The Effect of Resource Aggregation at Different Scales: Optimal Foraging Behavior of Cotesia rubecula

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1 University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications in the Biological Sciences Papers in the Biological Sciences The Effect of Resource Aggregation at Different Scales: Optimal Foraging Behavior of Cotesia rubecula Brigitte Tenhumberg University of Nebraska - Lincoln, btenhumberg2@unl.eu Michael A. Keller University of Aelaie, mike.keller@aelaie.eu.au Anrew J. Tyre University of Nebraska-Lincoln, atyre2@unl.eu Hugh P. Possingham University of Queenslan, h.possingham@uq.eu.au Follow this an aitional works at: Part of the Life Sciences Commons Tenhumberg, Brigitte; Keller, Michael A.; Tyre, Anrew J.; an Possingham, Hugh P., "The Effect of Resource Aggregation at Different Scales: Optimal Foraging Behavior of Cotesia rubecula" (2001). Faculty Publications in the Biological Sciences This Article is brought to you for free an open access by the Papers in the Biological Sciences at DigitalCommons@University of Nebraska - Lincoln. It has been accepte for inclusion in Faculty Publications in the Biological Sciences by an authorize aministrator of DigitalCommons@University of Nebraska - Lincoln.

2 vol. 158, no. 5 the american naturalist november 2001 The Effect of Resource Aggregation at Different Scales: Optimal Foraging Behavior of Cotesia rubecula Brigitte Tenhumberg, 1,2,* Michael A. Keller, 1, Anrew J. Tyre, 1,2, an Hugh P. Possingham 2, 1. Department of Applie an Molecular Ecology, University of Aelaie, Waite Campus, Private Bag 1, Glen Osmon, South Australia 5061, Australia; 2. Ecology Centre, Department of Zoology an Entomology, University of Queenslan, St. Lucia, Queenslan 4072, Australia Submitte October 24, 2000; Accepte May 16, 2001 abstract: Resources can be aggregate both within an between patches. In this article, we examine how aggregation at these ifferent scales influences the behavior an performance of foragers. We evelope an optimal foraging moel of the foraging behavior of the parasitoi wasp Cotesia rubecula parasitizing the larvae of the cabbage butterfly Pieris rapae. The optimal behavior was foun using stochastic ynamic programming. The most interesting an novel result is that the effect of resource aggregation within an between patches epens on the egree of aggregation both within an between patches as well as on the local host ensity in the occupie patch, but lifetime reprouctive success epens only on aggregation within patches. Our finings have profoun implications for the way in which we measure heterogeneity at ifferent scales an moel the response of organisms to spatial heterogeneity. Keywors: resource aggregation, optimal foraging behavior, Cotesia rubecula, stochastic ynamic programming, spatial scale. The importance of scale to ecological investigations has been emonstrate in many ifferent areas, incluing behavioral ecology, population ecology, community ecology, an plant physiology (Wiens 1989). Ecologists are increasingly intereste in ecological patterns an processes that are scale epenent, where patterns at one scale change when observe at other scales (Wiens 1989; Levin 1992; Schneier 1994). This article is concerne with a behav- * Corresponing author; btenhumberg@zen.uq.eu.au. mkeller@waite.aelaie.eu.au. tyre@zen.uq.eu.au. hpossingham@zen.uq.eu.au. Am. Nat Vol. 158, pp by The University of Chicago /2001/ $ All rights reserve. ioral process. In particular, we examine the scale-epenent effects of resource aggregation on optimal patch-leaving behavior an reprouctive success of consumers. Previous work in optimal foraging theory concentrates on the influence of resource ensity within patches on patch resience time (e.g., Charnov 1976; Cook an Hubbar 1977; Comins an Hassel 1979; Iwasa et al. 1981; Visser et al. 1992; Nelson an Roitberg 1995). In general, these moels are base on rate maximizing theory an preict that foragers leave a patch when its rate of fitness gain rops to a threshol value. However, the spatial arrangement of resource-containing patches is equally important for behavioral ecisions (e.g., Lima an Zollner 1996; Roitberg an Mangel 1997). The spatial arrangement of resources can vary at ifferent scales. Resource-containing patches may be aggregate in space, while at a smaller scale, there may be aggregation in the frequency istribution of resource ensity within patches. Aggregation of resource-containing patches implies a spatial correlation between patches; there will be clusters of patches in space. Mangel an Aler (1994) propose a escription of spatial pattern base on Markov processes that escribes aggregation at multiple spatial scales. Their structure function escribes the probability that a point some istance r away from a given point contains a resource, conitione on the resource state at the current point. When resources are aggregate in space, the expectation of fining a resource close to an occupie patch is greater than the average probability of fining a resource over the entire habitat an greater than the expectation of fining a host close to an unoccupie patch. Structure functions can escribe ifferent egrees of spatial aggregation (Roitberg an Mangel 1997). Aggregation of resource ensity within patches implies that the variance of the istribution is greater than the mean; there will be many patches with no resources an a few with a very high resource ensity. The frequency istribution of resource ensities within patches (fig. 1) summarizes the resource availability in the environment. Resource availability influences the behavior of consumers, in particular, ecisions concerning patch use an the is-

3 506 The American Naturalist Figure 1: Frequency istribution of resource ensities erive from a negative binomial istribution with mean m p 3 an aggregation inex k p tribution of consumers between patches (Stephens an Krebs 1986; Bernstein et al. 1991). For ecisions mae at the iniviual level, however, the frequency istribution of resources within patches alone might be insufficient because those ecisions might also be influence by the arrangement of resource-containing patches in space. For example, the resources escribe by the frequency istribution of figure 1 can be arrange ranomly or aggregate in space (fig. 2). Imagine a forager in a habitat where resource patches are aggregate in space (fig. 2A). In this case, the probability of encountering a resource patch nearby will be higher than average if the forager is alreay in a resource patch. The encounter probability is lower if the forager is in an empty patch. Therefore, the istribution of resources in space influences the travel time. In the presence of resource epletion, we expect a forager to spen less time in a patch when the average travel time is short because short travel times translate to a high average rate of encountering resource patches (the marginal value theorem of patch use; Charnov 1976). The spatial istribution of resources influences the variation of travel times experience by the foragers (e.g., travel time as a function of resource status of the current patch). Variation in encounter rates with resources can have consequences for life-history ecisions (Tenhumberg et al. 2000). In this article, we examine whether variation in travel times influences the patch-leaving behavior as well. We stuy how the two ifferent scales of aggregation influence the patch-leaving behavior of foragers. For the remainer of the article, we refer to ensity aggregation when escribing the aggregation of resources within patches an to spatial aggregation when escribing the aggregation of resource-containing patches in space. Increasing ensity aggregation arises from increase variance of the frequency istribution of resources within patches. Increasing spatial aggregation arises from increasing the spatial correlation between patches that contain resources. Consier a three species parasitoi-herbivore-plant system in which plants (i.e., resource patches) are istribute throughout the lanscape an in which zero, one, or more herbivores are present on each plant. How oes the spatial clustering of resource-containing plants (i.e., plants with herbivores) an the istribution of herbivores among plants influence the parasitoi behavior an reprouctive success? Our article aresses this question for the parasitoi wasp Cotesia rubecula. We evelop a stochasticstate-epenent optimal foraging moel of foraging behavior of the parasitoi wasp C. rubecula parasitizing the larvae of the cabbage white butterfly Pieris rapae. An unexpecte an novel result is that the effect of aggregation on patch resience time an lifetime reprouction epens on whether hosts are aggregate within or between patches. High spatial aggregation of host-containing patches causes shorter patch resience times at all host ensities, whereas ensity aggregation ecreases patch resience time on plants when host ensity is low an increases patch resience time on plants when host ensity is high. Lifetime reprouction is more sensitive to ensity aggregation than spatial aggregation. Whereas the wasp s patch-leaving behavior compensates completely the effect of spatial aggregation on lifetime reprouction, increase ensity aggregation still reuces lifetime reprouction. The Biological System Cotesia rubecula females are solitary parasitoi wasps that lay single eggs insie their hosts. After hatching, a wasp larva fees internally on the host s tissue an kills the host at the en of larval evelopment. The hosts are larvae of the cabbage white butterfly, which fee an live on cruciferous plants. Plants change the amount an composition of oor constituents in response to herbivore amage. These infochemicals are attractive to the wasps (Nealis 1986). It has been emonstrate for some parasitoi species that females can istinguish patches with ifferent host ensities base on the concentration of infochemicals (van Alphen an Vet 1986). When a wasp arrives on a plant, she starts searching for hosts in the vicinity of areas with host-feeing amage (Nealis 1986). However, feeing amage oes not always inicate the presence of hosts. Hosts move away from feeing sites while resting or molting to avoi preators.

4 Resource Aggregation at Different Scales 507 Figure 2: Two istributions of resources in space: (A) aggregate an (B) ranom. Each point in the gri is a patch; there are 10,000 patches in all. Patches with resource ensities 10 are isplaye by a circle. The iameter of the circle is proportional to the ensity in that sampling unit. In both cases, the frequency istributions of resource ensities can be escribe by the same negative binomial istribution (fig. 1). Wasps reaily attack hosts they encounter (M. A. Keller, personal observation), even if the hosts are alreay parasitize (superparasitism). As C. rubecula is a solitary parasitoi, only one egg can evelop within a single host (Gofray 1987). Therefore, the profitability of a plant ecreases with time because the probability of encountering an ovipositing in an unparasitize host ecreases with each oviposition. Eventually the wasp leaves the plant to search for hosts elsewhere. The patch-leaving behavior of parasitois has attracte a lot of attention by researchers (e.g., Hubbar an Cook 1978; Waage 1979; Iwasa et al. 1981; Haccou an Hemerik 1985; Haccou et al. 1991; Roitberg et al. 1992; Hemerik et al. 1993; van Alphen 1993; Wiskerke an Vet 1994; Driessen et al. 1995). The result of this extensive research inicates that the probability of a wasp leaving a patch (e.g., plant) generally epens on several factors, incluing the parasitoi s age, the patch resience time, the host ensity in the current patch, the ensity an istribution of hosts in the rest of the habitat, an the number of ovipositions or the oviposition rate in the current patch. A single-scale perspective characterizes all this research. Our moel incorporates all the aforementione factors, but in aition, it inclues the influence of host istribution on multiple scales. The Moel We use stochastic ynamic programming (SDP; Bellman 1957; Mangel an Clark 1988) to preict when a parasitoi shoul leave a patch. In SDP moels, behaviors maximizing the fitness of organisms are ientifie as a function of an organism s internal an external state(s), the remaining life expectancy, an physiological an ecological parameters. SDP moels start at the en of an iniviual s life an then work backwar in time, calculating for each combination of states the behavior that results in the highest fitness. After calculating the optimal behaviors, we use a Monte Carlo simulation to analyze the foraging behavior of female wasps using ecisions etermine by the SDP. For the sake of simplicity, we consier only mate female wasps. The wasps are foraging in a gri of uniformly istribute patches. The patches are either empty or contain up to 10 hosts. The location of empty patches etermines the egree of spatial aggregation, while the istribution of hosts within nonempty patches etermines the egree of ensity aggregation. The moel consists of two parts: searching for hosts on a patch (i.e., cabbage plant) an flying between patches searching for suitable laning sites. On the patch, there is the following hierarchy of events. If a wasp survives the next time step of length 2.5 min (survival probability p 1 m 1 ), she has to ecie whether to stay or to leave the patch. If she stays, there is the probability w that she encounters a host. She attacks all hosts she encounters, but successfully oviposits an egg only with probability s. She must hanle the host for h time steps regarless of whether oviposition is successful. Every time step a wasp survives outsie a patch (survival probability p 1 m 2 ), she encounters a new patch on which she may or may not lan. The number of patches encountere since last leaving a patch is the travel istance. The probability of laning b epens on the host ensity in that particular patch (M. A. Keller, unpublishe ata). Moel Assumptions Fitness Currency. We assume that Cotesia rubecula females ajust their foraging behavior to maximize their fitness

5 508 The American Naturalist (Mangel an Clark 1988). In our moel, we use the total number of surviving eggs oviposite by a female uring her life as fitness currency. Other moels use the expecte initial egg complement of aughters (the gran eggs ) as a fitness measurement (Heimpel et al. 1998), taking into account that larger hosts generally prouce parasitois with higher egg loa. This fitness currency is appropriate for parasitois that reject some hosts for oviposition. In contrast, C. rubecula females try to oviposit in all hosts they encounter (M. A. Keller, personal observation). As our moel is concerne only about patch-leaving ecisions, we ignore the influence of host size on a parasitoi s fitness, even though the instar of the host also influences the fitness of the parasitoi s offspring (Nealis et al. 1984). State Space. We assume that a parasitoi s patch-leaving ecision epens on the following state variables: her age, the host ensity an istribution in the habitat, the host ensity in the current patch, how long she has searche (patch resience time), an how many eggs she has lai in the current patch. We assume that wasps have perfect knowlege of all these states. The patch resience time an the number of ovipositions in that patch are the minimally sufficient statistics a forager nees in orer to estimate the number of unparasitize hosts remaining in the patch (Iwasa et al. 1981). As C. rubecula is probably not egg limite (Nealis 1990), we assume ovipositions in previous patches have no influence on her patch-leaving behavior. Patch Depletion. We inclue only epletion as a result of a wasp s foraging behavior, such that the probability of encountering an unparasitize host in a patch is a ecreasing function of the number of ovipositions. We assume a system with a high number of patches an a low number of parasitois, such that wasps never encounter an alreay eplete patch. The probability of reencountering a previously visite patch or a patch eplete by a conspecific is negligible. The effect of allowing other wasps Figure 3: Daily probability of survival as a function of age (m 1 (t)) for female Cotesia rubecula, estimate from laboratory ata (Nealis et al. 1984). to oviposit in the patch woul be to lower the expecte payoff from each host by a uniform amount if females cannot etect each other s ovipositions. Survival. In our moel, we erive the mortality risk of wasps in a patch from the survival curves measure uner laboratory conitions by Wäckers an Swaans (1993), with an average life expectancy of 23. The cost of searching for suitable laning sites involves time loss an an increase mortality risk. Roff (1977) showe that ipterans incur a higher mortality rate flying rather than walking, which in turn reuces the average lifetime expectancy. To account for the increase mortality risk uring flight, we multiplie the mortality risk in a patch by a fixe factor (table 1; fig. 3). The life expectancy of wasps in the real worl is probably much shorter than that of our moel wasps. Potential mortality sources for ault parasitois in the fiel inclue abiotic factors (DeBach et al. 1955; Roitberg et al. 1992; Fink an Völkl 1995), starvation (Wäckers an Swaans Table 1: Parameter values an functions inclue in the stochastic ynamic programming moel Parameter Value or formula Reference exp b Prob{laningF} 1 exp M. Keller, unpublishe ata m 1 (t) Mortality rate in patch 1 survival probability (fig. 3) Wäckers an Swaans 1993 m 2 (t) Mortality rate while flying 1 (survival probability #.99) w goo ( ) a Prob{encounter an unparasitize hostf} f(number of ovipositions, ) (table 3) w ba ( ) a Prob{encounter a parasitise hostf} 1 exp ( /7) p goo w( ) a Prob{encounter a hostf} w goo( ) w ba( ) s Prob{ovipositionFs}.84 Nealis 1986 Note: p host ensity in current patch, t p age of the parasitoi. a Assumes that host encounters in patch follow a Poisson istribution. See text for formula an erivation of w goo ().

6 Resource Aggregation at Different Scales ; Heimpel an Collier 1996; Jervis et al. 1996), an preation (Rees an Onsager 1982; Völkl 1992; Völkl an Mackauer 1993; Rosenheim et al. 1995). For example, the longevity of Trichogramma platneri in the laboratory is reuce 90% by unfavorable temperature an 75% by starvation (McDougall an Mills 1997). Similar rastic effects of starvation on the life expectancy of C. rubecula have been reporte (Wäckers an Swaans 1993; G. Siekmann, unpublishe ata). Actual fiel mortality rates from any of these factors are largely unknown (Heimpel et al. 1997). However, fiel experiments on Aphytis parasitois suggest that preation rivals starvation an extreme temperature conitions as sources of mortality (Heimpel et al. 1997). We o not expect that the qualitative results of our moel will be affecte by choosing a life expectancy that is too long. Simulations with a rastically reuce life expectancy (2 ) prouce a similar istribution of patch times; the lifetime offspring prouction was scale own without changing the relative ifferences in reprouction resulting from iffering egrees of aggregation. Activity Perio. In our moel, the wasps are active 12 h each ay. This activity perio is probably too high; real wasps spen at least 50% of their time resting an grooming an are inactive uring unsuitable weather (Nealis 1986; Geervliet 1997; M. A. Keller, personal observation). Unfortunately, no ata are available on the proportion of time wasps spen foraging in the fiel. A shorter activity perio woul reuce the life expectancy an consequently scale the reprouctive success own further. No Egg Limitation. In our moel, female C. rubecula are not egg limite an try to oviposit in all hosts encountere. A check on this assumption is the magnitue of the lifetime egg prouction of simulate wasps. At a mean host ensity of two hosts per plant, moel wasps lai fewer than 400 eggs uring their life. As Nealis (1990) foun an average of 100 mature eggs in the uterus of C. rubecula females, this egg prouction is above the natural egg complement of the wasps. However, our moel wasps probably have at least a tenfol longer life expectancy than real wasps (average life expectancy of moel wasps p 23, compare with 1 4 at 120 C in fiel cages; G. Siekmann, unpublishe observations). Therefore, we think that real wasps harly ever run out of eggs. Superparasitism. Our moel assumes that female wasps cannot istinguish parasitize from unparasitize hosts, an as a consequence, wasps attack every host they encounter. Acceptance of parasitize hosts coul result from one of two mechanisms: aaptive superparasitism (van Alphen et al. 1992; Visser et al. 1992; Papaj an Messing 1996; van Ranen an Roitberg 1996) or an inability to istinguish between parasitize an unparasitize hosts (Rosenheim an Mangel 1994). In our moel, all eggs oviposite in previously parasitize hosts ie. We assume that only a single wasp forages in each patch, so it oes not matter which egg survives when superparasitism occurs; all that matters is that the potential payoff from a patch ecreases with time as the number of unparasitize hosts ecreases. If aaptive superparasitism operates in C. rubecula, then ovipositing more than one egg in a single host woul increase the fitness payoff. However, eliberate self-superparasitism in solitary insects is rather unlikely (Gofray 1994). Host Distribution. The moel inclues three ifferent host istributions: the istribution of hosts within a patch, the istribution of host ensities among patches, an the spatial istribution of patches. We assume hosts are ranomly istribute within a patch. Therefore, we moel the probability that wasps encounter a host in any particular time step as a Poisson process. We assume that the frequency istribution of host ensities within plants (e.g., the istribution of circle sizes in fig. 2) comes from a negative binomial istribution. We assume the spatial istribution of host-containing patches (e.g., spatial arrangement of all circles in fig. 2) comes from a one-imensional structure function. Using a one-imensional structure function to escribe the spatial istribution of host-occupie patches assumes that the host ensities of neighboring patches are not correlate. In many species, host ensities on neighboring plants are highly correlate. Such a high correlation occurs particularly when reprouction an juvenile evelopment take place in ajacent places, such as in aphi populations. In this case, the escription of aggregation in space is more complicate than we have iscusse here (e.g., Possingham et al. 1994). However, there are populations, such as for larval Pieris rapae, where the assumption of uncorrelate host ensities in neighboring patches is realistic. Pieris rapae females are very mobile an fly ranomly among brassicaceous plants where they lay mostly one or two, but sometimes up to five, eggs per plant (Harcourt 1961; Kobayashi 1966; Jones 1977; Kareiva an Shigesaa 1983). The ranomly istribute plant visits compoune with the logarithmically istribute number of eggs lai at each visit generates a negative binomial istribution of eggs per plant in cabbage fiels (Kobayashi 1966). The feeing activity of preators, or heterogeneous light conitions (Kobayashi 1966), might result in a spatially aggregate istribution of host-infeste plants. Unfortunately, no ata on the spatial istribution of larval P. rapae in their natural habitat have been publishe.

7 510 The American Naturalist Characterizing Density an Spatial Host Distributions The probability of encountering a host-occupie patch epens on the istance r the wasp has covere since she left the patch. Every time step a wasp is outsie a patch, she encounters a new patch with hosts, an she lans with the probability b ; r is then the total number of patches encountere since last leaving a patch. The probability that this patch is host occupie (i.e., 1 0) is given by the structure function, that is, the conitional probability that a patch is host occupie given the last patch at istance r was host occupie ( p[rf1] ) or was not host oc- cupie ( p[rf0] ). Figure 4 (moifie fig. 1 from Mangel an Aler 1994) illustrates hypothetical structure functions for ranom an aggregate patterns similar to those in figure 2. We erive p(rf1) from a one-imensional first-orer Markov process with three parameters, p a, the average patch ensity, p 0 an p 1 (Mangel an Aler 1994): r p(rf1) p pa (1 p a)(p1 p 0), p a[1 p(rf1)] p(rf0) p, (1) (1 p a ) where p 1 is the probability that a patch immeiately next to the current host-occupie patch ( r p 1) is also occu- pie an p 0 is the probability that the next patch is unoccupie. The parameters p 1 an p 0 are relate by the equation pp (1 p )p p p. (2) a 1 a 0 a Equation (1) implies that p 1 ( 0! p 1! 1) an the egree of spatial aggregation are positively correlate. Hence, we can examine the effect of spatial aggregation on parasitoi behavior simply by varying p 1. For further etails on the structure function, see Mangel an Aler (1994). The spatial an ensity istributions are connecte by p a, which is set to 1 minus the zero term of the negative binomial istribution. In our moel, a patch can contain zero to 10 hosts. In the fiel, usually a cabbage plant contains between zero an two P. rapae larvae (Harcourt 1961; Kobayashi 1966; Jones 1977). The negative binomial istribution has two parameters: the mean host ensity, m, an the aggregation inex, k. In general, the istribution becomes more aggregate as k becomes smaller. The probability that a host-occupie patch contains hosts is k G(k ) m k!g(k) ]( m k)( m k) H p, (3) [ where G(k) is the gamma function (Krebs 1989). Figure 4: Probability of encountering a resource as a function of istance r from any given point in the habitat given the point contains a resource ( p[rf1] ). The average probability of fining a patch with resource (P a ) is ( 1 P 0 ; same parameters as fig. 2). Broken line, aggregate pattern (fig. 2A); soli line, ranom pattern (fig. 2B). If p 1 is very small ( p1 p 0.1), the istribution of host- occupie patches is completely ranom, an the spatial an ensity istributions coul be escribe by the negative binomial istribution alone. In this respect, we can consier the negative binomial istribution a special case of the combine istribution use in our moel. The ensity istribution of P. rapae eggs among plants is usually aggregate with k values!1 (Jones 1987). In this article, we examine k values between 1 an 0.01, which are consistent with k values foun in natural insect populations (table 2). The probability of fining hosts r patches away is l r,0 p p(rf0), l p p(rf1)h, (4) r, p 1,, 10. When both spatial an ensity aggregation are low, that is, p p p [1 n(0)], (5) 1 a where n(0) is the number of empty patches, then the struc-

8 Resource Aggregation at Different Scales 511 Table 2: Density aggregation in empirical stuies Flies (%) a Whiteflies (%) b resiues (%) c Beetles in granary Pieris rapae (%) 1! k ! k ! k ! k Fruit or culture meium Leaf Granary sample Plant Sample unit: Range of k Sample size Number of species Note: k is the aggregation inex of the negative binomial istribution. a Rosewell et al b Hassell et al c Barker an Smith Jones 1987 (we calculate the k values from the presente variance mean ratios [ v/m] an corresponing means using the 2 following formula: k p m /[v m] ). ture function we have use here yiels overisperse patterns. Combinations of parameters that meet this conition have been exclue from the results (i.e., p 1! 0.5 when k p 1). The type of spatial structure escribe by our combination of a structure function an a negative binomial istribution is similar to a marke-point process (Stoyan an Stoyan 1994), where a ranom process creates the points (i.e., occupie patches) an a secon ranom process creates marks (i.e., the number of hosts) attache to each point. The statistical methos for ealing with marke-point processes are goo at proviing top-own escriptions of pattern, but we require a bottom-up wasp s eye view of the spatial structure for our moel, an structure functions effectively provie this. Dynamic Programming Equation Given one time step is 2.5 min (both insie an outsie patches) an assuming that a wasp forages 12 h/ uring her maximum life span of 40 (Wäckers an Swaans 1993), then the maximum foraging time of a wasp is 11,520 time steps. The SDP moel inclues the state variable time ( t p 1, 2,, 11,520) an the patch-relate variables host ensity ( p 0, 1, 2,, 10), patch time ( tp p 1, 2,, 17), an number of ovipositions ( e p 0, 1, 2,, 17); therefore, the lifetime fitness function is efine as F(t,, t p, e). A wasp in a patch may leave or stay in the patch; each ecision translates in a particular fitness payoff (payoff leave, payoff stay ). After leaving a patch, the wasp encounters another patch every time step on which she may or may not lan. The probability of laning on a particular patch, b, epens on the host ensity of the patch. The parameter b is estimate from a logistic regression on unpublishe ata from M. A. Keller. He place 16 cabbage plants containing ifferent host ensities (0, 1, 2, an 5 hosts/plant) in the fiel an observe the laning behavior of C. rubecula foraging between those plants. The probability of fining a plant with a certain host ensity after covering the istance r, l, r, is etermine by the istribution of host ensities between patches an the spatial istribution of host-occupie patches (see eq. [4]). While in a patch, a wasp encounters a host with the probability of w( ) in which she successfully oviposits an egg with the probability s. Inepenent of the oviposition success, she spens an extra time step to hanle the host. The payoffs for encountering an unparasitize (host goo ), parasitize (host ba ), or no host (host 0 ) are as follows: host goo p s[f(t 2,, tp 2, e 1) 1] (1 s)f(t 2,, t 2, e), (6) p host ba p sf(t 2,, tp 2, e 1) (1 s)f(t 2,, t 2, e), (7) p host p F(t 1,, t 1, e). (8) 0 p Note that successful oviposition in an alreay-parasitize host changes the state space but provies no fitness increment. The scenario escribe above is escribe with the following ynamic programming equation where fitness is optimize over the biological choice, leave or stay:

9 512 The American Naturalist F(t,, t, e) p max (payoff, payoff ), (9) p leave stay payoff p [1 m (t r)] { 10 leave 2 rp1 r 1 10 ( j j, i) ip0 jp1 10 r, p1 payoff p [1 m (t)] 2 stay 1 # 1 bl (10) # F(t r,,1,0)l b, # [host goow goo() } host baw ba()] (11) [1 m (t)] 1 # host [1 w () w ()], 0 goo ba where m 1 (t) an m 2 (t) are the mortality rates in the patch an while flying, with m 1(t)! m 2(t) an w goo ( )orw ba () the probabilities of encountering an unparasitize or parasitize host. The values of these parameters an the references from which the values are erive are given in table 1. The probability of encountering an unparasitize host, w goo, epens on the number of eggs that have alreay been lai an the number of hosts into which they have been lai, assuming that eggs are ranomly istribute among the available hosts (table 3). The key is the probability istribution of the number of parasitize hosts on the plant A. This can be calculate iteratively for each host ensity using the observation that a particular number of parasitize hosts can arise only in two ways if hosts are parasitize one at a time. First, if the encountere host is unparasitize, then there will be one more parasitize host. Secon, if the encountere host is parasitize, then the number of parasitize hosts will not change. The probability of i parasitize hosts after e eggs have been lai is i i 1 A i, e p A i, e 1 A i 1, e 1, (12) where is the number of hosts in the patch. The initial istribution for e p 0 is 1, if i p 0 A i,0 p {. (13) 0, if i 1 We use this to calculate w goo for a patch with hosts an e eggs alreay lai as the sum of the probability of encountering an unparasitize host when i hosts have been parasitize multiplie by the probability that i hosts have been parasitize: ip1 i wgoo,, e p A i, e, (14) w p 1 G 1. (15) goo,,0 By applying equations (14) an (15) iteratively from e p 1 up to the maximum number of eggs lai in a patch, we get all the probabilities require. (See Rosenheim an Mangel 1994 for another metho of representing host epletion without recognition.) Results We analyze the effect of host aggregation at two scales on a parasitoi s patch resience time an lifetime reprouction: ensity aggregation (variance of the frequency istribution of host ensities) an spatial aggregation. We ran the moel for ifferent mean host ensities in the range of 0.1 an nine hosts per patch. The higher the mean ensity m use, the higher the lifetime reprouction of the wasps became an the more reaily the wasps left the patch. The influence of ensity aggregation an spatial aggregation on the patch-leaving behavior was qualitatively the same over the whole range of m. Therefore, we present the results only for a mean ensity of two hosts per patch, which is commonly observe in fiel populations. Figure Table 3: Probability of encountering an unparasitize host as a function of host ensity () in the patch (number of hosts) an the number of eggs alreay lai Number of hosts No eggs lai One egg lai Two eggs lai i ip1 A i, e

10 Resource Aggregation at Different Scales 513 Figure 5: Frequency istribution of host ensities within 1,000 host-occupie patches if the host ensities have a negative binomial istribution with a mean host ensity m p 2 an varying aggregation inex k. 5 shows example frequency istributions of hosts within occupie patches for ifferent egrees of ensity aggregation (k values). Patch Resience Time The response to aggregation is complex. In general, with increasing egree of aggregation, the average patch resience time ecreases, inepenent of the scale of aggregation (fig. 6). However, at k values 10.1, the effect of ensity aggregation is very small compare with spatial aggregation. The average time wasps spent on a patch uring their life is the result of the wasps patch-leaving behavior an the encounter frequency of patches with ifferent host ensities. Therefore, the average patch resience time oes not necessarily reflect the behavior of wasps on patches with ifferent host ensities. The influence of aggregation on the patch resience time epens on the host ensity of the current patch. Increasing host ensity reuces the magnitue of the effect of spatial aggregation but actually changes the irection of the effect of ensity aggregation (fig. 7). When a wasp is on a patch with low host ensity, patch resience time is shorter at higher egrees of spatial an ensity aggregation (fig. 7A). However at the extreme ens of spatial aggregation, that is, p1 p 0.1 or 0.9, there is no effect of ensity aggregation. On patches with higher host ensities, the effect of aggregation at both scales isappears in the range of ensity aggregation foun in natural Pieris rapae populations ( k 1 0.1; fig. 7B). However, at very high egrees of ensity aggregation ( k! 0.1), wasps remain longer in a patch rather than leaving earlier as they o in patches with low host ensity. This interesting behavioral switch occurs within a range of ensity aggregation typical for a lot of insect species (table 2). Increasing spatial aggregation still causes wasps to leave earlier, but the magnitue of the effect is small.

11 514 The American Naturalist how aing spatial structure to moels influences the range of parameters uner which host-parasitoi populations coexist. To istinguish when spatial istribution oes matter, it is critical to unerstan how ifferent representations of spatial heterogeneity influence preictions of iniviual behavior an, consequently, the ynamics of populations, communities, an ecosystems. This article contributes to this unerstaning by simultaneously examining the influence of aggregation at two ifferent scales on the iniviual behavior an lifetime reprouction of the parasitic wasp, Cotesia rubecula. As far as we know, this article is the first publishe work on the influence of resource aggregation at multiple scales Figure 6: Influence of ensity aggregation (k) an spatial aggregation (p 1 ) on mean patch resience time of simulate Cotesia rubecula females average over all patches. The arrows inicate increasing egree of aggregation. Lifetime Reprouction As a result of ajusting their patch-leaving behavior in response to the spatial istribution of host-infeste plants, the reprouctive success of female Cotesia rubecula is inepenent of spatial aggregation (fig. 8). In contrast, with increasing egree of ensity aggregation, lifetime offspring prouction ecreases. A high egree of ensity aggregation translates into a rastic increase in the frequency of empty patches (fig. 5). Therefore, with increasing egree of ensity aggregation, they waste an increasing amount of time visiting an resting on empty patches. Even though wasps always leave empty patches after one time step, patches with hosts are rare at higher egrees of ensity aggregation. Consequently, there are fewer opportunities to compensate for increase travel time. Discussion Aggregation of resources in space can occur at many scales. This article suggests that the impact of aggregation on foraging behavior, an, hence, on lifetime reprouction, may vary profounly epening on the scale of resource aggregation. Ignoring one scale of aggregation might not tell us the whole story, especially when stuying the behavior of iniviuals. The problem of when an how to incorporate spatial heterogeneity into moels is not straightforwar. Kareiva (1990) reviewe both theoretical an empirical results from a wie range of spatial problems an conclue that sometimes the spatial istribution matters an sometimes it oes not. Keeling (2000) showe Figure 7: Influence of ensity aggregation (k) an spatial aggregation (p 1 ) on mean patch resience time of simulate Cotesia rubecula females on patches with a small (A) anahigh(b) number of hosts within a population with a mean ensity of two hosts per plant. The arrows inicate increasing egree of aggregation.

12 Resource Aggregation at Different Scales 515 patches at very high egree of ensity aggregation (fig. 7). However, if wasps are on patches with very high host ensity an ensity aggregation is moerate to low, chances are low of encountering a similar goo one quickly, an the wasps o not respon to a change in spatial aggregation. Clarifying Apparent Contraictions to Optimal Foraging Theory Figure 8: Reprouctive success of simulate Cotesia rubecula females as a function of ensity aggregation (k) an spatial aggregation (p 1 ). The arrows inicate increasing egree of aggregation. on the patch-leaving behavior of iniviuals. Our moel explains the influence of aggregation on patch-leaving behavior through its effects on travel time an the relative profitability of host patches. This is entirely consistent with Charnov s (1976) marginal value theorem that preicts preators shoul stay longer in patches with high profitability an when the travel time is long. Density Aggregation Density aggregation influences the travel time an the relative profitability of patches. With increasing ensity aggregation, the frequency of both empty an high-ensity patches increases; the former increases the travel time, an the latter ecreases the relative profitability of low-ensity patches. So increasing ensity aggregation coul have both an increasing an ecreasing effect on patch resience time. Our moel preicts that wasps respon only at high levels of ensity aggregation by increasing patch resience time on high-ensity patches ue to increasing travel time through empty patches (fig. 7). At low to moerate levels of ensity aggregation, the moel preicts no influence of ensity aggregation on patch resience time. Here the effects of changing travel time an relative profitability counteract each other. Spatial Aggregation High spatial aggregation means that there is a high probability of fining a host-occupie patch nearby (short travel time); consequently, we expect a wasp to leave the current patch early. This is exactly what our moel wasps o on low-host-ensity patches an on high-host-ensity Most research on foraging behavior concentrates on the effects of ensity aggregation using functional response moels. In these moels, the number of prey attacke per unit time by a single preator is a function of prey ensity (May 1978; Crawley 1992). The proportion of prey that escape from preation is the zero term of the negative k binomial istribution: H 0 p (1 ap t/k), where a is the area of iscovery an P t is the parasitoi ensity at time t (Rogers 1972). The general conclusion of such moels is that increasing ensity aggregation (ecreasing k) ecreases resource consumption. Our simulation results confirm this preiction, as the lifetime reprouctive output of wasps ecrease with increasing egree of ensity aggregation. Mols s (1993) criticism of functional response moels confuse spatial an ensity aggregation. He simulate the foraging behavior of iniviual carabi beetles (Pterostichus coerulescens) searching for prey istribute either ranomly or aggregate in istinct clusters (20 prey per cluster). In his simulations, the preation rate was higher when prey were confine to clusters compare with when prey were ranomly istribute, which is opposite to the preictions of functional response moels. He conclue that using the zero term of the negative binomial istribution is not appropriate for estimating preation in aggregate prey istributions. In the light of our results, this contraiction is not surprising because Mols s (1993) work confouns the effects of ifferent scales of aggregation, spatial aggregation in his moel, an ensity aggregation in functional response moels. Conclusions In this article, we have investigate the effect of host aggregation at ifferent scales on the optimal foraging behavior an performance of the parasitic wasp C. rubecula. Our moel suggests that there is no effect of spatial aggregation on the reprouctive success of parasitois, which lens some support to population moels that largely ignore spatial resource aggregation. However, for unerstaning the foraging behavior of iniviuals, it is important to istinguish between ifferent scales of

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