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Ecology, 81(4), 2000, pp. 921–935 ! 2000 by the Ecological Society of America GENERALIST FEEDING BEHAVIORS OF AEDES SIERRENSIS LARVAE AND THEIR EFFECTS ON PROTOZOAN POPULATIONS JOSEPH N. S. EISENBERG,1,4 JAN O. WASHBURN,2 and SEBASTIAN J. SCHREIBER3 1School of Public Health, Division of Environmental Health Sciences, University of California, Berkeley, California 94720 USA 2Department of Plant and Microbial Biology, University of California, Berkeley, California 94720 USA 3Department of Mathematics, Western Washington University, Bellingham, Washington 98225 USA Abstract. The generalist feeding strategy of larvae of the western tree hole mosquito , Aedes sierrensis, is central to understanding the community-level effects of the tritrophic interactions among mosquito larvae, midsized organisms (such as protozoa), and lowerlevel organisms (such as bacteria and fungi) in west coast phytotelmata. Laboratory microcosm experiments were conducted to characterize the feeding strategies of Ae. sierrensis larvae in the presence of multiple resource types (free-swimming protozoa and substratebound particulate material). In our experiment, we quantified the effects of varying instar numbers and profile, resource type, and refuge size on predation of protozoa. Refugia were explicitly modeled in our microcosms, representing the interstitial spaces of leaf litter and the wood lining of natural tree holes. Results from these microcosm experiments suggested that: (1) Even in the absence of larvae, the majority of protozoa resided in the small-volume, resource-rich refugia. There was, however, a strong nonlinear and negative relationship between larval densities in the upper compartment and the protozoan densities in the refuge, suggesting that there was continual movement of protozoa between the two spaces. (2) Fourth instars harvested resources by filter-feeding at a higher rate than second instars. (3) As the level of substrate-bound particulate food was increased, the predation pressure by filter-feeding on the protozoa decreased. (4) As the refuge volume increased, the predation pressure on the protozoa decreased. We constructed a three-state-variable mathematical model describing the generalist feeding behavior of Ae. sierrensis larvae. The model system, with constant predator densities and two prey groups, exhibited full cooperativity; i.e., an increase in protozoa density resulted in a shift toward predation by filter feeding, while an increase in substrate-bound resources resulted in a shift toward predation by browsing. This indirect mutualism is mechanistically distinct from previously published systems and provides a potential mechanism for protozoan persistence in the presence of larval predation. Key words: Aedes sierrensis; biological model; filter feeding; generalist feeding; indirect mutualism; indirect positive effects; instar numbers and profile; microcosm experiments; mosquito larvae; Paramecium aurelia; resource type and refuge size; tree hole mosquitoes. INTRODUCTION Aquatic habitats residing on or in terrestrial plants (phytotelmata) provide a niche for a large number of microorganisms and invertebrates (Fish 1983). Many of these microcosms have been studied in order to address fundamental ecological problems, such as food web complexity (Kitching 1987, Pimm and Kitching 1987), competition (Livdahl 1982, Chambers 1985, Broadie and Bradshaw 1991), and succession (Kurihara 1983, Mogi et al. 1985). The majority of these investigations have focused primarily on the invertebrate (insect) populations, especially mosquitoes, while generally combining all microorganisms and detritus into a single ecological category. However, these habitats support rich protozoan populations that play a critical role in community organization (Kurihara 1983). FurManuscript received 19 June 1998; revised 16 February 1999; accepted 26 February 1999; final version received 26 March 1999. 4 E-mail: [email protected] thermore, protozoa occupy the midtrophic levels of phytotelmata ecosystems; they consume bacteria and are important prey for larger invertebrates, such as mosquito larvae (Fenchel 1987). Tree holes, a common microhabitat in tropical and temperate forest ecosystems, are filled with rainwater, and the communities they contain are based on an allochthonous energy source from leaves and other plant materials (Kitching 1971). Additional dissolved nutrients are supplied by material carried in stem flow (Carpenter 1982). Since leaf shed is the primary energy source for tree holes, these tend to be resource-limited habitats, resulting in a high degree of competition within and among resident invertebrate populations (Carpenter 1983, Livdahl 1982, Washburn et al. 1988a). In addition to resource limitations, periodic disturbances such as drying can limit the number of trophic levels; however, tree hole ecosystems generally include the larvae of one or more insect species, particularly immature mosquitoes (Pimm and Kitching 1987). These 921 922 JOSEPH N. S. EISENBERG ET AL. detritus-based, semipermanent systems are thus regulated from the bottom up, in that the community structure is largely based on the quality and quantity of lower-trophic-level resources. The effects of these resources on the feeding behavior of mosquito larvae have been widely studied. For example, Fish and Carpenter (1982) showed that Aedes triseriatus larvae can significantly reduce microbial abundance on decaying leaves, through a browsing feeding mode, while Carpenter (1983) showed that the amount of detritus per larva was an important factor regulating larval growth and adult fecundity. Field data and analyses of the midguts of larval Aedes sierrensis, the primary native predator in tree holes in the U.S. West Coast region, indicate that protozoa, free-swimming and sessile, are major food resources for populations residing in natural tree holes in California oak woodlands (Washburn et al. 1988b; Woodward, personal communication). Like other culicids, larvae of Ae. sierrensis are primarily considered browsers that harvest microorganisms from substrates, but they rarely are restricted to this mode of feeding. For example, Washburn et al. (1988b) showed that the rapid disappearance of the protozoan Lambornella clarki in tree holes of northern California was due to the presence of Ae. sierrensis. In addition, Walker and Merritt (1991) observed that larvae of the congener, Ae. triseriatus, both browse substrates and filter suspended nutrients from the water column. Furthermore, they demonstrated that the gut contents of Ae. triseriatus larvae contain significant numbers of free-swimming protozoa (Merritt et al. 1992). Plasticity in feeding modes appears to be an adaptive response to variations in the kinds and amounts of food available within resource-limited habitats. This flexibility in feeding strategies among larval Aedes mosquitoes can be regulated by environmental cues; for example, Aly (1985) found that Ae. vexans larvae reacted to waterborne phagostimulants and ingested food particles at a significantly higher rate than inert particles. Similar results were found with Culex quinquefasciatus, using nucleotides and amino acids as feeding stimulants (Barber et al. 1982, 1983). While these two distinct modes of feeding—browsing of detritus-based microbes and filter feeding of free-swimming protozoa—have been described, little is known about how these two feeding modalities are integrated in order to harvest resources in natural tree holes. In order to elucidate these processes, we conducted a series of laboratory microcosm experiments intended to model the trophic dynamics that occur in water-filled tree holes found in the forests along the west coast of North America. We subsequently developed a populationbased dynamic model in order to analyze our results and to develop both testable hypotheses of the dynamic process and a framework for generalizing our findings to a broader class of ecosystems. The communities found in water-filled tree holes in Ecology, Vol. 81, No. 4 California are divided into three broad trophic levels: (1) decomposing microorganisms, (2) rotifers and protozoa, and (3) insect larvae. The bacteria and fungi that constitute the decomposing microorganisms have not been characterized previously, nor were they in this study. The protozoan community is dominated by species in six families of the Ciliophora: Parameciidae, Erionellidae, Discocephalidae, Tetrahymenidae, Chilodonellidae, Colpodidae (Washburn and Gross, unpublished data). Many species of the protozoa in these families and the rotifers from California tree holes are 50–250 !m in length, a size range that coincides with waterborne particles ingested by filter-feeding Ae. triseriatus larvae (Merritt et al. 1992). Immature forms of nine insect species are common residents of California tree holes, including eight dipterans (two Ceratopogonidae, two Chironomidae, two Culicidae, one Syrphidae, one Psychodidae) and one coleopteran (Helodidae) (Woodward et al. 1988). The most common species, Ae. sierrensis and Culicoides spp. (Ceratopogonidae), were present in nearly all of the tree holes sampled by Woodward and coworkers, but Ae. sierrensis accounted for "90% of the biomass of adult insects emerging from these habitats. Gut contents of the remaining insect larvae showed that none of these species were predators of Ae. sierrensis (Woodward et al. 1988). Aedes sierrensis is primarily a univoltine species, whose immature stages are found almost exclusively within water-filled tree holes in western North America (Bohart and Washino 1978). Eggs are laid throughout the late spring and summer in moist tree holes, and they remain in diapause in a drought-resistant embryonic state until the onset of rains during the following fall and winter. Larvae begin hatching within a few hours of water accumulation and develop throughout the winter months. Once the fourth instar is attained, the larvae enter a developmental diapause and continue to feed until pupation is triggered in the spring by the increasing photoperiod and warmer temperatures (Jordan and Bradshaw 1978). After the adults emerge and mate, females seek blood meals from mammalian hosts, subsequently laying the eggs of the next generation. Previous investigations have shown that Ae. sierrensis larvae may not reduce populations of free-swimming protozoa if they acquire sufficient food resources by browsing, but they shift towards increasing levels of filter feeding as browsing resources decline, thereby cropping protozoan population levels (Washburn et al. 1991). Thus, it appears that some free-swimming microorganisms co-occurring with larval Ae. sierrensis are subject to predation, but the intensity of predation pressure is modulated by environmental factors, such as the amount of alternative food materials and the density of microorganisms. Census data from northern California tree holes have shown consistently that protozoan densities decline dramatically coincident with the growth and development of larval Ae. sierrensis FEEDING BEHAVIORS OF AEDES LARVAE April 2000 923 ora: Parameciidae). We chose this species as our model protozoan because congeners are common throughout California tree holes, and P. aurelia are comparable in size ("100 !m) to a variety of protozoan species seen in natural tree holes. The lowest trophic level consisted of microorganisms (primarily bacteria) that colonized and multiplied on the surfaces of both wheat seeds and ground rat chow (Purina; Ralston Purina, Saint Louis, Missouri, USA) that were placed in the microcosms. The bacteria growth on the wheat seeds served as the primary food source for P. aurelia, while the rat chow (with the associated bacteria) and P. aurelia served as two distinct resource types for the mosquito larvae. Our experiments focused on quantifying the population responses of protozoa living in simulated tree holes and subjected to the following conditions: (1) predation resulting from various larval densities; (2) a resource gradient of alternative resources for larvae available by browsing; and (3) the availability of refugia from larval predation. FIG. 1. (A) Seasonal relative abundance of free-swimming protozoa in a typical natural tree hole in Mendocino County, California, USA. For each sample point, a 100–250mL water sample was collected and filtered to remove insect larvae. All samples were returned to the laboratory and examined microscopically (10–50#) to determine the identity and relative abundance of protozoa. Samples initially were collected at 72-h intervals, beginning with first flooding with rainwater; later in the season, samples were collected at 14– 21-d intervals. (B) Instar profiles of the resident Ae. sierrensis collected from the same tree hole as shown in (A). Each profile was determined from a collection of 50–100 larvae. Data for first, second, third, and fourth instars are shown, along with data for pupae (P). (Fig. 1). Little is known, however, about the effects of shifts in feeding behavior of these predators on protozoan population dynamics. Here, we report results from laboratory experiments, and corresponding model analyses, which describe the dynamics of a protozoan population in the presence of a generalist predator such as Ae. sierrensis. Specifically, we explored the following question: Can the presence of a spatial refuge and the generalist feeding strategy of mosquito larvae enable protozoa to persist under strong larval predation, while also providing a stable and constant resource for the larval predators? By identifying resource regimes that favor larval filter feeding and the degree to which larvae can crop protozoan populations, we were able to demonstrate a mechanism by which protozoa persist in tree hole communities that are subjected to intense predation pressure. METHODS Background and rationale In our microcosm study, the top trophic level consisted of Aedes sierrensis larvae, while the middle trophic level consisted of Paramecium aurelia (Cilioph- Materials We collected first and second instar Ae. sierrensis from natural tree holes in an oak woodland site in Marin County, California. Larvae were maintained in a laboratory growth chamber for one week at 15$C in plastic trays, which were filled with a dilute solution of autoclaved, natural tree hole (15%) and deionized (85%) water; autoclaved, ground rat chow was provided as food. After one week, larvae were moved to an 11$C growth chamber with a short-day photophase (8 h), and they were maintained under these conditions until they reached the appropriate stage for experiments. Paramecium aurelia (Carolina Lab Supplies, Burlington, North Carolina) were cultured at room temperature (22$ % 3$C) in glass jars filled with 3 L spring water. We added 30 irradiated wheat seeds to each jar to provide a substrate for the growth of bacteria, which served as a food source for the protozoa. Paramecium densities were estimated biweekly by thoroughly mixing the contents of the jar, removing and staining (with 0.1 mL amido black dye) a 1 mL sample, and counting the cells with a Sedgewick-Rafter counting chamber. Cultures with densities &100 cells/mL were divided into two jars, which were returned to volume with spring water. To harvest P. aurelia for experiments, the lower portion of the water columns in the culture jars was removed by siphoning, and population densities were determined. To achieve the desired P. aurelia density for each experiment, we diluted cultures with the appropriate volume of spring water. Experimental setup The microcosms for all experiments consisted of 250 mL plastic containers with a screen barrier (0.25 # 0.25 mm) that divided the habitat into two compartments. The screen allowed free movement of the P. aurelia between compartments, but prevented the mos- 924 JOSEPH N. S. EISENBERG ET AL. quito larvae from entering and feeding in the lower compartment. Each container was first filled with 135 mL water (20% tree hole water, 80% spring water). When microcosms were full, the lower refuge compartment was either 24% (32 mL) or 6% (8 mL) of the total habitat, depending on the specific treatment. All microcosms were then inoculated with enough P. aurelia to obtain a mean density of 100 cells/mL. For each experiment, the appropriate number of larvae were placed in the upper compartment, and three irradiated wheat seeds, used as a substrate for bacteria growth, were placed in the lower compartment. During the experimental period, all containers were maintained in a growth chamber at 11$C with a short-day photoperiod (8 h). Initially and at two-week intervals, the contents of each container were mixed, a 1 mL sample was removed, and the P. aurelia densities estimated. Spring water was added weekly to each container in order to keep the volumes constant, and each day all larvae that had molted were removed and replaced with individuals of the appropriate developmental stage in an effort to maintain constant instar profiles. Experiments continued for 35 d, and, at termination, masses of the larval cohorts were measured in groups of 10, in order to obtain a mean biomass estimate for the mosquito populations developing under each experimental regime. This biomass estimate was used to standardize predation pressure over multiple trials, within and between experiments. Three experimental series were conducted: 1). The first experiment was designed to characterize the filter-feeding dynamics of Ae. sierrensis populations consisting of different age structures. Microcosms were established using all permutations of five larval density levels (0, 1, 5, 10, and 20 individuals/microcosm), two larval single-age structures (second and fourth instars), and two refugia conditions (24 and 6% of total habitat). Each treatment was replicated five times. 2). In the second experiment, we characterized protozoan movement between compartments under varying predation pressures. Treatments consisted of variable densities of fourth-instar larvae (0, 1, 2, 4, and 6 individuals/microcosm) and a standard refuge size (24% of total habitat). Twice weekly, independent protozoan samples were obtained from each compartment to determine the size of the resident P. aurelia population. These samples were taken by first carefully siphoning out all the upper compartment fluid. Independent densities were estimated both for the fluid from the upper compartment and for the fluid remaining in the lower compartment of the microcosm. To minimize mixing when reconstituting the microcosm, the upper compartment fluid was added back into the container by covering the screen barrier with a piece of paper and slowly adding fluid along the container sides. 3). The third experiment was designed to characterize the mode switching between browsing and filter- Ecology, Vol. 81, No. 4 feeding under five regimes of browsable food (0, 0.005, 0.010, 0.020, and 0.040 g of rat chow), administered weekly. The food was offered in small plastic cups placed in the upper compartment, excluding it from the lower compartment. For each food regime, we employed five treatments with 20 fourth-instar larvae and five controls with no larvae. The proportional difference between control density and treatment density was used as the metric to assess how each level of browsing substrate indirectly influenced P. aurelia population dynamics. This metric is shown here as ' ( (C)T)/C where C is the mean control density, and T is the mean treatment density of P. aurelia. Therefore, ' ( 0 indicates that there is no difference between the treatment and control densities; ' & 0 indicates that the treatment densities are lower, suggesting that larval predation was a significant mortality factor. Because the measured values of protozoan densities and the number of replicates used were small, the use of confidence intervals to assess the significance of our results was not viable. Instead we chose to use a rank-sum statistic (Dixon and Massey 1983) to calculate the probability that ' & 0; i.e., the treatment is different from the control. Model We developed a three-state, 12-parameter model to analyze the data from these experiments. The first state, Xl, describes the protozoan population levels in the refuge, where predators were absent. Within this lower compartment, the protozoan growth is assumed logistic with a maximum growth rate, r, and carrying capacity, K. The protozoa are assumed to move freely between this predator-free lower compartment and the upper compartment, in which they are susceptible to consumption by the larvae. The protozoa in the upper compartment inhabited by mosquito predators are described by the second state variable, Xu. Their movements from the upper to lower, and lower to upper, compartments are described by the rate constants dul and dlu, respectively. For simplicity, the protozoa within the upper compartment are assumed to have insignificant growth dynamics. This assumption is based on experimental trials using the protocol of Experiment 2. In these trials, we determined that P. aurelia densities in the upper compartment showed no growth, either in the presence or absence of mosquito larvae, over the time period of the experiment (35 d). Furthermore, densities in the upper compartment remained two orders of magnitude less than those in the lower compartment. These data are presented in the Results and shown in Fig. 2. The consumption of the protozoa in the upper compartment is described by a Lotka–Volterra functional response, with * representing the maximum per capita consumption rate. The appropriateness of this functional form was evaluated by analyzing preliminary data using a more general function proposed by FEEDING BEHAVIORS OF AEDES LARVAE April 2000 925 where . is the number of larvae per container, Mi is the biomass per larva for the ith instar, s is the exponent scaling factor, and +(Xu, B) is the proportion of larvae that are filter-feeding rather than browsing. The substrate-bound bacteria dynamics are written as ! " dB B ( rb B 1 ) ) (1 ) +)*b (Mi ) s .B dt Kb FIG. 2. Population abundance of Paramecium aurelia in both the upper compartment (circles, data; dashed line, fit) and lower compartment (squares, data; solid line, fit). These data were collected in the absence of larval predation over a 35-d period. The curves are the least-square fit of the data using Eqs. 1 and 2, and + ( 0. Parameter values are r ( 0.1, K ( 30 000, dul ( 0.44, dlu ( 0.02. DeAngelis et al. (1975). The analysis suggested that this general form simplifies to the Lotka–Volterra functional response; i.e., it is the only term necessary to explain the data statistically. This result is in agreement with the generally accepted idea that the Type 1 functional response is an appropriate representation of filter-feeding activity. To incorporate size-specific information, we have characterized the relationship between body size and prey capture. For suspension feeders, prey capture increases as a power (s) of mass (M), where 0.33 , s , 1.0 (Sebens 1982). Therefore, in our model, the term Msi is incorporated as a multiplier of the functional response, where i indicates instar. The third state, B, is the substrate-bound biomass (bacteria) present within the upper compartment and available for larval browsing. This biomass is primarily limited to the bacterial film that grew on the container surfaces for Experiments 1 and 2, but also includes bacteria associated with the rat chow in Experiment 3. There probably were also suspended organic particles with a film of bacteria that were available for consumption by filter-feeding, but this minor resource is not accounted for in our model, and filter-feeding activity is assumed to be limited to free-swimming protozoa. Bacteria growth is assumed logistic, with a maximum growth rate, rb, and carrying capacity, Kb; the consumption of bacteria by larvae is described by a linear functional response, with *b representing the maximum consumption rate of bacteria. Thus, the protozoan dynamics are written as follows: ! " dX l X ( rX l 1 ) l ) d lu X l - d ul X u dt K (1) dX u ( d lu X l ) d ul X u ) +*(Mi ) s .X u dt (2) (3) where 1) +(Xu, B) is the proportion of larvae that are browsing rather than filter-feeding. For predators foraging in a heterogeneous environment with two resources that are nutritionally substitutable, selection for optimal foraging can lead to nearly perfect switching in favor of the resource that maximizes the predator’s per capita growth rate (see, e.g., Tilman [1980]). The transition from the primary to the secondary resource can occur either in a discontinuous switching mode, or in a gradual, continuous shifting mode. Consequently, in our model, the switching values for + are a function of larval resources within the upper compartment, the free-swimming protozoa (Xu), and the substrate-bound bacteria (B). The functional form is written as +( (aX u ) / (aX u ) / - (bB) / (4) where a and b are scaling constants that allow for the comparison of the two resources, and / determines the rate of transition from consuming one food source to consuming the other, as a function of the relative abundance of the two food sources. The larger the value of /, the more abrupt the switching behavior of the predator. In particular, as / gets large (i.e., approaches infinity), +(Xu, B) approaches a discontinuous switching function (i.e., equals unity if aXu & bB and vanishes if aXu , bB). Using the control trials of Experiments 1 and 2, four of the 12 parameters of our model could be independently analyzed by using Eqs. 1 and 2 and setting + ( 0. The four parameters assessed were as follows: r, the intrinsic per capita growth rate of the protozoa; K, the carrying capacity; and dul and dlu, the respective transfer rates of the protozoa from the upper to the lower, and from the lower to the upper, compartments. A gradient search method (Press et al. 1988) was used to identify optimal parameter combinations. The time series data from the treatments of both experiments were then analyzed using the full model (Eqs. 1–4), in which the parameters r, K, and the ratio of dul/dlu were fixed from the analysis of the control trials. Confidence intervals, although used in some of the preliminary results, should be evaluated with caution, since they are univariate solutions, and the true confidence region is multivariate; however, the true confidence region will reside within the region formed by the interval ranges of each parameter. The goodness of fit was assessed through a least-squares statistic, by JOSEPH N. S. EISENBERG ET AL. 926 summing up all sum of square terms for each treatment (each with a unique larval density, .). Both the total sum of squares (data point minus mean value) and the residual sum of squares (data point minus model prediction) were calculated. An R2 value was defined as the proportion of data explained by the model, as compared with the mean value. The parameter combination with the minimum residual sum of squares was considered the optimal solution. Parameter significance was then assessed by evaluating the percentage change in either the residual sum of squares or the R2 value for a given percentage change in the parameter value. ANALYSIS Simulation studies have the potential to produce misleading results, unless the global behavior of the model is understood. Furthermore, they may miss the existence of a stability region, if the appropriate initial condition was not used. This issue becomes more relevant as a model becomes increasingly complex. For example, continuous, nonlinear, three-state models can exhibit complex behaviors (Hastings and Powell 1991, Eisenberg and Maszle 1995). In this section, we discuss the analysis that proves that our model exhibits simple global properties, providing increased confidence in our simulation results. The details of the analysis are shown in the Appendix, where we consider a generalized form of Eqs. 1–4. This model is easily seen to be a cooperative system (i.e., an increase in the density of one state variable results in an increase in the growth rates of the other). This cooperativity is mediated by the larval feeding behavior, which shows preference to the resource that is in greater abundance. To understand the behavior of the two-species model, it is necessary to understand the behavior of the onespecies subsystems. In the absence of protozoan dynamics (i.e., Xu ( Xl ( 0), the substrate-bound bacteria dynamics reduce to logistic dynamics: ! " dB B ( rb B 1 ) ) *b (Mi ) s .B. dt Kb (5) Hence, in the absence of the protozoa, the substratebound bacteria have a positive per capita growth rate at low densities provided that rb & *b·(Mi)s·.. Therefore, in the absence of the protozoa, the substrate-bound bacteria persist only if the larval predation rates are lower than their intrinsic rate of growth. In the absence of the substrate-bound bacteria (i.e., B ( 0), we can assess the role of refuge in protozoan persistence, independent of prey switching. This approximates the conditions of Experiment 1, in which substrate-bound microorganism density is low. Solving for the nonzero equilibrium of Eqs. 1 and 2 when B ( 0, we obtain the following: X*u ( d lu X* d ul - *M s . l X*l ( K d ul d lu r ) d lu . r d ul - *M s . ! (6a) " (6b) Ecology, Vol. 81, No. 4 Notice that the equilibrium density X* l is positive, if and only if r & (dlu*Ms.)/(dul - *Ms.). In the Appendix, we show that, when X* l is positive, then the equilibrium Xu*, Xl* for Eqs. 1 and 2 with B ( 0 is in fact globally stable. Alternatively, if r , (dlu*Ms.)/(dul - *Ms.), then the protozoan population is always driven to extinction. At low levels of predation (i.e., . # 0), the feasible equilibrium is given by Xl* # K and X* u # (dlu/ dul)X* l . In particular, the population in the absence of predation is able to exceed the carrying capacity of the lower compartment by reducing intraspecific competition. At high levels of predation (i.e., . very large), the protozoan population persists, provided that r & dlu. When this is the case, the globally stable equilibrium is given by Xu* # 0 and X* l # (r ) dul)K/ r. Thus, the analysis suggests that refuge alone can maintain protozoan persistence at high predation levels. The cooperativity of the two species model is shown in the Appendix to imply that Eqs. 1–3 have a globally stable equilibrium at which both species persist, provided that either subsystem is persistent (i.e., either rb & *b (Mi)s., or r & (dlu*Ms.)/(dul - *Ms.). Because the substrate-bound bacteria can not persist at high levels of larval predation, our analysis implies that both species persist at high levels of larval predation only if the rate of movement to the upper compartment is sufficiently small (i.e., r & dlu). A small dlu also corresponds to a large refuge area, since a decrease in the refuge size both increases the surface area:volume ratio of the refuge area and decreases the rate of movement to the upper compartment. In general, we see that, although prey-switching alone cannot guarantee persistence, under the condition of unbounded increases in predation, it can relieve the predation pressure on the protozoa allowing for smaller refuges to provide sufficient protection. RESULTS Protozoan growth in the absence of larval predation Data depicting the population growth of Paramecium aurelia in the absence of larval predation are shown in Fig. 3. Fig. 3 contains data from the microcosms used in Experiment 1 having different refuge sizes: one consisting of 24% of the habitat, and the other consisting of only 6% of the habitat. A logistic-growth model was fit to both of these time series, resulting in estimates for the per capita growth rate, r, and carrying capacity, K. The estimates of r ranged 0.15–0.19 per day for the standard refuge trial, and ranged 0.13–0.23 per day for the reduced refuge; estimates for K ranged 446–498 cells/mL and 407–493 cells/mL, respectively, for the two trials. These estimates of r and K were not statistically different, suggesting that refuge size did not impact protozoan population characteristics in the absence of larval predation. It should be noted here that this comparison was only valid for trials that were run with the same protozoan culture that were provided April 2000 FEEDING BEHAVIORS OF AEDES LARVAE 927 Protozoan growth in the presence of larval predation FIG. 3. The density of Paramecium aurelia, with means taken over both the upper and lower compartments. These data were collected in the absence of larval predation over a 35-d period. The line is the least-square fit of the data using a logistic growth model. Results are shown for Experiment 1 with the standard refuge size (squares, data; fit, dashed line; r ( 0.17 % 0.02, K ( 472 % 26) and for Experiment 1 with the reduced refuge size (circles, data; solid line, fit; r ( 0.18 % 0.05, K ( 450 % 43). with similar bacterial cultures (i.e., food for the P. aurelia). In fact, control trials for Experiments 2 and 3 yielded significantly different estimates for r and K than those observed for Experiment 1 (Fig. 3). Since parameter estimation algorithms are not tractable for more than approximately five parameters, we used these estimates of r and K from Experiment 1 control trials for subsequent analysis of Experiment 1 treatment trials. Experiment 2 provided additional information on the spatial distribution of the P. aurelia in the absence of larvae (Fig. 2). Logistic growth was observed in the lower, resource-rich compartment, whereas the protozoan population in the upper compartment remained constant. To estimate protozoan transfer rates between the two compartments, these data were fit to Eqs. 1 and 2 with + ( 0. On average, the density values from the lower compartment were "20 # larger then those from the upper compartment. Therefore, in order to allow data from the upper compartment to contribute equally to the fitting procedure, we scaled these data points by a factor of 20. In Experiment 2, the estimate for r was lower than that found in Experiment 1, which may be attributed to differences in the initial protozoan cultures used for each experiment. Similarly, the estimate for K, which was compared with results from Experiment 1 by dividing 30 000 cells by 135 mL (222 cells/mL), was also significantly smaller. The optimal transfer rates were found to be 0.44 and 0.002 per day for the movement from upper to lower and lower to upper compartments, respectively. Results from Experiment 3 verified that the shift from browsing to filter feeding was a function of the level of available browsing substrate material. In these trials, 20 fourth instar larvae were placed in each treatment microcosm, whereas none were placed in the control microcosms. Data from these trials suggested that Aedes sierrensis larvae became more effective predators of free-swimming P. aurelia as the availability of browsing substrate material was reduced. This shift in feeding behaviors is shown in the P. aurelia population trends for three different food regimes (Fig. 4) and is consistent with results from previous studies (Washburn et al. 1991). There was rapid divergence in P. aurelia population densities between the treatment with no browsing substrate added and the treatment in which 0.02 g of browsing substrate was added. It took longer, however, for the treatments with differing levels of browsable food to equilibrate and show separation; this is perhaps due to the discrete nature of applying food in the experiment. For example, after each addition of particulate, browsable food, we observed a pronounced increase in protozoan populations, suggesting that larval predation was relaxed (Fig. 4). Nonetheless, by 21 d, a clear differentiation was apparent between the 0.005-g and 0.020-g food regimes. After two weeks, in the absence of any browsing substrate for larvae, P. aurelia density was 85% higher in the controls, as compared with the treatment with predators. In the food regime with 0.005 g of browsing substrate, P. aurelia density was 62% higher; and, with 0.020 g of browsing substrate, P. aurelia density was 53% higher. By day 25, protozoan densities among the three food regimes were differentiated (Eq. 1): the control densities were 60% higher than the treatment densities for the low- FIG. 4. Percentage change of P. aurelia density for treatment groups consisting of 20 larvae, relative to control groups with similar substrate resources but no larvae; 0.00 indicates that the control and treatment groups were equal. Time trends associated with 0.0 g browsing (diamonds), 0.005 g (squares), and 0.02 g (triangles) are shown. Arrows indicate times when resources were added. 928 JOSEPH N. S. EISENBERG ET AL. Ecology, Vol. 81, No. 4 food regime (' ( 0.81; P[' ( 0] ( 0.026), 35% for the medium-food regime (' ( 0.48; P[' ( 0] ( 0.115), and nearly equal for the high-food regime (' ()0.09; P[' ( 0] ( 0.421). These trends continued through day 35 when the experiment was terminated. While there is a clear statistical distinction between the highand low-resource regimes, the statistical significance of the medium-resource regime is less clear. The data do suggest, however, that given a choice, Ae. sierrensis preferred to feed on resource-rich browsing substrates and shifted progressively towards more filter-feeding as such resources became limiting. The impact of predation by five densities of two developmental stages of Ae. sierrensis larvae on P. aurelia populations is shown in Fig. 5. In these experiments, no browsing substrate was added, although it was assumed that the container surfaces acted as a substrate for a small amount of bacteria growth. Treatments contained 0, 1, 5, 10, or 20 second- or fourthinstar larvae in the upper compartment. When subject to predation by second-instar larvae, differences in P. aurelia populations between treatments with zero and one larvae were not significant, resulting in equilibrium levels of 450–550 cells/mL (Fig. 5A). There was an intermediate response for treatments with five larvae, resulting in equilibrium levels of 300–350 cells/mL; whereas, in treatments with 10 or 20 larvae, the predators cropped the P. aurelia densities to ,200 cells/ mL. The results using fourth-instar larvae were similar, except that predation levels were higher. For example, five fourth-instar larvae cropped the P. aurelia densities to ,200 cells/mL, a prey density similar to the treatments with 10 and 20 larvae. Fig. 5B shows the P. aurelia response to fourth-instar larvae using our standard refuge size, whereas Fig. 5C shows the response using the reduced refuge size. The effect of FIG. 5. Population abundance of Paramecium aurelia over a 35-d period for five treatments: 0 larvae (circles), 1 larva (squares), 5 larvae (diamonds), 10 larvae (upward-pointing triangles), and 20 larvae (downward-pointing triangles) per microcosm. The curves are the least-square fit of the data using Eqs. 1–4. The curves associated with zero and one larvae coincide and are the uppermost curves, while the curves for 5, 10, and 20 larvae are the successively lower three curves. Parameter values are r ( 0.2, K ( 50 000, dul ( 0.4, r. ( 0.8, K. ( 1.0, . ( 10, s ( 0.5, * ( 1, a ( 1.5 # 10)6, and b ( 100. Data are shown for microcosms with (A) second-instar larvae and standard refuge size, *b ( 4, dlu/dul ( 0.34; (B) fourth-instar larvae and standard refuge size, *b ( 1, dlu/dul ( 0.34; and (C) fourth-instar larvae and reduced refuge size, *b ( 4, dlu/dul ( 0.38. FIG. 6. Comparison between standard (shaded) and reduced (unshaded) refuge sizes of the average Paramecium cell density (geometric mean) for different fourth-instar larval densities. April 2000 FEEDING BEHAVIORS OF AEDES LARVAE different refuge sizes is evident by comparing the protozoan populations from the 5, 10, and 20 fourth-instar larvae treatments (Fig. 6). At all three predator densities, P. aurelia populations were higher (as much as 40%) when the refuge size was larger. In addition, Fig. 5B shows higher density levels by day 35, than does Fig. 5C, indicating that increasing the refuge space had a positive effect on the prey levels. We next analyzed the data from Experiment 1, using the complete model (Eqs. 1–4). For numerical tractability, we limited the number of optimized parameters to five. We used the results of the previous experiments, the control trials of Experiment 1 and Experiments 2 and 3, as well as preliminary analytical results, to provide estimates for the remaining seven parameters. In preliminary optimization runs, we determined that four of the 10 parameters were insensitive to the sum of squares values: (1) K, the carrying capacity for P. aurelia in the lower compartment; (2) rb, the per capita growth rate for bacteria; (3) Kb, the carrying capacity for bacteria; and (4) /, the food-switching scaling factor. Therefore, K was set to 50 000 cells (based on results from the control studies; i.e., 490 cells/mL and 102 mL in upper compartment), rb was set to 0.8, Kb was set to 1.0, and / was set to 10. The biomass exponent scaling factor, s, was adjusted so that the treatment using second-instar larvae and the treatment using fourth-instar larvae could optimally fit their respective data sets using the same parameter set. These estimates required biomass values. Based on a series of dry-mass measurements, we assumed a biomass of 0.02 mg/larva for the second instars and 0.70 mg/larva for the fourth instars. Using these biomass values, a second set of preliminary runs estimated s at 0.5 for both filter-feeding and browsing. Finally, the ratio of the protozoan transport rates between the compartments was assumed to be constant, leaving five parameters, along with the ratio of two parameters, to be fit to the data: (1) r, the P. aurelia per capita growth rate; (2) *, the maximum larvae consumption rate of protozoan; (3) *b, the maximum larvae consumption rate of bacteria; (4) a, the scaling constant for the switching function; (5) dul, the protozoan transfer rate from the upper to the lower compartment; and (6) the ratio dul/dlu. The trajectories associated with the optimal parameter estimates are shown in Fig. 5 for second instar with standard refuge (R2 ( 0.62), fourth instar with standard refuge (R2 ( 0.66), and fourth instar with reduced refuge (R2 ( 0.61). The analysis produced values for four of the six parameters that were indistinguishable among the three treatments. The parameter r was estimated at 0.2 per day, * at 1.0, a at 1.6 # 10)6, and dul at 0.4. The one parameter required to distinguish the second-instar treatment from the two fourth-instar treatments was *b. The maximum larval consumption rates of bacteria were estimated at 4.0 bacteria·d–1·mL–1 for second instars and 1.0 bacteria·d–1·mL–1 for fourth instars. Shifting the value of *b for the second-instar treatments from 4.0 to 1.0, 929 or the fourth-instar treatments from 1.0 to 4.0, increased the sum of squares by &200%. Therefore, the ratio of fourth-instar to second-instar maximum consumption rates, assuming the larval biomass are 0.02 and 0.7 mg for the second and fourth instars respectively, was 5.9 P. aurelia and 1.5 for bacteria. Analogously, only one parameter, the ratio dul/dlu, was required to distinguish between the standard and reduced refuge treatments. This ratio was 0.34 for the standard refuge treatments and 0.38 for the reduced refuge treatments. Shifting the value of dul/dlu for the standard refuge treatments from 0.34 to 0.38 increased the sum of squares by 27%. Therefore, the decrease in cell density was associated with a decrease in refuge size from 30 to 7 mL and was characterized in the model by a parametric shift in the ratio dul/dlu from 0.34 to 0.38. We used Eq. 6 to calculate the theoretical predictions of the asymptotic protozoan density levels for increasing larval densities. The maximum-likelihood value for dlu was 0.136 for the standard refuge condition and 0.152 for the reduced refuge condition. Therefore, the prediction of the asymptotic refuge density for protozoa under the conditions of Experiment 1 was 157 cells/ mL for the standard refuge case and 118 cells/mL for the reduced refuge case. DISCUSSION In natural tree holes, the feeding dynamics of Aedes sierrensis larvae are controlled by many factors that are not well characterized. Our analysis suggests two mechanisms that provide these larvae with stable and continual resources. First, layers of leaf litter on the bottom and interstitial spaces within the wooden sides of the tree hole cavity serve as important refugia for naturally occurring protozoa and other microorganisms (e.g., rotifers), preventing the larvae from eliminating prey populations. Although the census data presented here from northern California show that protozoan abundances decline dramatically coincident with the growth and development of larval Ae. sierrensis (Fig. 1), these data were derived from samples of the water column and do not reflect protozoan diversity and abundance either within leaf litter or in the interstitial spaces. Second, Ae. sierrensis larvae switch between two broad resources types, based on their relative abundance, i.e., this prey-switching mechanism relieves predation pressure for a given prey with a decreasing density, relative to the other prey type. This is consistent with the empirical demonstration that Aedes larvae are effective predators of both free-swimming and substrate-bound microorganisms (Washburn et al. 1991, Merritt et al. 1992), and that they are highly effective at finding and exploiting resource patches (Aly 1985; J. Washburn, unpublished observations). Role of refuge and prey-switching in protozoan persistence Both our experimental and analytical results suggested that refugia are critical for protozoan persis- 930 JOSEPH N. S. EISENBERG ET AL. tence, and that sufficient refuge space can guarantee persistence, even when predation pressure is high. Data collected from the two compartments of our experimental microcosms demonstrated that there was strong spatial heterogeneity in protozoan densities, even when mosquitoes were absent; i.e., a large proportion of Paramecium aurelia resided near the bottom of the microcosm where the food resource associated with the wheat seeds was greatest, and a lower density existed in the upper portion of the water column, where there was a higher oxygen content. Furthermore, our experiments suggested that, in the compartment with larvae, the protozoa existed at constant low densities, independent of the presence of larval predators. However, density levels in the predator-free compartment varied with larval density, exhibiting decreasing density levels with increasing larval densities. In Experiment 1, we assumed that there was a very low (but nonnegative) level of larval browsing on the bacterial surface film on the container walls. Therefore, in the presence of only one fourth-instar larva, consumption of P. aurelia was negligible, because browsing apparently satisfied the majority of larval food requirements. However, in the presence of two or more fourth instars, P. aurelia densities were cropped to the refuge level (Fig. 5B), suggesting that the browsing material was depleted to a level that triggered larvae to switch to filter-feeding mode. Second instar larvae produced a more gradual response (Fig. 5A). In the presence of five larvae, the P. aurelia densities were cropped somewhere between the control and refuge levels, suggesting that, due to a lower consumption rate, second-instar larvae could be supported by bacteria growth on walls at a greater density than fourth instars. At higher predator densities, we showed experimentally that P. aurelia can coexist with densities of !40 fourth-instar larvae/container, and we observed a mean asymptotic protozoa density within the entire microcosm of "100 cells/mL. This coexsitence was due to the refuge, an essential stabilizing feature of this ecosystem that can provide a source for a sustainable protozoan population. Indeed, in experiments utilizing microcosms without refugia, P. aurelia populations were eliminated completely within just a few days (data not shown). However, even with a refuge, protozoan movement into the upper water column resulted in a relationship between the protozoan densities in the refuge and the larval densities in the upper water column. These experimental findings agree with analysis of our model, and they suggest that persistence can be ensured, given sufficient refuge as defined by the ratio of the rates of protozoan movements between the two compartments. Less critical than refugia, prey switching also plays a role in protozoan persistence. In systems with a dynamic predator that switches among prey, persistence of all species often occurs (Gleseson and Wilson 1986, Huston and Schmitt 1992). However, for microcosms in which the predator is not dynamic, in the time frame Ecology, Vol. 81, No. 4 of interest, switching does not have such a strong effect on determining protozoan persistence. We demonstrated this theoretically using our model, which is in agreement with our experimental findings. Results from Experiment 3 suggested that prey switching significantly reduced predation pressure, potentially decreasing the amount of refuge required to maintain persistence. A highly nonlinear relationship between larval and protozoan densities indicated that, for densities &1 fourthinstar larva/container (0.007 larva/mL), or &5 secondinstar larva/container (0.04 larvae/mL), Ae. sierrensis significantly cropped free-swimming protozoan populations. This relationship held unless the density of browsable substrate material increased, in which case the protozoa density also increased. Our experimental results are consistent with the nonlinear cooperative behavior exhibited by our model, which contained a switching function that depended on the relative concentrations of the two food resource types raised to a power greater than 1. The higher the value of the exponent, the closer the behavior is characteristic of a switch. Although the exact value of the exponent was highly uncertain, the maximum-likelihood estimate was significantly greater than 1, suggesting that the shift is more abrupt than gradual. In addition, our findings are in agreement with results from Washburn et al. (1991), which showed that Ae. sierrensis larvae cropped free-swimming trophonts of the ciliate, Lambornella clarki, by differing amounts depending on the level of substrate-bound resources. This is in contrast to the predominant paradigm in the literature that categorizes Aedes larvae as primarily substrate browsers (e.g., Merritt et al. 1992). The Washburn et al. (1991) laboratory-based study suggested that L. clarki trophont densities were initially suppressed with the addition of both larvae and a substrate-bound resource, and they were further suppressed when the substratebound resource was absent. Our experimental results extend the Washburn et al. (1991) findings by quantifying the resource shift at a finer resolution, including a substrate-bound resource level that produced a complete shift from filter-feeding to browsing. The nonlinear relationship that we observed in our laboratory experiments may be significant in only a portion of larval populations in tree holes, because natural densities are often higher than those used in our experiments; e.g., Hawley (1985) measured Ae. sierrensis densities of 0.03–3.2 larvae/mL in Oregon tree holes, compared to our experimental range of 0.007–0.3 larvae/mL. Indirect mutualistic effects The cooperative nature of our model is the consequence of a nondynamic predator, whose functional response for either prey depends on the densities of both prey species. In our model, this density dependence is achieved by combining a switching mechanism between a protozoan-based resource and a bacteria-based resource, with a linear functional response April 2000 FEEDING BEHAVIORS OF AEDES LARVAE that describes larval consumption rates. Therefore, the observed indirect relationship is mediated via a common predator, in that an increase in protozoa density resulted in decreased predation on the substrate-bound microorganisms, while an increase in the substratebound microorganisms resulted in decreased predation on the protozoa. The latter relationship was measured in Experiment 3, whereas our model inferred the former relationship. We are not modeling the numerical response of the predator, because the time frame of interest, when studying the population dynamics of protozoa within tree holes, is a single season. Thus, the predacious larvae do not increase in number as a result of resource dynamics. The long-term effect of predator dynamics between seasons may lead to the opposite conclusions. For example, when accounting for the dynamics of the predator in a two-prey, single-predator model, Holt (1977) showed that apparent competition may occur, in which an increase in the density of one prey results in an increase in the predator density, and this gives rise to a concomitant decrease in the density of the alternative prey. The long-term effect of predator dynamics between seasons may also lead to similar results. For example, Levine (1976) and Vandermeer and Boucher (1978) studied indirect mutualistic behavior in systems that consisted of highly competitive prey sharing common predators. In this model ecosystem, any increase in one species resulted in an increase in predation, relieving the interspecies competition of the prey. Whether either of these predictions extends to seasonal systems, such as the water-filled tree hole habitats of California, is not clear and certainly requires further investigation (Hambäck 1998). Modeling assumptions This indirect mutualism between prey populations may be modulated by the fact that bacteria are consumed by both larvae and P. aurelia within the upper compartment, thus adding an additional consumption term to the protozoan equation. Under these conditions, an increase in bacteria populations will result in both an increase in predation by larvae via switching, decreasing the predation pressure of the protozoan population, and an increase in consumption of bacteria by P. aurelia. Thus, the positive indirect effect of the bacteria population on the protozoan population is either maintained or strengthened; however, the implications of an increase in the protozoan population are more complex. In addition to larvae switching towards feeding on the protozoa, and thereby relaxing the predation pressure on bacteria, the higher protozoan population also may result in an increase in bacteria consumption. A more detailed look at the equation for the bacteria population (Eq. 3), when an additional term of the form )cBXu (where c is the consumption rate of the protozoa on the bacteria) is added to describe the upper compartment protozoan consumption of bacteria, suggests 931 that the criterion for a competitive vs. cooperative relationship depends both on c and +(Xu, B), the switching function. With this additional term, a small increase in the protozoan population increases the growth rate of the bacteria population only if B !0X * M . ) c" & 0. 0+ s b u Therefore, a positive indirect effect will be maintained only if 0+/0(Xu) is large compared with c. This rate of change is greatest when aXu # bB (i.e., conditions when small changes in density result in large shifts of predatory behavior), and the rate of change smallest at either small or large values of Xu (i.e., conditions when the larvae are not switching resources for small changes in Xu). In the development of the model, we assumed that the dynamical effects of this modification are negligible, due to the fact that the protozoan population within the upper compartment is small. Hence, the rate at which the protozoa consume bacteria (i.e., c) is most likely small. Two additional process that complicate community interactions in natural tree holes were not included in our analysis: (1) larvae increase in size with each instar, changing their predation rates as they develop; and (2) staggered installment hatching recruits new first instars into the population throughout the season. Two attributes of the system suggest that the former complication should not be significant with respect to the positive indirect effects. First, the allometric relationship between larval size and predation is less than linear (Sebens 1982), which is consistent with our analysis suggesting consumption scales with the 0.5 power of biomass. Second, the larval growth saturates to a maximum value; i.e., when larvae attain the fourth instar and enter diapause. The latter complication will not affect the steady-state behavior nor the effects of the positive indirect behavior, because installment hatching is not a function of the larval densities and can be treated as an input variable that decreases to zero over the course of the season. A major assumption of our model is that the Ae. sierrensis larvae exhibit a Type I functional response in their consumption of P. aurelia. Because increasing the complexity of the model to a Type II or a more general form proposed by DeAngelis et al. (1975) did not improve the maximum-likelihood fit, our data did not support use of a more complex function. This finding suggests that, within the range of protozoan (prey) densities of 100–400 cells/mL, we observed no evidence of handling time. Furthermore, within the range of 1–40 larvae/container (0.007–0.3 larva/mL), we observed no evidence of interference. We used the fact that protozoan densities within tree holes are actually ,100 cells/mL, to justify the use of the linear functional response over a more general form (DeAngelis et al. 1975). Our estimates of total protozoan popula- JOSEPH N. S. EISENBERG ET AL. 932 tions in the 50–250 !m size range were ,100 cells/ mL, and even ,1–10 cells/mL when increased larval numbers were in residence (J. Washburn, unpublished data). It is interesting to note that the combination of an explicit switching mechanism (Eq. 4) with the Lotka–Volterra functional response produces a functional form that differed from the traditional Holling type III functional response. Recall, in the generalized Holling type III function, the exponents in the numerator and denominator are equal (Real 1977, Nunney 1980), f (X ) ( aX n b - Xn so that as prey densities increase, predation becomes constant, independent of the prey density. The functional response used in this study contains a term in the numerator whose exponent is one greater than the term in the denominator. This difference between the functional forms has important dynamical consequences. Namely, it has been previously shown (Ludwig et al. 1978) that a system with logistic growth and a Hollings III functional response exhibits multiple stable states. We have shown that the increased exponent value in our model results in the loss of this multiple stable-state property. Conclusion By using our model to analyze the data from Experiment 1, we were able to find an optimal set of parameter values that could fit data across the three treatment runs from Experiment 1, by accounting for the differences between second and fourth instar feeding characteristics and the size of the refuge. The distinction between second and fourth instar was addressed in two ways: (1) the maximum consumption rates of protozoa by second-instar larvae and fourthinstar larvae were assumed to differ by a biomass scaling factor of *·M0.5, where * is the maximum consumption rate and M is the biomass of the larvae; and (2) the parameter 1 had the same value for second and fourth instars, with respect to protozoan consumption, but differed by a factor of 4 for browsing. The size of the refuge was accounted for by the ratio dul/dlu. A change in ratio of 0.34–0.38 adequately described the different protozoan response between the standard and reduced refuge. One discrepancy that needs further discussion is the fact that dul/dlu was 1/3 for Experiment 1 and 1/20 for Experiment 2. This is a significant difference as measured by the sum of squares term, and it was possibly due to the fact that these experiments were run at different times using different protozoan and bacteria cultures. Our model characterizes the population dynamics of a protozoan species within a microcosm designed to mimic tritrophic interactions occurring in natural tree holes. It describes this tritrophic system as fully cooperative, a highly adapted feature that is necessary in such a resource-limited system. Most phytotelmata Ecology, Vol. 81, No. 4 contain allochthonous resources that must be sufficient to support a protracted period of larval development. In most areas, this protracted period is across multiple generations, often of different species. However, on the West Coast of North America, instead of having a succession of many species with short developmental times, there exists a single species, Ae. sierrensis, with a long ("6 mo) developmental period, resulting in particularly severe resource limitations. Moreover, owing to the highly seasonal pattern during winter rainfall, the temperature of these aquatic habitats are generally ,10$ C, lowering rates of microbial decomposition and nutrient turnover. Therefore, the Ae. sierrensis larvae have evolved a flexible feeding strategy, whose plasticity is shaped through environmental cues of resource quality and quantity. ACKNOWLEDGMENTS We would like to thank Roger Nisbet and two anonymous reviewers for their insightful comments. This work was supported by a National Science Foundation (NSF) postdoctoral fellowship (DEB 9303260) given to J. N. S. Eisenberg, a Faculty Development Grant given to S. J. 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Regulatory role of parasites: impact on host population shifts with resource availability. Science 253:185–188. Woodward, D. L., A. E. Colwell, and N. L. Anderson. 1988. The aquatic insect communities of tree holes in northern California oak woodlands. Bulletin for the Society of Vector Ecology 13:221–234. JOSEPH N. S. EISENBERG ET AL. 934 Ecology, Vol. 81, No. 4 APPENDIX d2 y d2 ! a In this appendix, we justify some of the assertions regardf )(x) # 2 # ing the long-term behavior of Eqs. 1–4. Consider the followx x ing model on the positive orthant !3! of !3: and, consequently, is strictly negative in the interior of the dx positive quadrant !2!. Hence by Dulac’s criterion (Perko " xf (x) # d1 x ! d2 y dt 1998) there are no periodic solutions for Eq. A.3 contained in !2!. Solving for the equilibria of Eq. A.3, we get the origin dy (0, 0) and any solutions (x*,y*) to the following equations: " d1 x # d2 y # $( y, z)ay dt dz " zg (z) # (1 # $( y, z))bz dt where f and g are continuously differentiable and strictly decreasing functions satisfying f (K1) " 0 and g(K2) " 0 for some K1 % 0 and K2 % 0; $:!2!\(0, 0)→[0, 1] is defined by $( y, z) " y& y & ! (cz) & with & ! 1; the constants d1, d2, a, b, and c are all positive. Although the function $( y, z) is not continuous at the origin, the functions ( y, z) ı→ y$( y, z) and ( y, z) ı→ z(1 # $( y, z)) are continuous but not differentiable at the origin. In general, continuity does not ensure that Eqs. A.1 generate a flow. Fortunately, it can be shown that the switching function $( y,z) is such that the functions ( y, z) ı→ y$( y, z) and ( y, z) ı→ z(1 # $( y, z)) are Lipschitz in a neighborhood of the origin. Consequently, the model of Eq. A.1 does generate a welldefined flow. For the interested reader, Chapter 2 of Perko (1998) provides a discussion of these issues. To understand the conditions under which both species in the model represented by Eq. A.1 are able to persist, we need to understand the behavior of the subsystems of Eq. A.1 corresponding to either x " y " 0 or z " 0. When x " y " 0, Eq. A.1 reduces to the one-dimensional system dz " zg (z) # bz. dt (A.2) It is straightforward to prove the following claim about Eq. A.2. Claim 1. If g(0) % b then Eq. A.2 has a unique equilibrium z* % 0 such that every solution z(t) to Eq. A.2 with z(0) % 0 satisfies z(t) → z* as t → '. If g(0) ( b, then every solution z(t) to Eq. A.2 with z(0) ! 0 satisfies z(t) → 0 as t → '. When z " 0, Eq. A.1 simplifies to the two-dimensional system dx " xf (x) # d1 x ! d2 y dt dy " d1 x # d2 y # ay. dt (A.3) Proof of Claim 2. First, we note that the solutions of Eq. A.3 with initial conditions in the positive orthant are bounded in forward time. The divergence of the rescaled vector field 1 dx 1 dy is given by d1 a d2 ! a y* " d1 x* . d2 ! a (A.4) Since f is strictly decreasing, there is a solution x* % 0, y* % 0 to Eq. A.4 if and only if f (0) % d1a/(d2 ! a). Furthermore, when it exists, the equilibrium determined by Eq. A.4 is unique. Since the variational matrix of Eq. A.3 at the origin equals [ ] f (0) # d1 d2 d1 #a # d2 the origin is linearly unstable if f (0) % d1a/(d2 ! a). Since there are no periodic solutions and all solutions are bounded, the Poincaré-Bendixson Theorem (Perko 1998) implies that every solution to Eq. A.3 with nonnegative initial conditions approaches (0, 0) in forward time when f (0) ( d1a/(d2 ! a). Alternatively, when f (0) % d1a/(d2 ! a), the Poincaré-Bendixson Theorem and the linear instability of (0, 0) implies that every solution to Eq. A.3 with positive initial conditions approaches the equilibrium defined by Eq. A.4. This completes the proof of the claim. Now we are ready to look at the asymptotic behavior of the full system of Eq. A.1. Claim 3. The system represented by Eq. A.1 has at most one equilibrium lying in the interior of !3!. Proof of Claim 3. At any equilibrium (x, y, z) lying in the interior of !3!, we have that (1/z)(dz/dt) " 0. Solving for y as a function of z in the expression (1/z)(dz/dt) " 0, we get y (z) " cz ! b #1 g (z) " 1/& which is well defined and nonnegative provided that 0 ( g(z) " b. Since g)(z) ( 0, it follows that, when & % 1, y)(z) " c Claim 2. If f (0) % d1a/(d2 ! a) then Eq. A.3 has a unique equilibrium x* % 0, y* % 0 such that every solution (x(t),y(t)) to Eq. A.3 with x(0) % 0, y(0) % 0 satisfies x(t) → x*, y(t) → y* as t → '. If f (0) ( d1a/(d2 ! a) then every solution (x(t),y(t)) to Eq. A.3 with x(0) ! 0, y(0) ! 0 satisfies x(t) → 0, y(t) → 0 as t → '. !x dt , x dt " f (x*) " (A.1) !g (z) # 1" b 1/& # cz !g (z) # 1" (1/&)#1 b bg)(z) & g (z) 2 (A.5) is strictly positive whenever 0 ( g(z) ( b. Similarly, it follows that y)(z) % 0 when & " 1. Define $̂ (z) " $(y(z), z). Since g(z) " b(1 # $̂(z)) and g)(z)( 0, it follows that $̂)(z) " #g)(z) 1 b (A.6) is strictly positive whenever 0 ( g(z) " b. At any equilibrium (x, y, z), we also have dy/dt " 0. Solving for x in dy/dt " 0, we get x (z) " y (z) (d2 ! $̂(z)a). d1 Eqs. A.5 and A.6 imply that x(z) is an increasing function of z. Plugging x(z), y(z), and $̂ (z) into dx/dt ! dy/dt " 0 and dividing out by x(z), it follows that 0 " f (x (z)) # $̂(z)ad1 . d2 ! $̂(z)a (A.7) FEEDING BEHAVIORS OF AEDES LARVAE April 2000 Taking the derivative of the right-hand side of Eq. A.7 with respect to z, we get $̂)(z)ad2 d1 f )(x (z))x)(z) # . [d2 ! $̂(z)a] 2 935 Let F (x, y, z) " (xf (x) # d1 x ! d2 y, d1 x # d2 y # $( y, z)ay, zg (z) # (1 # $( y, z))bz). Since $̂)(z) and x̂)(z) are positive and f )(x(z)) is negative, the right-hand side of Eq. A.7 is a strictly decreasing function of z. Hence, there exists at most one equilibrium in the interior of !3!, which completes the proof of the claim. Claim 4. If g(0) % b or f (0) % d1a/(d2 ! a), then there exists a unique equilibrium x* % 0, y* % 0, z* % 0 to Eq. A.1 such that every solution (x(t), y(t), z(t)) to Eq. A.1 with x(0) % 0, y(0) % 0, z(0) % 0 satisfies x(t) → x*, y(t) → y*, z(t) → z* as t → ' (i.e., the equilibrium (x*, y*, z*) is globally asymptotically stable). Proof of Claim 4. Let int !3! denote the interior of the positive orthant !3!. To prove this claim, it suffices to verify the conditions of the following theorem. Recall that F (x, y, z) is continuously differentiable in int !3!. The derivative matrix DF(x, y, z) for points in int !3! is of the form * ! 0 ! # ! 0 ! * where an asterisk (*) indicates an entry that can be positive or negative. Hence F (x, y, z) is cooperative. To show that the system of Eq. A.1 is permanent, we first show that all solutions to Eq. A.1 with initial conditions in !3! are bounded from above in forward time. Notice that dx/ dt ! dy/dt " xf(x) and dz/dt " zg(z). Therefore, any solution x(t), y(t), z(t) to Eq. A.1 must satisfy lim sup [x (t) ! y (t) ! z (t)] " K1 ! K2 t→' Theorem (Ji-Fa 1991). If F " (F1, F2, F3): int!! → ! is a continuously differentiable function that satisfies the following conditions: 3 3 1) Cooperative.—The off-diagonal terms of the derivative matrix DF(x, y, z) are nonnegative for all points (x, y, z) ∈ int !3!. 2) Permanence.—There exist constants M % 0 and m % 0 such that given any solution (x(t), y(t), z(t)) ∈ int !3! to dx " F1 (x, y, z) dt dy " F2 (x, y, z) dt dz " F3 (x, y, z) dt there is a T % 0 such that m " x(t) " M, m " y(t) " M and m " z(t) " M for t ! T. 3) There is not more than one equilibrium in int !3!. Then there exists a unique equilibrium point (x*, y*, z*) ∈ int !3!, and it is globally asymptotically stable. where K1 equals the unique positive solution of f (K1) " 0 and where K2 is the unique positive solution of g(K2) " 0. To show that the system of Eq. A.1 is permanent, we verify the conditions of Theorem 2.21 in Hutson and Schmitt (1992). This requires some extra terminology that we do not define here but which is presented in detail in their excellent review article on permanence. Let * be the union of the x-y plane and the z-axis. Notice that * and !3!\* are forward invariant for the flow of Eq. A.1. Let +(*) denote the union of the omega limit sets for points in *. Claims 1 and 2 imply that +(*) is acyclic and consists only of isolated equilibria. The form of $ and our assumption that g(0) % b or f (0) % d1a/ (d2 ! a) imply that stable set of +(*) does not intersect !3!\*. Theorem 2.21 in Hutson and Schmitt (1992) implies that the system represented by Eq. A.1 is permanent. Claim 3 asserts that the system of Eq. A.1 has at most one equilibrium in int !3!. Applying the theorem of Ji-Fa (1991) completes the proof of Claim 4.