Download generalist feeding behaviors of aedes sierrensis larvae and their

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
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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. Schreiber by Western Washington University, and by the Division of Insect
Biology, University of California at Berkeley.
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JOSEPH N. S. EISENBERG ET AL.
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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.