Behavioral Ecology Vol. 11 No. 6: 577-586
© 2000 International Society for Behavioral Ecology
Patch leaving decision rules and the Marginal Value Theorem: an experimental analysis and a simulation model
a INRA, Ecologie des Parasitoïdes, 37 Boulevard du Cap, 06600 Antibes, France b INRA, Biométrie, Domaine de Vilvert, 78352 Jouy-en-Josas Cedex, France
Address correspondence to E. Wajnberg. E-mail: wajnberg{at}antibes.inra.fr .
Received 2 February 1999; revised 14 September 1999; accepted 23 January 2000.
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ABSTRACT |
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The patch exploitation strategy of females of the insect parasitoid Trichogramma brassicae was studied on patches containing different proportions of hosts that were previously attacked by conspecific females. On average, T. brassicae females spent more time on patches of higher quality, and all patches were reduced to the same level of profitability before being left. This appeared to be in accordance to the optimal predictions of the Charnov Marginal Value Theorem. The proximate leaving mechanisms involved were analyzed by means of a Cox proportional hazards model. Each oviposition in a healthy host appeared to have an incremental influence on the patch residence time, whereas each rejection of a healthy host or of a host that was previously attacked by the same female (i.e., self-superparasitism) had a decremental effect. These patch leaving mechanisms did not change according to the quality of the patch the females were exploiting. A Monte Carlo simulation was developed around the results of the Cox regression model. The results suggest that this set of patch leaving rules seems to provide the females with a sufficient way to reach the predictions of the Charnov model. Among the different mechanisms involved, the incremental effect associated with each oviposition in a healthy host appeared to play the most important role. The relationship between the proximate mechanistic rules adopted by the females and the ultimate prediction of the Charnov model is discussed.
Key words: patch leaving rules, Marginal Value Theorem, Cox regression model, parasitoids, Trichogramma, Monte Carlo simulation.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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Foraging behavior in insect parasitoids has long been a favored subject in behavioral ecology. Among the different behaviors studied, the time allocated to patches of hosts of different profitability is likely the one that has received most of the interest in the last few decades, particularly in the context of optimal foraging theory (see Stephens and Krebs, 1986
, for
a review). Hosts of most of the parasitoid species occur in discrete patches
in the environment (Godfray,
1994) and, in order to maximize their reproductive success, female
wasps have to adjust accurately their foraging time in each of the patches
they encounter. A number of theoretical models have been proposed in order to
predict the optimal residence time a female should allocate to each visited
patch. The Marginal Value Theorem (MVT) developed by Charnov
(1976) is based on the
assumption that females, foraging on depleting patches of hosts, should
experience diminishing returns. In such a condition, this model predicts that
each patch should be exploited until the rate of fitness gain within the patch
has decreased to a marginal value. This is the mean rate of fitness gain that
can be achieved in all the available patches in the environment. Factors
having an influence on the rate of fitness gain on the patch under
exploitation should lead to a change in the optimal time a female should
remain on the patch. Consequently, three main predictions can be drawn from
the MVT: (1) Females should spend more time on each patch when travel time
between patches is high (Charnov,
1976; Godfray,
1994); (2) females should remain a longer time on patches of
higher quality (Bonser et al.,
1998; McNair,
1982); and (3) patches of different quality should be reduced to
the same level of profitability before leaving
(Bell, 1991;
Cook and Hubbard, 1977). The
last two predictions have been frequently verified experimentally on different
host-parasitoid systems (e.g.,
Hassell, 1978;
Hubbard and Cook, 1978;
Waage, 1979).
These predictions are based on the implicit assumptions that foraging
females have a perfect knowledge of the spatial distribution and quality of
all the patches in the habitat (Iwasa et
al., 1981; McNair,
1982; van Alphen and Vet,
1986; Waage,
1979). For this, foragers are supposed to visit sequentially a
relatively large number of patches and to assess quickly their profitability
(Galis and van Alphen, 1981;
Godfray, 1994;
Krebs et al., 1974). Such
assumptions are not realistic, especially in a stochastic, unpredictable
environment (Green, 1980;
McNamara, 1982). Thus, there
is a need to link the functional, optimality predictions with the actual
proximate mechanisms used by the foragers
(van Alphen, 1993;
van Alphen and Vet, 1986;
Vos et al., 1998). These
mechanisms should be relatively simple, leading the females to adopt an
optimal behavior without having to know the spatial distribution and quality
of all the patches in the habitat (Iwasa
et al., 1981; Shaltiel and
Ayal, 1998).
Despite the fact that they are usually considered to be difficult to
analyze (van Alphen, 1993),
several patch leaving mechanisms have been proposed. They are all built on the
assumption that the females, while foraging on a patch, are likely to sample
their environment in order to collect the information needed to trigger the
patch leaving decision (Green,
1984; Li et al.,
1993; Yamada,
1988). Among the possible mechanisms, the rule suggested by Waage
(1979) is probably the most
well known one attempting to relate a proximate, rule-based to an ultimate,
goal-based model. In his mechanistic approach, Waage
(1978,
1979) supposed that a female
enters a patch with an initial level of responsiveness (i.e., an initial
tendency to remain on the patch) that is related to the number of hosts
available. Then, this responsiveness decreases with the time spent on the
patch, until a critical threshold value is reached whereby the patch is
abandoned. Each oviposition in a host has an incremental influence on the
responsiveness, thereby increasing the total residence time. Such an
incremental effect has been observed in several parasitoid species. However,
it has been shown more recently that in some cases, each oviposition may have
a decremental effect on the responsiveness to the patch, thereby decreasing
patch residence time. This decremental effect, called a
"count-down" mechanism
(Driessen et al., 1995), has
been observed with several other wasp species (see
Driessen and Bernstein, 1999;
Wajnberg et al., 1999, for
recent reviews). Incremental mechanisms are supposed to be adaptive when there
is a large heterogeneity in patch quality in the environment. On the other
hand, a count-down mechanism is generally thought to perform better when the
patches contain an uniform number of hosts
(Iwasa et al., 1981;
Vos et al., 1998).
Using a simple simulation procedure, Waage
(1979) showed that his
incremental mechanistic model could generate patch exploitation strategies
that are in accordance with some of the predictions of the MVT. Using such a
proximate rule, (1) females would stay longer on patches of higher quality,
and (2) the rate of oviposition in the last few min before leaving the patch
would be similar for a range of patch qualities. This suggested that all
patches would be reduced to the same level of profitability before being left.
As far as we know, this is the only study, based on experimental data, trying
to relate patch leaving mechanisms to the ultimate predictions of the MVT.
Therefore, there is a need for a more detailed analysis aiming to relate
accurately the patch leaving rules adopted by the females to theoretical
predictions.
This was the aim of the present study. Females of the egg parasitoid
Trichogramma brassicae Bezdenko (Hym.; Trichogrammatidae) were
offered patches of one of their hosts, the eggs of the Mediterranean flour
moth Ephestia kuehniella Zeller (Lep.; Pyralidae). Patches of
different quality were compared and the females' patch exploitation strategy
was analyzed in the light of the predictions of the MVT. Besides this, the
experimental data were also analyzed by means of a Cox
(1972) proportional hazards
model, a specific statistical method whose features seem to be particularly
appropriate when analyzing data on time allocation (see
Hemerik et al., 1993;
van Roermund et al., 1994;
van Steenis et al., 1996;
Vos et al., 1998;
Wajnberg et al., 1999). In
turn, the results of the Cox model were used to build a Monte Carlo model
simulating the exploitation of a patch by a T. brassicae female. The
results indicate that the females' behavior was in accordance with some of the
predictions of the MVT, and their patch leaving mechanisms, revealed by the
Cox model, seem to provide them with a simple and efficient proximate way to
achieve this ultimate goal.
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MATERIAL AND METHODS |
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Insects and experimental set-up
T. brassicae is a polyphagous parasitic wasp that attacks the eggs of several moth species. It is one of the most intensively produced insects over the world for inundative biological control programs (Wajnberg and Hassan, 1994
).
Like all Trichogramma species, T. brassicae females are
known to mark their hosts after each oviposition
(Salt, 1937). Therefore, the
females of this species can discriminate between healthy and already attacked
hosts (Klomp et al., 1980). To
this end, olfactory cues are sometimes involved during host antennal
inspection (Bruins et al.,
1994); but, in most cases, sensory receptors are used during
probing of the host with the ovipositor
(Le Ralec and Wajnberg,
1990). The strain used in the experiment originates from a population that was collected several years ago from parasitized European corn borer (i.e., Ostrinia nubilalis Hübner; Lep. Pyralidae) eggs in Alsace, France. From the time of capture onwards, the strain was maintained under laboratory conditions on E. kuehniella eggs at 25 °C, LD 12:12.
For this study, less than 24 h old T. brassicae females were each
offered a single patch of nine UV-killed E. kuehniella eggs. These
were arranged on a 3 x 3 regular square grid pattern, with a distance of
one mm between the hosts aligned in rows and columns. The host patches, built
just before the experiment, consisted of different proportions of non-attacked
hosts and hosts that were already attacked by a conspecific female 24 h before
the experiment. After being attacked, and before being used to build the
patches, the hosts were stored at 25°C. The healthy hosts were also stored
under the same conditions. Four treatments were compared (number of replicates
in parentheses): patches with only healthy hosts (31), patches with six
healthy hosts and three attacked hosts (30), patches with three healthy hosts
and six attacked hosts (31), and patches with only already attacked hosts
(29). For the two intermediate cases, the two types of hosts were placed along
the diagonal of the square pattern. This set-up was used in order to modify
the quality of the patch without changing its surface. This appeared to be
important since it has been shown that the time allocated by some parasitoids
could vary according to the patch area, without a change in host density or
quality (van Lenteren and Bakker,
1978). Females were all virgin, honey-fed, and naive (i.e.,
without previous experience with hosts) and were isolated at random, before
adult emergence, from the mass-reared population. This protocol was chosen in
order to run the experiment under conditions that were as standardized as
possible. All experiments were carried out during daytime at 25±1°C
and 60±5% RH. Females were used only once and were free to leave the
patch whenever they wanted. Hosts were not replaced during the observation, so
patches may have suffered a continuous depletion.
The behavior of each female was observed continuously from the moment the
parasitoid entered the patch for the first time up to the moment it left the
patch for more than 60 s. On some occasions, the females left the patch and
walked a few millimeters away before returning to the hosts. Like in Driessen
and Bernstein (1999), these
short excursions were included in the computation of the patch residence
time.
Using an event recorder, the beginning and the end of the following behaviors were recorded during the whole observation with an accuracy of 0.1 s: (1) entering or (2) leaving the patch, (3) antennal drumming on a host, (4) drilling a host with the ovipositor, (5) ovipositing in a host, and (6) walking between hosts. The location of the host on which behaviors 3 to 5 were observed was also recorded. A drilling behavior followed by an oviposition behavior was considered as a successful host attack behavior. When there was no oviposition after a drilling behavior, the host was considered to be rejected.
These data were used to test whether the females' patch exploitation
strategy was in accordance with the predictions of the MVT. The total patch
residence time was thus computed, and also the level of the patch
profitability before leaving. This last parameter should ideally correspond to
the rate of reproductive success just before leaving the patch, and is usually
estimated by the rate of encounters with hosts during the last min before the
patch is abandoned (e.g., Cook and Hubbard,
1977; Hubbard and Cook,
1978; Waage,
1979). However, in the present study, both healthy and already
attacked hosts were present on the patch, therefore attacking a host does not
necessarily produce the same level of reproductive success to the forager. We
thus preferred to estimate patch profitability by the actual rate of progeny
production during the last min before the patch is left. For this, a
complementary experiment was performed with two T. brassicae lines
originating from the same population, but differing by a single neutral random
amplified polymorphic DNA marker (i.e., marker N4, located on linkage group
number III, Laurent et al.,
1998). Using these two lines, we estimated that the attack of a
healthy host or of a host already parasitized 24 h before by an other female
leads to an adult progeny with a probability of 0.8634 (n = 205) and
0.0519 (n = 77), respectively. When a healthy host is attacked twice
during the same patch visit (i.e., self-superparasitism), the total progeny
production reaches an average of 1.2952 (n = 105). These three
parameters were used to estimate accurately the reproductive success of all
females before the patch is left.
The proportional hazards model
The patch leaving mechanisms used by the females were analyzed by means of
a Cox proportional hazards model (also called a Cox regression model). A
thorough description of this model can be found in the literature dealing with
survival analysis (e.g., Collett,
1994; Kalbfleisch and
Prentice, 1980). Briefly, in our case this model expresses the
data in terms of leaving tendency (the so-called hazard rate), which is the
probability per unit of time that a female leaves the patch, given that she is
still on it. This leaving tendency can be modified by some pre-defined
explanatory factors (i.e., covariates) according to the following equation:
 | (1) |
Usually, the number of hosts successfully attacked and the number of hosts
rejected during the patch visit are the main covariates whose influence on the
females' leaving tendency are tested with such a model. This enables the
quantifying of the associated incremental or decremental effect on patch
residence time (e.g., Haccou et al.,
1991; Hemerik et al.,
1993; van Roermund et al.,
1994; Wajnberg et al.,
1999). However, in the present study, the quality of each host on
the patch was known throughout the experiment, so that more accurate
covariates could be defined. Six time-dependent covariates were used: the
number of successful attacks or rejections of: (1) a healthy host, (2) a host
previously attacked by the same female (self-superparasitism), or (3) a host
already attacked by another female 24 h before the experiment
(conspecific-superparasitism). The initial quality of the patch was added as a
fixed categorical covariate, the case corresponding to nine healthy hosts
being arbitrarily assumed to be the reference level corresponding to the
baseline hazard with a parameter set to zero. Thus, only three parameters need
to be estimated for this factor (see
Collett, 1994;
Wajnberg et al., 1999, for a
detailed explanation). Finally, in order to check whether there was a change
in the patch leaving mechanisms with the quality of the patch, the
interactions between all the time-dependent covariates and the fixed covariate
were also added to the model. Thus, the full model included 27 parameters.
They were estimated from the data by partial likelihood maximization
(Cox, 1975).
The significant effects of the covariates were tested using a standard
likelihood ratio test. Like in Wajnberg et al.
(1999), this test was used
through an iterative procedure to identify the parameters having a significant
influence of the females' patch leaving tendency. The adequacy of the final
fitted model can be assessed by making residual plots (see
Wajnberg et al., 1999, for an
example of such a plot). This showed that nothing was amiss. Thus, the final
model was considered to properly describe the patch leaving mechanisms used by
T. brassicae females under all the conditions tested. All
computations were done in S-Plus (Venables
and Ripley, 1994).
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RESULTS |
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As can be seen in Figure 1, T. brassicae females spent more time on patches containing a higher initial proportion of healthy hosts (Log-rank test: chi square = 22.92, df = 3, p <.001). Moreover, Figure 2 shows the number of progeny produced per time unit, and Figure 3 the distribution of the rates of reproductive success just before the patch is left. These final rates of reproductive success appeared to be statistically similar, whatever the initial quality of the patch the females were foraging on (Kruskal-Wallis test: chi square = 5.45, df = 3, NS; see Figure 3). An important proportion of the data plotted in Figure 3 are null values. This seems to suggest that the strategy adopted by the females consists of depleting completely the patch before leaving it. However, as can be seen on Figure 2, that shows average values, some progeny can still be produced after the patch is left. Moreover, an average of 3.74, 2.73 and 1.36 healthy hosts remained unattacked when the females left patches containing nine, six and three initially unparasitized hosts, respectively. Therefore, the patch exploitation strategy adopted by T. brassicae females appears to be more complex than a simple complete exploitation before leaving. The rates of reproductive success that are plotted in Figure 3 were computed over the last 5-min period spent on the patch. The Kruskal-Wallis test used to compare the final rates of reproductive success in the four situations remains non-significant if the estimate was done over an interval of time ranging from 1 to 10 min before the patch is left. Therefore, females spent statistically more time on patches of higher quality and seem to reduce them to the same level of profitability before leaving. Thus, their patch exploitation strategy appeared to be in accordance to some of the predictions of the MVT.
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A Cox regression model was fitted to the behavioral data in order to identify the patch leaving mechanisms used by the females under the different conditions compared. Table 1 gives the estimated effect of all the covariates having a significant influence on the patch leaving tendency of the wasp females.
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The tendency of the females to leave was clearly related to the initial quality of the patch they were foraging on. The leaving tendency on a patch containing only hosts already attacked by a previous female was 2.85 higher than on a patch containing only healthy hosts. Furthermore, each successful oviposition on a healthy host decreased the patch leaving tendency by a factor of 0.78. This indicates that T. brassicae females are using an incremental mechanism to control their patch residence time. On the other hand, when a female rejected a healthy host or a host that it previously attacked, its leaving tendency significantly increased by a factor of 1.22 and 1.07, respectively. These two types of host rejection have thus a decremental effect on patch residence time. In contrast, ovipositing a second time (or more) in a healthy host (i.e., self-superparasitism), and attacking or rejecting a host that was previously attacked by another female did not show any influence on the patch leaving tendency (all chi square at p >.05). Finally, no significant interactions were observed between the effect of the patch quality and the effect of each oviposition in a healthy host, rejection of a healthy host or a host that was previously attacked by the same female (all chi square at p >.05). Therefore, the patch leaving mechanisms discussed so far did not change according to the quality of the patch exploited by the females.
A simulation model
Are the different leaving rules discussed so far a sufficient set of
mechanisms leading the females to adopt a patch exploitation strategy that is
in accordance to the theoretical predictions of the MVT? If this is true, what
is the importance of each of these rules in fitting to such optimal
predictions? In order to answer these two questions, a Monte Carlo procedure,
based on the results of the Cox regression model, was developed in order to
simulate the patch exploitation behavior of T. brassicae females on
patches of different quality. The idea of using a Cox regression model as a
basis for building a simulation model of patch exploitation in parasitic wasps
has been already suggested by several authors
(Driessen and Bernstein, 1999;
van Roermund et al., 1994;
van Roermund et al., 1997;
Vos et al., 1998). Some of
them actually proposed detailed procedures simulating specific situations
(Driessen and Bernstein, 1999;
van Roermund et al., 1997).
The model developed in the present study is built on a more general procedure
that can be easily adapted to any kind of situation.
The Monte Carlo model was developed according to two successive steps. The
first one was based on a six-states Markov chain, and used to generate the
different behavioral events occurring during the patch exploitation. These
events were then used as inputs for a second step that was built around the
results of the Cox regression model. The six behavioral states used in the
Markov chain were a successful attack or a rejection of a healthy host, a host
previously attacked either by the same female or by a conspecific. The patch
was assumed to be exploited randomly by the females, so the probability of
starting the behavioral sequence in each of these states was set according to
both the initial quality of the patch and the rejection rates of the three
types of host, as estimated from the experimental data. Then, the inter-state
transition probabilities were modified according to the type of host
successfully attacked. Therefore, this procedure takes patch depletion into
account through a monotonous decrease in the probability of encountering a
healthy host each time such a host is successfully attacked. Based on the
experimental data, the times between the moment the patch is entered and each
of the behavioral states, and the inter-state transition times were drawn from
two-parameter exponential distributions (i.e., exponential distributions with
a minimal time-lag), whose features were estimated from the experimental data
(see Haccou and Meelis, 1992,
for a detailed description of such statistical distributions). The behavioral
sequence was generated up to the maximal observed patch residence time (i.e.,
5835.9 s). For a given simulation, and from the moment the female entered the
patch, the behavioral sequence was used at each time step to compute both the
waiting time t1 before the next coming event, and the
waiting time t2 before the patch is left. This second
waiting time was drawn randomly using the results of the Cox regression model.
The detailed procedure is described in the Appendix. If t1
was smaller than t2, the female was supposed to keep on
foraging on the patch, experiencing the next coming event, and the procedure
was resumed at this next event. In the other case, the patch was supposed to
be left, and both the patch time duration and the rate of reproductive success
during the last 5-min period were outputted.
The simulation model was first used with the regression parameters estimated from the experimental data (see Table 1). This was done only for verification, and, despite the fact this was not a real validation done on independent behavioral data, it would reveal whether some important factors were omitted. As can be seen in Figure 1, which gives an example of the simulated patch time durations obtained on the four types of patches, the model generated patch residence times that were in close agreement with the experimental observations. In order to study the effect of different patch exploitation strategies on the ability of the females to fit the predictions of the MVT, the simulation model was then used with different combinations of the regression parameters. Five different sets of patch leaving rules were compared: (1) the real patch leaving mechanisms, with parameter values estimated from the experimental data (i.e., with an incremental effect associated to each successful oviposition, and decremental effects associated to the rejection of a healthy host or a host already attacked by the same female, see Table 1); (2) the same set of mechanisms but with the incremental effect of each successful oviposition set to its opposite value, leading to a decremental mechanism; (3) and (4) the observed set of patch leaving mechanisms (see Table 1) but with the decremental effects of the two types of host rejection set to their opposite values, respectively; (5) all the three patch leaving mechanisms observed set simultaneously to their opposite values. In all the cases, parameters related to the quality of the patch remained unchanged. In each case, 200 simulations were done, each of them simulating the same number of replicates on the different patch qualities than the number of females observed experimentally. Finally, as in Figure 3, the ability of these different patch leaving strategies to produce results that are in accordance to the predictions of the MVT was assessed. For each simulation, the final rates of reproductive success on the different patch qualities were compared by means of a Kruskal-Wallis test. In each case, the frequency distribution of the 200 tests will thus indicate whether the females behave in an optimal way or not. Figure 4 gives the frequency distributions of these tests for the five patch exploitation strategies compared.
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More than 54% (i.e., 109 out of 200) of the simulations done with the regression parameters estimated from the experimental data led to Kruskal-Wallis tests below the 5%-risk threshold value of 7.81 (Figure 4a). However, if each successful oviposition had a decremental effect on the patch residence time instead of the incremental effect observed (see Table 1), only one Kruskal-Wallis test out of 200 remained non-significant (Figure 4b). Thus, the observed incremental effect of each oviposition in a healthy host seems to be an important mechanism ensuring the ability of the females to reach theoretical predictions of the MVT. On the other hand, Kruskal-Wallis tests were not affected by substituting the decremental mechanisms associated to the rejection of a healthy host or a host already attacked by the same female by opposite incremental effects. In both cases, more than 54% of the chi square tests were non-significant (Figure 4c,d). Therefore, these two patch leaving decision rules do not seem to have an important influence on the ability of the females to behave according to the MVT predictions. Finally, when all these three patch leaving rules were changed simultaneously into their opposite effects, only 4 out of 200 chi square tests remained non-significant (Figure 4e). According to the previous results, this is likely due to only the effect of each oviposition in a healthy host.
These results indicate that, within the set of the patch leaving mechanisms studied, the incremental effect of each successful oviposition in an healthy host appeared to play the most important role for the females to behave according to the MVT predictions. In order to understand this result more accurately, another set of simulations was carried out in order to perform a sensitivity analysis on the only parameter showing a significant influence. The same procedure was used, but only with different values (i.e., from -1.50 to 0.25) of the regression parameter describing the effect of each successful oviposition. All the other parameters remained unchanged. The results are presented in Figure 5.
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As long as a regression parameter lower or equal to the real observed value of -0.25 is used, more than 50% of the simulations led to Kruskal-Wallis tests below the 5%-risk threshold value. This indicates that, as long as each successful oviposition had an incremental effect, and whatever its intensity, the patch exploitation strategy adopted by the foraging females remains in accordance to the theoretical predictions of the MVT. On the other hand, if higher values of the regression parameter are used, and especially when positive value are considered (i.e., when each oviposition led to a decremental mechanism), more and more Kruskal-Wallis tests became significant, indicating that females behave in a less and less optimal way.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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T. brassicae females stayed statistically longer on patches containing a higher initial proportion of healthy hosts. On the other hand, the rates of progeny produced just before the patch is left remained statistically the same. This suggests that patches were all reduced to the same level of profitability before being left. Thus, the patch exploitation strategy adopted by the foraging females appears to be in accordance with some of the predictions of the MVT. This theorem implicitly assumes that foragers have a thorough knowledge of the quality and distribution of all the patches in the habitat. For this, they are assumed to sample a large number of patches in order to rapidly assess their profitability (Cook and Hubbard, 1977
;
Galis and van Alphen, 1981;
Godfray, 1994). However, the
results presented in this study were obtained on naive females, and only their
first patch visit was considered. This suggests that, as soon as they emerge,
females should have some innate, not necessarily accurate patch leaving rules
leading them to adopt a patch exploitation strategy that is on average in
accordance with the quality of the patches they will encounter. As soon as the
first patch is discovered, their foraging strategy would thus be already in
accordance with the predictions of the MVT, without prior knowledge of the
quality of all the available patches in the habitat. This ability could then
be improved during the following patch visits
(Vos et al., 1998).
The proximate mechanisms involved deserve to be analyzed and several
hypotheses can be proposed. According to Waage's
(1979) model, the patch
residence time is supposed to depend on the initial degree of responsiveness
to the patch that is related to the number of hosts available. However, in the
present study, patches offered to the females were all constituted of nine
hosts, and the amount of kairomone provided by a healthy host or a host
previously attacked by another female should be the same. Thus, in the present
case, the change observed in the patch residence time with the quality of the
patch should not be related to a change in the initial responsiveness of the
females. Sometimes, it has also been demonstrated that females could spend
less time on patches that were previously marked by a conspecific foraging
female (Bernstein and Driessen,
1996; Galis and van Alphen,
1981; van Lenteren and Bakker,
1978; Waage,
1979). However, in the present study, patches were built just
before the experiment, and the previous conspecific females used to produce
the hosts already attacked did not have any chance to mark the patch itself.
Other mechanisms should thus be proposed to explain the reduction in the time
spent on patches containing a higher initial proportion of already attacked
hosts. A third hypothesis has been suggested by Rosenheim and Mangel
(1994). An imperfect ability
to discriminate between healthy and already attacked hosts within the same
patch should lead the females to experience a higher risk of superparasitism,
with a corresponding reduction in reproductive success. Under such a
hypothesis, the presence of already attacked hosts within the patch may lead
the females to leave earlier. In the present case, this hypothesis is unlikely
because Trichogramma females are known to be able to accurately
discriminate between healthy hosts and hosts that are already parasitized
(Salt, 1937). Therefore, more
complex patch leaving rules are likely to be involved, and we decided to
analyze these by means of a Cox regression model.
Using such a regression model, we first found that each successful
oviposition in a healthy host significantly reduced the patch leaving tendency
of the females. This indicates that T. brassicae females were using
an incremental mechanism similar to the one described by Waage
(1979). This incremental
effect is likely to be related to the fact that, following oviposition,
Trichogramma females are known to increase their turning tendency
together with a reduction in their walking speed. Such an arrestment response
leads to an increasing tendency to remain on the patch after each oviposition
(Gardner and van Lenteren,
1986; Yano, 1978).
This incremental mechanism is supposed to be adaptive when the hosts exhibit a
clumped distribution (i.e., with a large variance in patch density), or if
foraging females are unable to assess accurately the number of hosts that are
expected in the patch under exploitation
(Iwasa et al., 1981).
Ovipositing a second time (or more) in a healthy host or in a host that was
previously attacked by another female did not have any influence on the patch
leaving tendency of the females. Thus, neither self- nor conspecific
superparasitism seem to provide the forager information regarding the future
value of the patch and thus does not seem to modify its readiness to leave. On
the other hand, rejecting a healthy host or a host previously attacked by the
same female led to a significant increase in the tendency to leave the patch.
As a general rule, the rejection of a host has repeatedly been considered to
provide the female with some information regarding the decreasing value of the
patch. Thus, the resulting increasing tendency to leave is usually considered
to be adaptive (van Alphen,
1993; van Alphen and Vet,
1986; van Lenteren,
1991; Wajnberg et al.,
1999). However, no significant effect was observed when the host
rejected was one that was previously attacked by a conspecific female. This
surprising result seems to indicate that indirect interference
(Visser and Driessen, 1991)
would not have any influence on the tendency of the females to leave the
patch, and this is in contradiction with what is usually expected
(van Alphen and Vet, 1986). In
the case of Trichogramma, only direct mutual interference would have
a significant influence (Kfir,
1983).
Finally, the patch leaving rules that have been discussed so far (i.e.,
incremental effect of each oviposition in a healthy host and decremental
effects of each rejection of a healthy host or a host previously attacked by
the same female) do not seem to change significantly according to the quality
of the patch exploited by the females. The only significant effect was that
the patch leaving tendency was higher on patches containing a higher
proportion of already attacked hosts. In turn, this led to an increase in
giving-up times (i.e., the time period from the last successful oviposition
until the patch is abandoned) with an increase in the quality of the patch
(Log-rank test: chi square = 10.50, df = 3, p <.02). Such a result
has been already observed on other parasitoid species (e.g.,
Nelson and Roitberg, 1995) and
is in accordance with the optimal predictions of the model developed by McNair
(1982). The variation in the
patch leaving tendency with the quality of the patch can thus be considered to
be adaptive.
As pointed out by Godfray
(1994), there is currently a
lack of studies relating the type of proximate patch leaving rules that are
revealed by means of the statistical approach used here with the ultimate
predictions of the MVT. Indeed, studies showing the ability of the foragers to
follow the predictions of the MVT are usually not focusing on the mechanistic
rules involved (e.g., Bonser et al.,
1998). Reciprocally, the analyses of the patch leaving rules
adopted by the females are not necessarily done within the framework of the
optimal predictions of the MVT (e.g.,
Driessen and Bernstein, 1999;
Waage, 1979). In the present
study, we thus tried to relate these two complementary approaches. By using
the results of the Cox regression model discussed above, we developed a Monte
Carlo procedure simulating the patch exploitation strategy of a wasp female on
patches of different profitability. Several other methods were already
proposed by some authors in order to simulate patch exploitation strategy in
parasitic wasps (Driessen and Bernstein,
1999; Roitberg and Prokopy,
1984; van Roermund et al.,
1997; Waage,
1979). The model presented here is the only not fully parametric
one allowing to take into account accurately and simultaneously: (1) the
effects of numbers and timings of the different events appearing on the patch
(e.g., host attacks or rejections), (2) the stochastic nature of these events,
(3) the effect of time-dependent mechanisms, and (4) the fact that the leaving
tendency of the females can be a function of time (i.e., with patch residence
times not necessarily distributed exponentially). Using this simulation model,
we demonstrated that the patch leaving rules adopted by the females seem to
provide them with an efficient and rather simple way to fit to the optimal
predictions of the MVT. Within the set of the leaving mechanisms that are
used, the incremental effect of each successful oviposition in a healthy host
appeared to be, at least qualitatively (see
Figure 5), the most important
one to achieve this optimal goal. On the other hand, the two other mechanisms
(i.e., decremental effects of rejecting a healthy host or a host previously
attacked by the same female) did not appear to play an important part (see
Figure 4). Using another type
of simulation procedure, Driessen and Bernstein
(1999) developed a model of
patch departure parameterized for Venturia canescens (Gravenhorst)
females foraging in a habitat containing an important amount of patches of
hosts. The inter-patch travel time was set to zero so that females could visit
all the available patches within a single foraging bout. Under such simulated
conditions, these authors showed that a decremental effect of each oviposition
would lead to a higher rate of progeny production because such a mechanism
would lead the female to visit a higher number of patches per time unit. On
the other hand, when travel time between patches is infinite, so that only one
patch is visited, an incremental mechanism was shown to produce the highest
rate of progeny production. In the present study, only one patch visit is also
considered, and we found an incremental mechanism to perform better according
to an other criterion, the terminal rate of reproductive success before
leaving the patch. Our results are thus in agreement with those found by
Driessen and Bernstein (1999)
and provide a more general framework suggesting that, when only one patch is
visited, an incremental effect of each successful oviposition would lead the
females to behave in a more optimal way.
The patch exploitation strategy of T. brassicae females is likely to change during successive visits to patches of host of different profitability. Thus, experiments are now being performed in order to identify any kind of change in the patch leaving rules adopted by the females that are offered in succession patches of different quality. Any modification observed will then be included in the simulation model developed in the present study in order to see whether the foraging females are also sampling their environment in order to get closer to the optimal predictions of the optimal foraging models.
|
APPENDIX |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIAL AND METHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
|---|
According to the Cox regression model, the hazard function for the i'th female is:
 | (A1) |
 | (A2) |
 | (A3) |
For this, the so-called Breslow's
(1974) estimator is usually
used. It leads to a step function whose jumps are, if
t(j) is the j'th patch time duration
among all the observed patch residence times, arranged in increasing rank
order:
 | (A4) |
be the estimated vector of
the regression parameters. Thus, the cumulative hazard function for the
i'th female, conditional on the covariates
Zi(t), can be estimated by:
 | (A5) |
 | (A6) |
), these values can be
used to estimate the corresponding distribution function:
 | (A7) |
Let us suppose that the last behavioral event occurred at time X,
and let Y = T - X the time elapsed between
X and the patch time duration T. Then, using the change of
variable t = s + X, the Cox regression model
presented in (Equation A1) for T is equivalent to the following one
for Y:
 | (A8) |
i,Y is thus the same as
the one described in (Equation A7), except that only the jumps of the
cumulative hazard function that appeared after the time of the last event are
taken into account. For each female, and after each event appearing on the
patch, this distribution function was estimated using the behavioral sequence
generated with the Markov chain described in the text.
Finally, a value u is drawn randomly from a Uniform distribution
over the interval [0, 1]. The simulated patch residence time x,
measured form the last behavioral event that appeared on the patch at
X, is then
i,Y-1(u)
which is the smallest value satisfying
i,Y(x) 
u (Rubinstein, 1981):
 | (A9) |
|
ACKNOWLEDGEMENTS |
|---|
We thank C. Bernstein and G. Driessen for their continuous encouragement and criticism. For their critical reading of the manuscript, we thank C. Bernstein, G. Driessen, L. Lapchin and J.S. Pierre. We thank N. Ledger for reading the English version of the manuscript, C. Curty and J. Pizzol for their valuable help in experimental work, and P. Chavigny for performing the analysis with the molecular markers.
|
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