measurements. We assessed the spatial distribution of the progeny of the released

parasitoids three weeks after the releases. For this, we counted the number of wasps

emerging from sampling units at fixed spatial coordinates surrounding the release

points. In 1994, we used sentinel plants as sampling units. Sentinel plants consisted of

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potted wheat infested with more than 500 initially unparasitised aphids. The pots were

placed every 2 m along 18×18 m grids centred on the release points, but excluding three

coordinates at each of the four corners (i.e., 88 sampling units per sample). Sentinel

plants were placed in the fields at the time of release, and were watered once a week. In

1996, we used field plants naturally infested with D. noxia as sampling units, and

sampled every 3 m over 18×18 m grids (49 sampling units per sample). Three weeks

after the releases, each sampled plant was clipped and placed in an emergence canister

(1 litre cardboard tubes covered with inverted plastic funnels and sample vials into

which adult parasitoids could move after emergence). Every day following the

collection of field samples, each emergence canister was checked and emerging

parasitoids were collected and counted. Because some parasitoids did not move into the

sample vials, we also searched the contents of each emergence canister for adult

parasitoids after emergence ceased.

Data analysis. The null hypothesis from the simulation model is an absence of

statistical interaction between the number of parasitoids released and the distance from

the release point on the proportion of progeny captured (Fig. 1). To test this hypothesis,

we computed, for each field, the proportion of progeny captured at each distance from

the release point. For this, we cumulated the number of parasitoid offspring captured at

each distance, and divided this number by the total number of progeny captured. Given

the sampling grids, this resulted in 12 different distances in 1994 and 10 distances in

1996. Such data are structured as a repeated-measure design where observations are

repeated within fields over distances (and are, for this reason, spatially autocorrelated).

However, standard repeated-measure analysis of variance assumes a normal distribution

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of error variance and is not suited to proportions that follow a binomial distribution.

Hence, we used a statistical method based on the generalised estimated equations (GEE;

Liang and Zeger, 1986). This method allows logistic regression, i.e., generalised linear

models designed for binomial data using a logit link function (McCullagh and Nelder,

1989), on autocorrelated data. Deviations of the error variance from that assumed by a

binomial distribution were also taken into account. All computations were done with

PROC GENMOD in the SAS Statistical software (SAS Institute Inc., 1999). To

complement this analysis, we summarised the distribution of dispersal distances by

computing, for each field, the mean dispersal distance (i.e., the mean distance between

the release point and each sampling unit weighted by the number of parasitoids

captured). The effect of number released on this mean dispersal distance was tested via

an analysis of covariance in which release size (large versus small) was the class

variable and the number actually released was the continuous covariate.

Results. Over the two years, we captured an average of 136 (range 9-448) parasitoids

per release site. The number captured increased with increasing number released

(Spearman rank correlation r=0.66, p=0.0031). In both years, we found a similar overall

pattern in the distribution of captured progeny: the proportion of progeny captured

peaked at around 3-4 m and decreased monotonically for higher distances (Fig. 2).

However, the interaction between the number released and distance from the release

point differed between years. In 1994, the interaction was marginally significant (Wald

χ²=-2.024, p=0.0430) and revealed no clear pattern (Fig. 2). In 1996, the interaction

between number released and distance was very significant (Wald χ²=13.545, p<0.0001)

and revealed a clear spatial pattern (Fig. 2). That year, as predicted by the simulation

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model, the proportion of captured progeny was higher close to the release point for low

numbers of wasps released, and higher further from the release point for large numbers

of wasps released. These results were confirmed by the analysis of average dispersal

distance (Fig. 3). In 1994, the average dispersal distance did not vary with the number

released (F1, 7=1.44 p=0.27). Average dispersal distance was small overall (mean and

standard error across the 10 replicates: 5.02 ± 0.40 m). In contrast in 1996, dispersal

distance was about twice as high in large releases (4.12 ± 0.15 m) than in small releases

(2.33 ± 0.10 m) (F1, 5=82.40 p=0.0003).

Discussion

Our model based on individual optimal patch leaving decision rules predicted

that the spatial distribution of released parasitoid populations should depend on the

number released: females from large releases should reproduce further from the release

point than females from small releases. Among two data sets from manipulative field

experiments, one did not fit the prediction (the 1994 data) while the other did (the 1996

data). Such a discrepancy between some, but not all, of the data and the model is of

interest, because if we understand which of the differences between experiments

explains the differences in results, we may in turn understand under which conditions

the decisions of individuals significantly affect the spatial structure of populations.

Three main hypotheses may explain the differences in result between 1994 and

1996. First, it is obvious from our model that a higher difference in the number released

results in a higher difference in the post-release spatial pattern (Fig. 1). In the field

experiments, the average difference between small and large releases was much higher

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in 1996 (812 versus 9416) than in 1994 (425 versus 2721). This difference could

account for statistically significant results in 1996 but not in 1994. Second, it is possible

(although it is not predicted by the model) that dispersal becomes affected by density

only above a certain threshold density. Because we released higher numbers of

parasitoids in 1996 than in 1994, this threshold may have been attained in 1996 but not

in 1994. Third, experimental designs differed between 1994 and 1996. Before the 1994

experiment, the dispersal capacity of A. asychis was unknown. Each parasitoid

population was thus released in a different wheat field to prevent cross-contamination

among releases. The small dispersal distance observed in 1994 allowed us to make all

releases in the same wheat field in 1996, but at sites far enough apart to prevent from

cross-contamination. For this reason, the variability among replicates was much higher

in 1994 than in 1996. This can be seen on Figure 3 and is also well illustrated by

coefficients of variation of mean dispersal distance among fields, which were about

three times larger in 1994 (27.2 and 19.2 for small and large releases respectively) than

in 1996 (8.4 and 7.3). Such a difference in variability among replicates could again

explain why the post-release distribution observed in 1994 was less precisely estimated

and did not fit the prediction, contrary to that observed in 1996.

The observed differences in results between the two experiments, along with the

three hypotheses proposed to explain these differences, raise some limits to the

predictive value of models aimed to link individual behaviour and population-level

phenomena in the field. In the case analysed here, it seems that the predicted pattern

emerges from experimental populations when the difference in treatments is large (i.e.,

a tenfold increase in the number of parasitoids), and when replicated populations are

studied in environments with low variability (i.e., in the same cultivated wheat field).

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Under naturally occurring higher levels of environmental variability, the predicted

effect of density on population spatial distributions may vanish because of interactions

with other unknown effects, as it may have happened in the 1994 experiment. This

conclusion supports some published reservations concerning efforts to link behavioural

ecology and population dynamics (Tony's comments in Ives and Hochberg, 2000).

The population model that we developed from individual optimal foraging

decisions predicts a form of density-dependent dispersal, characterised by a positive

relation between population density and the distances separating emergence and

reproduction. Such a relation is not exclusive to our model, as it is in general agreement

with both theory and data on dispersal. Density-dependent dispersal was primarily

viewed in early group selection models as a means for reducing population density

below a threshold, for the benefit of individuals that did not disperse (Wynne-Edwards,

1962). More recently, the concept of condition-dependent dispersal, where dispersal rate

depends on predictors of individual fitness like population density, has led to a new

family of evolutionary dispersal models (e.g. McPeek and Holt, 1992). Such models

contrast with classic approaches which usually assumed dispersal to be a constant

fraction of individuals leaving a habitat (Johnson and Gaines, 1990). Although our

simulation model is based on a different modelling philosophy, it shares the underlying

reasoning of condition-dependent dispersal: high densities lead to intraspecific

competition which in turn results in decreased fitness and may for this reason trigger

dispersal. In the wild, a positive relation between density and dispersal (i.e., dispersal

rate or dispersal distance) has been documented in organisms as diverse as collembolans

(Bengtsson et al., 1994), mites (Li and Margolies, 1993), insects (Crespi and Taylor,

1990; Denno et al., 1985; Fonseca and Hart, 1996; Herzig, 1995; Roitberg et al., 1979;

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Strong and Stiling, 1983), lizards (Léna et al., 1998) and mammals (Gaines and

McClenaghan, 1980). For parasitoids, only one other study suggests that dispersal

depends on density as predicted by our model: Anderson and Paschke (1970) analysed a

series of fifteen releases of the egg parasitoid Anaphes flavipes, and observed a positive

relation between the number released and dispersal distance.

Such a consistency of density-dependent dispersal among theoretical and

empirical approaches reinforces our findings. However, the ubiquity of density-

dependent dispersal in models and data also raises some questions concerning the

specific relation between our model and data. How much is the observed post-release

distribution of Aphelinus asychis explained by the assumed optimal foraging decisions?

To answer this question, we must look at how tight are the relations between (1) the

model assumptions and the behaviour of the parasitoid we studied, and (2) the model

predictions and the distribution pattern observed in the field.

First, considering the behavioural decisions, the fact that A. asychis patch

residence time increases with increasing number of hosts (Li et al., 1993) is indeed

consistent with predictions from Charnov's (1976) marginal value theorem. However, it

is also in agreement with predictions from alternative models based on state and

assuming other fitness currencies than maximisation of intake rate (Nonacs, 2001).

Hence, to justify the use of the marginal value theorem as an assumption for patch use

in A. asychis, more specific predictions from this model, such as the equality of

marginal rates of fitness gain (Wajnberg et al., 2000) should be tested. More

experiments are needed on learning. Laboratory experiments have shown that in A.

asychis, patch residence time could be affected by oviposition experience (Li et al.,

1997), and that females were arrested in odour plumes of the host-plant complex that

18

they had experienced in the past (De Farias and Hopper, 1997). However, the relation

between such learning and optimal patch time allocation is still unclear. Only in small

mammals (Cassini et al., 1990) and birds (Cuthill et al., 1990; Todd and Kacelnik,

1993) has the relationship between learning and the marginal value theorem been

documented.

Second, it is possible that developing complementary predictions from the

simulations and collecting additional field data to test these predictions could have

improved the relationship between model and data. For example, correlating the

distribution of parasitoids to the distribution of their hosts, as in other BKK models

(Beauchamp et al., 1997; Bernstein et al., 1988; Bernstein et al., 1991; Ward et al.,

2000) would have refined the analysis. However, generating data is easier in computers

than in the fields. For this reason, many analyses of parasitoid dispersal are based on

observations of the post-release distribution of only one or a few populations (e.g.

Corbett and Rosenheim, 1996; Floate et al., 2000; Goldson et al., 1999; Greatti and

Zandigiacomo, 1995; Krause et al., 1991; Messing et al., 1995; Tobin and Pitts, 1999;

Weisser and Wölkl, 1997). Here, the difficulty was circumvented in two ways. First, we

did not consider the distribution of the aphid hosts but privileged that of the parasitoid.

Second, we used a rapid method for estimating parasitoid density over up to 880

sampling units simultaneously. These methods allowed the replication of mass mark-

recapture trials, and, in turn, highlighted the importance of environmental variability in

approaches aiming to link individual behaviour to population phenomena.

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Note : to ease the review, legends are associated with figures, but they will be

compiled on a different page before final submission.

29

Figure 1. Proportion of parasitoid progeny expected versus distance from the release

point, for different numbers released: 1 (dotted line), 10 (short dash), 100 (long dash) or

1000 (continuous line). The spatial distribution of parasitoids is displayed after 50, 100,

150 and 200 time steps from the release, for an environment consisting of randomly

distributed patches with 100 hosts at the beginning of each simulation, and the

following parameters values for attack rate: m=0.5, Q=0.1, th=0.1, α=0.1.

30

Figure 2. Proportion of Aphelinus asychis offspring collected at different distances

from the release point for low (white symbols) and high (black symbols) numbers

released. Error bars are standard error of means. Displayed means and standard errors

are back-transformed from computations made on arsine-square-root transformed data.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

distance from the release point (m)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

pro

port

ion o

f in

div

idu

als

1994

1996

31

Figure 3. Average dispersal distance of Aphelinus asychis versus number of individuals

released (white: small releases; black: large releases) in 1994 (triangles) and 1996

(circles). Each point represents the average dispersal distance of progeny in a single

released population.

32

Acknowledgements

This research was funded in part by the United States Department of Agriculture NRI

Grant 93-37302-9432 to K.R.H. This experiment would not have been feasible without

the help of Nathalie Ramualde, Richard Turcotte, Aaron Spriggs, Jody Smith, Matthew

Harbey, Aaron Chavez and Adam Roush. Their unbeatable sense of humour despite

torrid summer Coloradoan conditions was also appreciated. Tom Holtzer, Barry Ogg,

Franck Peairs, and Robert Hammon at Colorado State University provided facilities and

crucial support in locating field sites and making contacts. Abdel Berrada and his

colleagues at the South-western Colorado Research Center provided facilities, and

Walter and Heidi Henes kindly offered their hospitality. Finally, we thank Carlos

Bernstein, Emmanuel Desouhant and Laurent Lapchin for helpful criticisms on a

previous version of this manuscript.