Behavioral Ecology VoL 8 No. 1: 57-45
The geometry of search movements of insects
in plant canopies
Jerome Casas* and Martin AIujab
•Department of Biological Sciences, University of California, Santa Barbara, CA 9316(M)610, USA and
b
Instituto de Ecologia A. C , Apartado Postal 63, 91000 Xalapa, Veracruz, Mexico
The aim of this study was to provide aframework,for describing and understanding the geometry of movement of insects
foraging within complex plant canopies where the insect is exposed to varying stimuli. We used the apple maggot fly, Rhagolttis
pomontUa (Walsh) (Diptera: Tephritidae), foraging in apple trees devoid of fruit as our model system. The framework provides
the null hypothesis required for inferring the influence of external stimuli, such asfruitcolor and odor, on the paths of foraging
flies. We mapped trees into cells, released preconditioned flies in caged trees, and recorded their behavior and location. Flies
moved mainly to the nearest neighbor cells, but displacement within a wide range of distances was observed. The model closest
to observations is a random walk with a position-dependent bias in the vertical component of movement. Four other models
were built, spanning a range of simplifications in the rules determining the vertical component of movement We used the
concept of avoidance of self-crossing in a searching path for defining efficiency of movement Flies were quite efficient at visiting
almost as many different cells as possible. Comparisons of assumptions and predictions of the five models revealed that this
efficiency is due to the small number of steps, the location of the starting cell, and a strong tendency to move upward in its
vicinity. We discuss the selection pressures on movement rules: pressure from predators may explain the short hops, while the
sensory ecology of fruit finding and the avoidance of sites already visited by other flies or by the same fly may explain the
position-dependent upward bias. Strong similarities between the rules for die vertical component of movement of one simplified
model and the observations lead us to believe that canopy architecture influences insect movement not only by defining the
set of locations that the insect can visit using predefined rules for movement but also by defining die rules of movement Kty
words: apple fruit fly, canopy, foraging behavior, fruit fly, movement, plant architecture, RhagoUHs pomontUa, searching efficiency,
search theory. (Behav Ecol 8:37-45 (1997)]
I
nsects typically search for food, hosts, or sexual partners in
geometrically complex environments. Examples include
the flight of insects within structurally variable pheromone
plumes (Baker, 1989; David, 1986; Elkinton et aL, 1987; Murlis
and Jones, 1981), the movement of foraging coccinellids and
parasitoids on plants (Ayal, 1987; Brodeur and McNeil, 1991;
Casas, 1991; Frazer and McGregor, 1994; Hoffmann, 1990),
and die movement of darkling beetles in semiarid grasslands
(Johnson et al., 1992). Nevertheless, modeling insect movement has mainly been restricted to continuous homogeneous
environments (Kareiva, 1990). The reasons for assuming a homogeneous environment when modeling and analyzing insect
movement have their origins in two decisive advantages. First,
a body of analytical solutions can be relied on, mainly centered around the simple diffusion equation. The issues addressed by Okubo (1980) .provide a good representative example of this approach. Second, insect movement can also be
studied without paying attention to the fine structure of the
environment This practice rests on the lesuicuve assumption
of deterministic homogeneous geometry of the texture of the
medium at die microscale. Its corollary is that die large-scale
results of diffusionary processes are virtually independent of
die fine texture of die medium (Skellam, 1979:67-68). Incorporation of even mild disorder in die environment can lead
to unexpected results about die dispersion of moving particles, such as a nonlinear increase of the mean-square displacement with time, which constitutes a severe violation of die
basic laws of die diffusion equation.
Only a few ecological studies do justice to heterogeneity in
J. Cuas is now at IRBI-CNRS UPRES A 6035. Avenue Monge, UnJverrite de Touri, F-37200 Tours, France.
Received IS February 1995; accepted 20 March 1996.
1045-2249/97/J5.00 O 1997 International Society for Behavioral Ecology
the environment mainly by varying food abundance or quality in space. Good examples, placed within die diffusion equation framework, include die work of Kareiva and Odell
(1987), Turchin (1991), and Lewis (1994). These authors
used limiting arguments about lattice spacing, time between
successive steps, and distribution of step size relative to die
size of die environment to go from a random-walk description
to a reaction-diffusion equation description. However, modeling insect movement in canopies requires another approach
because too many assumptions of die reaction—diffusion equation are violated. First, die movement of flying insects in canopies is often either by flying or hopping from one leaf to
another, so that single steps are clearly identified. Second, a
single move may easily cover half or more of die maximal
distance between two points of die canopy. Finally, die number of steps before finding the target or moving out of die
canopy may be quite low (Aluja and Prokopy, 1993).
The need to consider die fine geometry of die canopy in
order to understand insect movement within vegetation was
recognized some time ago (Ahija et aL, 1989; BeU, 1991). But
analysis of data collected under such conditions still lacks a
theoretical framework to distill die essence from die detail
available for each path. Hence, die data available are underexploited. The summary statistics extracted are typically mean
values over die whole observation period, such as die number
of cubes visited or die sum of individual displacements (Aluja
et aL, 1989; Bell, 1991). As a consequence, some basic questions in insects and other biological systems are still awaiting
answers because they require die understanding of the geometry of a sequence of moves. Examples range from the
definition of efficient search paths to applied pest management problems (Roitberg and AngeriHi, 1986; Prokopy and
Goli, 1978). The estimation of "trapping probabilities" for
insects moving in an environment with "traps" is still in its
Behavioral Ecology Vol. 8 No. 1
38
infancy. The best models (see Mangel, 1989; Mangel and Adler, 1994; Perry and Wall, 1984) make simplistic assumptions
about the movement of insects, such as random movement
(but see Williams, 1994). For example, a thorough understanding of landing points and foraging paths of die cherry
fruit fly (R. ctrasi L.) is of critical importance to the efficient
deployment of the costly host-marking pheromone that suppresses further opposition (Aluja and Boiler 1992a,b). Because spraying whole trees is far too costly, the question of the
exact location of the spray, and therefore the foraging path's
geometry, becomes unavoidable.
The ultimate aim of our work is to provide a framework for
describing and understanding the geometry of insect movement in complex plant canopy structures where the insect is
exposed to varying stimuli. The basic ingredients of our approach are random walks for modeling movement and lattice
structure for modeling canopies. The core of our approach is
similar to the one used in modeling anomalous diffusion
(Bouchaud and Georges, 1988; Hughes and Prager, 1983): the
basic laws governing the dynamics of movement are intentionally kept simple. By testing die outcome of die process
with appropriate statistics, the correct prediction of die dynamics of the process is ensured, while some loss of realism
in die description of die rules of die movement is accepted.
The model system we use is die apple fruit fly, Rhagoittis
pomoneUa (Walsh) (Diptera: Tephritidae), moving in apple
trees after having been conditioned to search for oviposition
sites (i.e., apples; Aluja, 1989; Aluja and Prokopy, 199S). Here
we focus on developing a model that predicts the fly's movement in trees devoid of fruit The analysis is restricted to die
first steps followed by an insect after landing in a tree and is
conditional on foraging. Therefore, it does not deal with leaving mechanisms. Subsequently, we estimate die influence of a
fruit's characteristics on the path of a fly by comparing the
predictions of the null hypothesis model presented here with
paths of flies observed foraging in trees harboring fruits of
varying qualities, such as color, • smell, etc This is done by
comparing die expected probability to reach a given lattice
point in a tree devoid of fruit widi the frequency of visits diat
are observed for real flies in a tree with an apple at die same
location.
This created dense regions in localized parts of the tree where
flies could be lost from sight We think diat manipulated trees
can be considered equivalent to unmanipulated ones for our
purposes, as die landing surface in every cube remained high
after die manipulations (more than 60% of die original surface, as we removed mainly small leaves) and because die
models use only die presence of vegetation within cubes for
representing canopy architecture.
Individual female flies were released on a predetermined
leaf in die central-lower part of die tree. This release leaf was
clearly marked and was die same in all experiments. Once
released, we observed any given fty for up to 20 min, recording all movements and behaviors. Every time a fty changed its
location, die plant pan number to which it moved was immediately recorded. The observation period was terminated
only when die predetermined time period elapsed or when
die insect left die tree and flew to die cage wall. If the fly was
lost for more than 1 min, die record was discarded. To ensure
diat all flies released were in roughly the same physiological
state with respect to fruit foraging, each fly was allowed to lay
an egg in a host fruit just before being released, as in Roitberg
et aL (1982). In die few cases where a fry landed on a tag, we
recorded die cube number as usual (more details about data
collection can be found in Aluja et al., 1989).
For modeling purposes, we discarded cubes devoid of vegetation and concentrated only on cubes diat could be used as
landing points for the fry, irrespective of die kind of vegetation parts within the cubes. The tree used for testing the model (tree A) had 800 cubes out of 2744 diat contained vegetation (Figure 1). This structure is called a lattice structure.
Anything within die cube is considered to lie on its lattice
point We use the lattice when referring to height levels and
movement from one location to another. The corresponding
distances and heights in meters are not given, although they
can be inferred from die lattice points. For example, given
diat die lattice points are located in die center of die cubes,
a height level of 3 is the same as a height of 50 cm. In terms
of our models, it is more important to understand diat a fly
moving vertically to height level 3 from height level 1 made
a move of 2 units rather than to know that die move was 40
MATERIALS AND METHODS
Core of die models
The experimental setup is detailed in Aluja et aL (1989) and
Aluja and Prokopy (1993). Only relevant information is given
here. Two apple trees diat had been planted 6 years earlier
and had canopies of approximately 2.8 m diam X 3 m height
were used. Each tree was enclosed in a 3.5 m diam X 3.5 m
tall cylindrical cage. The tree volume was divided up into
imaginary cubes. Every tree part falling within a particular
cube (20 X 20 X 20 cm) was marked with a distinctive number, corresponding to x, y, and z coordinates. The number
was written on a small tag.
The architecture of die tree was manipulated by pulling
branches or twigs of varying sizes in horizontal or vertical directions and attaching diem with string or clear nylon cord
to adjacent branches. We removed 40% of all leaves, paying
particular attention not to vary die number of cubes with vegetation in diem (i.e., all cubes with leaves at the outset had
leaves after leaf clipping). Pruning was performed 1 week before the initiation of experiments and did not affect canopy
shape. These mamptdatieBs were needod to (1) keep die tree
architecture constant over the period of the experiments
(about 8 weeks) and (2) allow die researcher a clear view of
almost any point within the tree. Pulled branches were only
slightly displaced from dieir original locations. The removed
leaves were often small ones, clustered around biggeT ones.
Since we were interested in die sequence of moves between
cubes and the geometry of die path made by a foraging fly,
we discarded both die time spent in die cubes and any movement within cubes. A move, or time step, is defined as a
change of cubes.
We developed five different probabilistic models. The core
is the same for all models: flies move for a given number of
steps, and each move is defined by its length and its direction
in three dimensions. The step length is the same in all five
models, but die rules deftning die move direction differ between models. The vertical component of movement requires
special attention, as analysis showed diat flies moved upward
rather than downward in die lower two-thirds of die tree.
Therefore, we focus on die probabilities diat the vertical component of movement is positive (Le., die insect moves upward); negative (i.e., die insect moves downward), or null
(Le., the insect moves horizontally). In die first two cases, a
move can contain both a vertical and a horizontal component
Expressed in spherical coordinates, die question is whether
die angle a that the vector makes with the vertical axis is
positive, negative, or null. If die vertical component is null,
die set of locations for die next landing is denned by die
locations on die perimeter of a horizontal circle with radius
equal to die step length. In both other situations, die set is
39
Catas and Aluja • Search movement! in canopies
• •
10
• • • • • • • • • • • a
11
• • • •
12
11
14
Figure 1
Tree used for model evaluation divided into slices, along its vertical axis. Each dot represents a cube that contains vegetation. Level 1
represents the bottom of the tree. The release point is highlighted at level 4.
the surface of a hemisphere with radius also equal to the step
length. Once the step length is determined, the fly may move,
for example, upward one unit and horizontally a relatively
large distance or horizontally a short distance but upward several units. We searched in the data for a dependency on horizontal position in the movement parameters, step length and
direction. For example, we expected flies at the tree periphery
to move toward the center of the tree. We did not detect this
anisotropy and so did not pursue this aspect further.
Once the vector length and its vertical component are chosen, the direction of the next location is made at random
from the set of available lattice locations defined by these two
conditions. The models predict the position of the next landing site given the current qne. The output of one simulation
of any of the models is a time series of locations. The number
of steps is given at the beginning of die simulation. The algorithm for the entire sequence is represented in Figure 2.
Differences bet*
•deb
The five models we developed vary only in the rules for determining the vertical component of movement The most
complete model contains every major aspect of movement
identified during the data analysis: it is a random walk with a
position-dependent anisotropy in the vertical component of
movement, and referred to hereafter as "die complete model." We used four other models that span a range of simplifications in the rules for die vertical component of movement
We have two reasons for formulating alternative rules for the
vertical component of movement radier dian for the step
length. First, sample size is not a problem for estimating die
step length distribution, but it is a problem for estimating die
anisotropy at some of die height levels (see below). Second,
we want to know which level of complexity in the definition
of the vertical component of movement has to be retained to
make correct predictions.
The first simplification is defining die anisotropy in the vertical component of movement as a constant radier than as a
variable. To this end, two models have a position-independent
anisotropy that differs in die way die anisotropy is estimated.
A further simplification uses isotropic random walk with an
equal probability of die vertical component being positive,
null, or negative. There is no ranking in importance or in
precedence between step length and direction for these models. Finally, a "random-choice model" picks die next position
randomly from die set of available positions denned by die
surface of die sphere with radius equal to die step length. In
that model, die step length is die only condition.
Test of assumptions common to sdl mooefa
Two assumptions common to all models are a lack of correlation between step length and vertical component of movement and a lack of correlation, in length and direction, between consecutive steps. We tested these assumptions using
die same data set as for parameter estimation.
We tested for correlation between vertical component and
step length using contingency table analysis (Table 1) (Bishop
et aL, 1975). The assumption of homogeneity among die
three groups of angle values was not rejected (likelihood ratio
X* 9.45, 6 df, p = .15, n » 413). Thus, step lengdi and direction are uncorrelated.
We tested for correlation between steps by modeling vertical displacement as a first-order Markov chain (Table 2). We
pooled data over height levels to attain a sufficient number
of observations. There was a weak correlation between steps
40
Behavioral Ecology Vol. 8 No. 1
Choi ce number
of steps N
T«ble 1
Relationship b
it'
i the value* of the angle a formed by the
etc - and the vertical axis (aee Figure 4) and die step
Step length
n-0
Starting location
as i n experi ments
n-N?
1. Choose step
length
2.choose
direction
1 denti fy al 1
1 ocati ons f i 11 i ng
conditions 1 & 2
Any such
location ?
Null
Positive
Negative
Total
1 Sx<2 2Sx<3 3 3 i x<4 x* 4
Total
139
103
62
304
174
155
84
413
16
26
12
54
9
14
7
30
10
12
3
25
(likelihood ratio x* 9.8, 4 df, p = .066) but none of the standardized residuals was high enough to be considered as an
outlier (the highest was 1.89), nor did the outliers form conspicuous patterns. For example, a strong positive correlation
between steps produces outliers in the cells positive-positive
and negative-negative. There was also a weak, but significant,
correlation between the length of two consecutive steps (n =
392, Spearman correlation corrected for ties r = .263,
/rC.001). We consider this correlation too weak to warrant
inclusion of one step-memory effects in the models, given the
associated complexity and difficulty in interpretation. For the
same reason, we did not proceed with the analysis of higher
order correlation.
All parameters are estimated from 413 moves of 21 flies in
tree B. We used the observed distribution of step length (Figure 3) in the simulations (Bratley et aL, 1987; Shanker and
Kelton, 1991). By definition, the smallest step length is one.
The observed vertical component of movement (Figure 4)
fitted a linear model in the range of available data (the number of observations were, for each level, 4:25, 5:60, 6:54, 7:40,
8:36, 9:50, 10:26, 11:44, 12:42,13:23, and 14:4). There are no
data for levels 1-3 because all flies were released at level 4
and always moved upward, as shown for representative paths
in Figure 5. The extrapolation from level 1 to level 4 was based
on the consideration for smoothness and monotonidty in the
curves and on the need for a summation of probabilities to
one. At level 14, a fly can only stay at the same level or move
downward. The estimation of position-independent anisotropy was done in two different way* (Table 3). We summed all
cases in the first model, irrespective of the fly's vertical location. We did the analysis per level in afirststep and combined
results for the second modeL
Testing the models
Pick one at
random
We used a second set of flies foraging in another tree (tree
A) to test the model. We chose two statistics for testing the
models: the number of different cubes or sites (.$,) visited
Table 2
Fim-order Marker chain for the values of the angle a between the
movement vector and the vertical axb (lee Figure 4)
n«
Figure J
Model flow chart
Positive
Null
Negative
Total
Negative
Null
Positive
N
10
37
75
56
168
33
58
51
142
80
163
148
391
so
41
81
Casas and Aluja • Search movement* in canopies
41
300
200
11
100-
18
21
28
91
number of steps
Representative movements of three observed flies.
after n steps, called the range, and the vertical position after
n steps. The range is an interesting measure of the "efficiency" of the fly: with a mesh size of 20 cm, revisiting the same
site is unlikely to lead to the discovery of an apple previously
unnoticed in that cell. From this point of view, revisiting the
same cell is a waste of time. While the predictions about the
range could potentially arise from several models, the prediction of the vertical position after a given number of steps is
expected to depend strongly on the rules defining the vertical
component of movement Furthermore, the number of possible values for the range after a few steps is low, while the
vertical position can take values over the whole tree after a
single step. Therefore, we expected the vertical position to be
more sensitive than the range in changes in the models.
We carried out 1000 simulations of the complete model at
different numbers of steps (n = 5, 10, 20, 30, 40, 50). For
each value of the number of steps, we obtained a distribution
of the range from the 1000 simulations (Figure 6A). We calculated 95% confidence bands by linear interpolation between the 95% confidence intervals obtained by simulation at
the six selected numbers of steps (Casas, 1990). The 33 observations are the final values of the range for each fly, either
because the maximal time was attained or because the fly left
the tree. Only four observations lie on the bands or slightly
outside (Figure 6B). These four cases do not produce any
obvious pattern of residuals and the model is not rejected.
We carried out 1000 simulations for the random-choice
model and 100 simulations for the other three models. The
predicted range and therefore the rate of visit to new sites
[S(n)/n] of the random-choice and complete models are
astonishingly similar (Figure 7). Hence, the range cannot be
used to discriminate between the two models. The difference
between the rates predicted by the complete and isotropic
models increases in the first 20 steps and remains constant
thereafter. This is to be expected: the first steps bring the fly
from a region where the difference in probabilities to move
upward is strong into a region with no difference. The differ-
Flgure 3
Observed distribution of step length.
8
7
9
height within the tree
5
7
9
11
13
liskjlu wtthbi the tree
Figure 4
Probability that the angle a between the movement vector and the
vertical axis is positive (i.e., the insect moves upward) (solid line),
negative (I.e., the insect moves downward) (dashed line), and null
(i.e., the insect moves exclusively horizontally) (dotted line), as a
function of the vertical location of a fly. Observed value* are given
in the top graph. Most observations were made at the height levels
5-12. See text for the exact number of observations at each level.
The bottom graph displays the probabilities as used in the complete
model.
TableS
Probability that the angle a b e f t m the movement vector and the
vertical axb H positiv6f oegaovCf and nullffaxthe two positlonmdependent blaacU random walk nn^^lf*
Model 1
Model2
Nun
Positive
Negative
0.424
0.38
0.373
0.334
0.203
0.286
' Positive, insects move upward; negative, insects move downward;
null, insects move only horizontally.
42
Behavioral Ecology Vol. 8 No. 1
40-
30 •
A
f'
to-
10-
c
16
20
A
X
40
50
20
30
numbar of steps (n)
Figure 7
Rate of visit to new sites for the complete (solid line), randomchoice (dashed line) and isotropic (dotted line) models.
I
10
20
number of steps
40
50
Figure 6
Test of the complete model, based on the number of different site*
visited, SfnJ, as a function of the number of step*, «. (A) Frequency
distribution of 1000 simulations of the complete model. The
diagonal line represents the maximum possible value of S{n) - n
(i.e., the fly never revisits any site). (B) Observed values at the end
of each path and 95% confidence bands (lines) estimated by linear
interpolation between values from simulations.
ence in probabilities decreases sevenfold from the starting level to level 9, where it is only 0.02. The rates predicted by the
anisotropic models (not shown) are within the boundary set
by the rates of the complete and the isotropic models.
The Kolgomorov-Smirnov tests of the observed and predicted vertical locations after 5, 10, 15 and 20 steps (Figure 8;
the number of observations at 5 steps is 33, at 10 steps is 26,
at 15 steps is 16 and at 20 steps is 10) did not reject (at the
5% level) the complete model at any number of steps. The
random-choice model was rejected at steps 15 and 20. Thus,
the complete model is better at predicting the vertical location. We did not use a likelihood ratio test to discriminate
betweea medols. This type of index may be sensitive to uninteresting features and often obscures the strengths and
weaknesses of a particular model (Bush and Mosteller, 1961).
We followed the advice of Bush and Mosteller and paid special
attention to the occurrence of movement below the releasing
height level (height level 4), where the complete model is
much better than the random-choice modeL The better fit of
the complete model in this region of the tree is also biologically meaningful, as flies do not often use the main trunk for
their displacements (Aluja M, personal observation). Flies
have a higher vertical location over time than the location
predicted by die complete model. This trend was not statistically significant, but the sample size was small. The number
of flies that moved more than 20 steps was too low to carry
this analysis further. We did not perform the test with the
predictions of the isotropic and position-independent anisotropic models, as their predicted distributions were even more
distant from the observations than the ones predicted by the
random-choice model (Le., they are stochastically smaller).
A good fit in the tree in which the rules for movement were
estimated (tree B) is a necessary condition for extending the
model to other trees. However, this condition is not sufficient,
and multiple uses of the same data sets is problematic, even
if we did not test die same characteristics of movement as
those extracted from the data. All but three values of the
range observed for 21 flies foraging in tree B were in the 95%
confidence intervals given by 100 simulations of the complete
model. Furthermore, the distribution of the vertical location
at 5, 10, 15, and 20 steps predicted by the complete model
was not rejected at any number of steps (Kolgomorov-Smirnov
tests at the 5% level). The visual agreement of predicted vertical location with the observed values was better in tree B
than in tree A.
DISCUSSION
Efficiency of search paths)
Measuring the efficiency of a search path is a daunting task
that depends on the assumptions made about target space,
target motion, divisibility of the searching effort, detection
law, number of decision time points, and definitions of efficiency (see, e.g., Iida, 1992). This may be the main reason
why so few studies (but see Hoffmann, 1990) have tackled the
problem of efficiency of movement in insects. Our approach
relies on the sensory ecology of host finding in this species
and on the avoidance of self-crossing in searching paths.
Casas and Aluja • Search movement* in canopies
14
An apple in a cell volume of 20 cm3 is unlikely to remain
unnoticed by a fly visiting that cell. Revisiting a cell is therefore a waste of time and energy, except if the information
gathered on the second visit increases die likelihood of finding fruit This could be the case if die fly perceived plumes
of fruit odor in a cell devoid of fruit If we discard this complication, the most efficient way to move is to avoid self-crossings and to produce "space-filling" paths (Lalley and Robbins,
1989). A natural definition of efficiency is then the ratio of
die number of different sites visited and die number of steps,
Syn. For the fruit fly, efficiency means visiting as many different cells as possible, thereby keeping die ratio to its maximum value of 1. The observed values of die range show diat
flies rarely revisit cells and are indeed very efficient This behavior was previously noticed by Roitberg et al. (1990), who
describe it as a form similar to "systematic search."
Selection pressures on movement rules
A quantitative estimate of a surrogate for fitness for the different models, such as die number of cells visited or the time
spent before finding a fruit, is premature. This study focuses
on trees devoid of fruit, and movement patterns do change
in die neighborhood of fruit (a change can be measured from
four cells away). Furthermore, die host shift of R pomontUa
is recent, and a comparison of movement patterns and resource distributions in the ancestral host, hawdiorn (Gratagtus spp.) and die apple tree is lacking. At this time, die best
we can do is to identify die possible selection pressures and
discuss die performance of die different models in die light
of diese pressures.
The two components of movement, die step lengdi and die
direction of movement, may have been subject to different
selection pressures. Difference in predation pressure may be
a reason flies do not fly within die tree but take short hops.
A careful field study by Monteidi (1972) demonstrated diat
adult R pomontUa were not preyed upon while on die foliage.
Figure 8
Cumulative probability of the
vertical location of the fly after
(A) 5, (B) 10, (C) 15, and (D)
20 steps. Prediction! of the
complete model (open circlet), random-choice model
(open Kjuaret), and obterrations (filled circles).
Theseflieswere very alert and were even avoided by jumping
spiders, maybe because of visual mimicry. Flying individuals
may be caught in spider webs or preyed upon by odler insects
specializing on flies, as observed for another tephritid (Hendrichs and Hendrichs, 1990).
As first observed in die field for odier Rhagoletis species
(Prokopy, 1976; Prokopy and Coli, 1978) and later reported
for R pomontUa (Aluja, 1989; Aluja and Prokopy, 1992,1993;
Roidjerg, 1981, Roitberg et al., 1982, 1990), a female searching for fruit displays a typical search pattern, which it may
repeat a few times. It starts foraging in die lower part of die
canopy, eidier because it landed diere or because circling
brought it from die upper part of die tree back to die bottom.
The sensory ecology involved in fruit finding may have biased
die direction of evolution of diis bottom-up scanning of trees
(die "sensory drive"; Endler, 1992). These insects search for
and detect fruits widiin trees principally, if not solely, by vision, by using intensity contrast of die fruit against background. Visual contrast is dependent on die position of die
fly relative to die fruit and background. Owens and Prokopy
(1986) showed diat an optimal position for die fly is looking
at die fruit from below, against a background of skylight or
light transmitted through foliage. The fruit dien appears dark
against die background. A second possible explanation for die
upward trend is die low probability to cross paths associated
widi a high bias. However, die probability of self-crossing by
die female is low even La die worst model (die isotropic model). Thus, die negative effects associated widi self-crossing
seem too weak to explain die upward trend. The probability
of multiple visits to a site increases strongly if several flies forage in a tree. We ran die complete model widi a varying number of flies, all starting in die same location and moving for
varying numbers of steps. For die same total number of steps,
die number of revisits is die lowest when flies are few and
when each one moves several steps. For example, die rate of
visit to new sites, S(n)/n, is 0.634 for 10 flies each moving 50
steps and 0.46 for 50 flies each moving 10 steps. In die latter
Behavioral Ecology Vol. 8 No. 1
chi
td movement
Canopy
Testing models in trees different from those in which they
were developed is a particularly stringent test: models may fail
because they are tree specific. If they pass the tests, then we
know something about movement in canopies in general, as
the outcomes of the models are independent of the details of
canopy geometry, such as the spatial arrangement of gaps in
the canopy or the geometry of branching, for example. The
influence of canopy-to-canopy variations in geometry is best
explained using the mean value of the range. The mean value
of the range in tree B, denoted (Sfn))^ is dependent on die
configuration of tree B. As our aim is to understand insect
movement in apple trees in general rather than in die apple
tree B, our objective is to characterize movement over trees
(Le., we are interested in mean ( S(n)), the mean value over
an apple trees.) The major difficulty is ensuring thai the trees
used for averaging are similar (Le., they can be considered as
realizations of the same stochastic process with the same parameters). If they are not, models may fail because they are
applied to environments produced by other rules. Given the
lack of studies of the geometry of apple trees in terms of
stochastic processes, we opted for a second tree of the same
age, with a similar total number of cells with vegetation, and
similar vertical distribution of foliage.
In the complete model, the influence of canopy architecture determines which of the cells belonging to the set of
potential sites have vegetation and can be used as landing
sites. The probabilities of upwards, downward, or horizontal
movement are given and determined independently of die
number of cells with vegetation in each direction. The set of
landing sites for die random-choice model is usually bigger
than for the complete model because we do not impose rules
for die vertical component of movement. All cells widi vegetation at a given distance have the same probability of being
chosen. The probability of moving upward, downward, or horizontally then equals die proportion of cells above, below, or
at the same level as die fly. respectively. Because die vertical
distribution of cells with vegetation is not uniform between
die bottom and die top of d\e tree, canopy architecture biases
die vertical component of movement (size-based sampling).
Specifically, flies tend to move toward die bulk of die canopy,
where die density of vegetation is high. The vertical component of movement as used by die random-choice model (Figure 9) shows strong similarities with die movement observed
in flies (Figure 4). Thus, we suggest diat die bias in die vertical component of movement as observed for flies, and as
used in die complete model, was partly die result of an influence of canopy architecture on fly movement.
The models do not incorporate scale effects (leaves versus
leaf clumps versus branches), discrimination between different types of substrate (leaves versus bark, for example), or
preferential movement along die same branches. Our models
make further assumpti»RS oenc«raing fb/ movement within
canopies. For example, die movement distance is independent of die distribution of vegetation within die canopy, die
horizontal displacement is independent of die current location, and die vertical displacement is dependent only on die
height within die canopy. We do know diat, to different de-
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case, the paths are long enough to escape from the sparse
canopy at the bottom of the tree, where the likelihood of
crossing its own path and the path of others is high, into the
bulk of the canopy. Thus, it is advantageous for a fly to have
a strong directional component leading it quickly to the bulk
of the canopy on two counts: avoiding sites already visited by
others and by itself. This is achieved in only two models, the
complete and the random-choice models.
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13
Figure 9
Probability that the insect moves upwardi (solid line), downward
(daihed line), and exclusively horizontally (dotted line), as a
function of the vertical location of a fly for the random-choice
model. See Figure 4 for the definition of the probabilities. These
probabilities were estimated from 100 simulated flies moving for 50
steps in tree A.
grees, diese assumptions violate reality. Our models are simple
models representing insect movement in plant canopies.
More complex models might have a realistic representation
of a tree as an insect perceives it and more sophisticated behavioral rules. The power of die random walk on lattice models diat we advocate lies in its simplicity and in its ability to
simulate biologically important features of foraging padis in
canopies diat differ in detail but are similar in their general
appearance.
We thank R. J. Prokopy for his help in designing the experiments and
in interpreting the data. We also gratefully acknowledge J. Miller, W.
Bell, R. Carde, and J. Elkinton for stimulating discussions on experimental design. We acknowledge outstanding assistance during field
work by M. AveriU and M. Y. Quoada. We thank S. Diehl, W. Murdoch, R. Nisbet. D. Papaj, R. Prokopy, B. Roitberg, A. Stewart-Oaten,
B. Wermelinger, and W. Wilton for their helpful comments on earlier
drafts. We also dunk M. Mangel and two reviewers for the care with
which they scrutinized the manuscript. The experimental pan of the
project was funded by U.S.-Israel Binational Agricultural Research and
Development grant US-607-84, by the Science and Education Administration of the U.S. Department of Agriculture (USDA) under grant
850060S from the Competitive Research Grants Office, by USDA Cooperative Agreement no. 58-S204-3-615, and by Massachusetts Agricultural Experiment Station project 604. M. A. was an International
Atomic Energy Agency Fellow during this research. The modeling
part of the project was supported by two grants from the Swiss National Science Foundation, 31.28608.90 and 823A-037138, to J. C
P k'l»VBFN< IKK
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