Feeding mechanisms and possible resource partitioning of the caprellidae (Crustacea: Amphipoda) from Puget Sound,USA

Caprellid amphipods feed by browsing, filter-feeding, predation, scavenging, and scraping. Food acquisition is related to the presence or absence of plumose setae on the second antenna; those species with such setae obtain a significant amount of their diet from filtering and scraping periphyton while those species without such setae usually rely on predation. Two primary predators and 8 primary filter-feeders and scrapers were investigated. Substrates inhabited by the predators did not overlap, but some filter-feeders were found together. Feeding overlap has been decreased among filter-feeders that occur together, as they either feed on different sized particles or they filter at different heights from the substrate. Species filtering the same sized particles at the same heights that were found together utilized alternate feeding mechanisms, e.g. browsing on algal filaments.     

Particle size,flow velocity,and suspension-feeding by the erect bryozoansBugula neritina andB. stolonifera

Patterns of feeding by the erect bryozoansBugula neritina andB. stolonifera were studied by assessing ingestion rates of mixtures of polystyrene particles of three sizes present in equal densities at two ambient water-flow velocities. Particle size was found to influence feeding byB. neritina, while feeding byB. stolonifera was influenced by an interaction between particle size and flow velocity. Large and mediumsized particles were ingested at rates disproportionate to their numerical abundance, and small particles were always ingested in low numbers. Disproportionate feeding did not appear to be due to the greater likelihood of directly intercepting or of detecting particles of larger sizes, but may be explained by other size-dependent particle behaviors, active selection or rejection by the bryozoans, and/or the utilization of different feeding techniques. Comparison with a parallel study indicated that patterns of feeding on single-sized suspensions cannot be used to predict patterns of feeding on mixed suspensions. This is one of the few studies to test the combined influence of variation in both suspended particulate matter and in properties of the fluid medium. Such investigations will provide more realistic views of suspensionfeeding performance.     

Some relationships between size,food availability and energy balance in the ribbed musselAulacomya ater

The effects of body size and suspension density on filtration rates, assimilation efficiencies and respiration rates in the ribbed musselAulacomya ater (Molina) have been determined by means of short-term laboratory experiments. Filtration rates accelerate rapidly in response to increasing algal concentration up to approximately 10×10⁶ cellsDunaliella primolecta l^-1, beyond which a plateau is approached. Percentage increments are greatest in small individuals. Assimilation efficiencies are independent of body size, but decline rapidly with increasing ration to approach zero above 32×10⁶ cells l^-1. Increases in respiration rate accompany increments in filtration rate in all but the smallest size class tested. Filtration, assimilation efficiency and respiration measurements are used to calculate ingestion rations, assimilation rations and scope for growth for mussels of different sizes over a range of algal concentrations. Scope for growth, expressed as percentage change in body energy per day, is a declining function of body size, but larger individuals achieve their maximum growth rates at lower ration levels than smaller ones. Growth efficiency is independent of body size, and is maximal at 5×10⁶ cells l^-1, where 29 to 43% of ingested ration is converted into body energy. The applicability of these experimental results to natural ecosystems is discussed.     

Food selectivity of the rotifer Brachionus plicatilis feeding on phytoplankton

Food selectivity was examined in amictic female rotifers, Brachionus plicatilis, fed Chlamydomonas sp. and Olisthodiscus sp. Filtering and ingestion rates of the rotifers in a single food suspension were higher with Chlamydomonas sp. than with Olisthodiscus sp. In every mixture of two food species, the apparent filtering rate on Chlamydomonas sp. was much higher than that on Olisthodiscus sp. Total filtering rate and apparent filtering rate in food suspensions containing Chlamydomonas sp. decreased with increasing cell concentration of Chlamydomonas sp. Total and apparent filtering rates were calculated from the decrease in total food concentration as well as from decreases in concentrations of each of the two food algae, respectively. B. plicatilis ingested Olisthodiscus sp. at an extremely low constant rate in all mixtures. The degree of food selectivity of the rotifers fed Chlamydomonas sp. (i.e., selective filtration) from the mixture of two food algae decreased with increasing cell concentration of Chlamydomonas sp. Filtering and ingestion rates of rotifers fed senescent Chlamydomonas sp. were relatively lower than those fed Chlamydomonas sp. in the exponential phase. This indicates that Brachionus plicatilis displays selectivity in regard to condition of cells as well as type of food.     

On detritus as a food source for pelagic filter-feeders

Data on the amount of organic detritus within the particle size fraction 1 to 150 are presented for the Western Kiel Bight. Grouped into a mixed surface layer and a stratified lower layer in accordance with the main hydrographic features of the Western Baltic Sea, the rounded-off values show a seasonal variation between 100 and 600 mg m^-3 expressed as dry weight of organic matter. The overall average for both layers is about 200 mg m^-3. Organic detritus thus comprises more than 40% of total organic matter in the above size class, which is the class most easily accessible to the relatively small filter-feeders in this area. Attempting to trace the origin of organic detritus, a positive correlation to phytoplankton standing stock was found in some cases, suggesting the predominance of autochthonous detritus. Proceeding from the assumption that pelagic filter-feeders select their food mainly by size and not by taste, it is concluded that organic detritus plays an important role as a supplementary food source, being ingested together with phytoplankton and small nonmotile heterotrophs. The nutritive value of detritus is increased by the adsorption of dissolved organic matter and above all through the subsequent colonization by bacteria, which utilize the dissolved substances. Detritus particles serving as a substratum for bacteria thus form a means whereby dissolved organic substances reenter the food chain. The ingestion of detritus by filter-feeders is, therefore, thought to be instrumental in increasing the effectivity of energy transfer from the primary to the secondary food-chain level.     

Filtering rates of the juvenile Atlantic menhaden Brevoortia tyrannus (Pisces: Clupeidae), with consideration of the effects of detritus and swimming speed

From July to September 1982 feeding experiments were conducted with 138-mm fork length Atlantic menhaden Brevoortia tyrannus (Latrobe) (Pisces: Clupeidae) to determine their particle size-specific feeding abilities. Monoculture clearing-rate experiments showed that the minimum size of particles filtered, the minimum size threshold, for 138-mm fish is 7 to 9 m. Filtration efficiency for three species of phytoplankton below the minimum size threshold. Pseudoisochrysis paradoxa, Monochrysis lutheri, and Isochrysis galbana, averaged 1.0% (n=14). Tetraselmis suecica, Prorocentrum minimum, and 2-celled Skeleionema costatum, phytoplankton which are larger than the minimum size threshold and smaller than the 20-m upper limit for nanoplankton, were filtered at efficiencies averaging 21% (n=24). S. costatum chains of 3 to 6 cells, prey particles exceeding the size limits of nanoplankton, were filtered at average efficiencies ranging from 22 to 84%. The mean filtration efficiency for Artemia sp. nauplii (San Francisco Bay Brand) of 36% (n=7) was lower than for smaller phytoplankton prey. The presence of detritus at concentrations usually encountered in nature enhanced filtering efficiency and lowered minimum size thresholds at which phytoplankton were retained. For small food particles, filtering efficiency decreased as swimming speed of the menhaden increased. As menhaden grow, their feeding tepertoire shifts to larger planktonic organisms.Contribution No. 1201 Virginia Institute of Marine Science     

Significance of small-scale spatio-temporal heterogeneity in phytoplankton abundance for energy flow in Mytilus edulis

The effect of small-scale heterogeneity of phytoplankton abundance on mussel (Mytilus edulis L.) energy flow was studied by comparing filtration rate, consumption and assimilation efficiency in two groups of mussels fed under constant (control) and fluctuating (24 min) seston regimes of the same average. The experiment was carried out twice. Mussels appeared to regulate their filtration rate in response to varying seston concentration in one experiment, but not in the other. This behavioral difference may have been caused by differences in mean food level. Mean consumption was not different in the two feeding regimes, but assimilation efficiency was probably lower under fluctuating seston concentration. This suggests that small-scale heterogeneity in phytoplankton abundance would not affect the total transfer of energy between plankton and benthos. However, since a higher proportion of consumed energy was probably lost as faeces and/or pseudofaeces under the fluctuating seston regime, it may be expected that growth rate of mussels is depressed by small-scale heterogeneity in phytoplankton availability.Contribution to the programme GIROQ (Groupe interuniversitaire de recherches océanographiques du Québec)     

Enchytraeid population dynamics: Resource limitation and size-dependent mortality

Enchytraeids are regarded as keystone soil organisms in forest ecosystems. Their abundance and biomass fluctuate widely. Predicting the consequences of anthropogenic disturbances requires an understanding of the mechanisms underlying enchytraeid population dynamics. Here I develop a simple model, which predicts that the type of dynamics is controlled by resource input rate. If fungal resource input is a discrete event once a year, an exponential growth phase is followed by starvation and sharp decline of enchytraeid abundance. Model simulations with three different forcing functions were compared to field data. Initial parameter values were obtained from various independent sources, and parameters were estimated by minimizing the residual sum of squares. The best fitting model with resource addition once a year explained 39% of the variation in enchytraeid biomass over an 8-year study period. Further, variation in rainfall explained 59% of the variation in R² of the exponential phase models, which is also an index of the stability of population size-structure. The results emphasize the importance of resource limitation for enchytraeid population dynamics and support the hypothesis that the mortality during the decline phase is size-dependent.     

Selective particle ingestion by oyster larvae (Crassostrea virginica) feeding on natural seston and cultured algae

I investigated selective particle ingestion by oyster larvae (Crassostrea virginica) feeding on natural seston from Chesapeake Bay and laboratory-cultured algae of different sizes or chemical content. In 15 of 16 experiments with complex natural suspensions as food, small(<150 m) and large (>150 m) larvae selected most strongly for small (2 to 4 m) food particles, but in the presence of a large (>10 m)-cell dinoflagellate bloom, large larvae strongly selected much larger (22 to 30 m) food material (presumably dinoflagellates). When fed simplified mixtures of four cultured algal species (Synechococcus bacillaris, Isochrysis sp., Dunaliella tertiolecta, and Prorocentrum minimum) ranging in size from 1 to 11 m, small larvae preferred 1 m algae while large larvae preferred 11 m algae. In experiments with algal mixtures, and with suspensions of natural particles and added algae, large larvae preferred algal species harvested from exponential-phase cultures over other species from stationary-phase cultures. Larval ingestion rates of the cultured alga Thalassiosira pseudonana were about three times higher for cells with a low carbon:nitrogen ratio (7.2:1) than for high C:N ratio (16.2:1) cells when these cells were offered separately in suspensions of equal concentration. As a result, more algal cells, algal C, and algal N was ingested by larvae fed low C:N cells. However, larvae did not show a significant preference for either type of cell when they were offered in a 1:1 cell mixture. Feeding patterns of C. virginica larvae in natural food suspensions can vary with the composition of these complex suspensions, and ingestion seems dependent not only on the size, but on the growth rate and chemical quality of food particles.     

Influence of flow speed on the functional response of a passive suspension feeder,the spionid polychaete <Emphasis Type="Italic">Polydora cornuta</Emphasis>

Passive suspension feeders rely on surrounding flow to deliver food particles to them. Therefore, the classic conception of functional response (feeding rate vs. food concentration) may require modification to account for flow speed as a second independent variable. I compared the functional response of Polydora cornuta at different velocities and determined whether food capture was proportional to particle flux (concentration × velocity). To understand feeding responses at a mechanistic level, I measured the functional responses in terms of contact and capture rates and determined particle retention efficiency. Experiments were run separately with two sizes of food particles, and with juvenile or adult worms. For both worm sizes and both particle sizes, capture rate in weak flow was directly related to concentration, but in strong flow it was constant. Worms were therefore unable to benefit from abundant food when in strong flow. The critical velocity at which the capture rate became constant was lower for adult worms than for juvenile worms, and it was lower for small particles than for large particles. Retention efficiency was constant among all treatments, and the results for contact rate were essentially the same as for capture rate. Therefore, the mechanics of particle contact must explain the effects of velocity on the functional response. Contact rate was not a constant proportion of particle flux; treatments with similar fluxes yielded different contact rates depending on the strength of flow. The results appeared to be caused by a velocity-induced behavioral change in appendage posture that affects contact rates: in moderate flow, worms form their feeding palps into helical coils, which they tighten as the velocity increases. I suggest this behavior constrains suspension feeding rates and the mechanical selection between particle sizes when worms are in strong flow, and that the effect changes with ontogeny. Because the results are consistent with patterns in measured growth rates of P. cornuta, I hypothesize that this influence of velocity on the functional response can constrain growth and population dynamics in this species.     

An individual based model of bearded pig abundance

We develop an individual based model of bearded pig abundance which predicts population dynamics based on the processes of energy accumulation and expenditure, reproduction and mortality of individual pigs. Because fatness is a key indicator of condition and reproductive potential in bearded pigs, processes are represented in terms of a fatness index variable. Only a small number of parameters are used in this simple model; these were chosen on the basis of fatness index data and qualitative observations of bearded pig population dynamics. The model was found to be accurate in predicting the timing of observed pig eruptions, and robust in that model results were unaffected by moderate variation in parameter values. There was insufficient quantitative data to obtain precise predictions of fatness and abundance, but qualitative insights about the effects of the size and timing of fruiting events on pig abundance were obtained. The results showed that a single fruiting peak will not produce a bearded pig eruption, no matter how large the fruiting event is, because the duration of the fruiting is too short to allow exponential growth of the population. Consecutive masting events are necessary for an eruption, because if events are separated by more than 1 year, the population will decline to its minimum fatness and abundance levels. It is also necessary for at least one of the fruiting events in a consecutive sequence to be a large event, as consecutive small fruiting events do not increase the fatness enough to cause an eruption. These insights help to explain and predict the effects of changes in mast fruiting patterns on bearded pig populations, such as the predicted increase in frequency and reduction in size of masting fruiting events as a result of climate change.     

Zooplankton feeding and queueing theory

Three of the most commonly used mathematical relationships between zooplankton ingestion and food concentration are derived from a queue model, by assuming different feeding strategies of the individual predators or grazers. The model suggests that the abruptness of the transition between maximum search or filtration rate and maximum ingestion rate depends on the size ratio between the animals and their food items. The results show that the stochastic nature of predation and grazing cannot be neglected when the food particles are relatively big.     

Influence of marine microalgal metabolites on the feeding behavior of the blue mussel Mytilus edulis

Effects of microalgal ectocrines on the feeding behavior of the blue mussel Mytilus edulis L., collected from Delaware Bay, Lewes, USA, were studied. Unialgal cultures of six species of marine microalgae were grown to the end of their exponential growth phase. Cells were removed by gentle filtration, leaving dissolved microalgal ectocrines in culture filtrates. Filtrates were then tested in filtration rate and particle selection bioassays. In filtration rate bioassays, dissolved ectocrines were delivered to mussels in a flow-through apparatus and removal of beads (6 m) used to determine rates. Filtration rates were significantly reduced by dissolved ectocrines from two of the microalgal species tested (Olisthodiscus luteus Carter and Dunaliella tertiolecta Butcher). None of the microalgal filtrates stimulated filtration rates of mussels. In particle selection bioassays, ectocrines were adsorbed onto either reverse phase (lipophilic) or normal phase (hydrophilic) microparticles (10 m) and delivered to mussels with an equal concentration of control treated particles. Mussels showed significant selection, either rejection or preferential ingestion, for ectocrine treated spheres depending upon microalgal species and sphere type used. This research shows that pre-ingestive chemical cues from microalgae can influence mussel feeding behavior. We suggest that epicellular ectocrines are more important in mediating this behavior than dissolved ectocrines.     

Effect of low salinity on the activity,feeding, growth and absorption efficiency of Luidia clathrata (Echinodermata: Asteroidea)

Exposure of Luidia clathrata (Say) to low salinity (17 S) results in a decrease in activity and rate of feeding, but has little effect on the absorption efficiency. Growth, as indicated by changes in body size and energy content of the body wall and pyloric caeca, is reduced in low salinity due to a decrease in feeding rate (energy consumption) and to a decrease in efficiency of utilization of material and energy absorbed. The low efficiency indicates that either additional energy is required for maintenance or energy is used inefficiently in low salinity. Thus, L. clathrata is limited energetically in low salinities, although it lives and reproduces there.     

Influence of algal concentration and temperature on the filtration rate of Mytilus edulis

The filtration rates of Mytilus edilis (=galloprovincialis; 40 mm) were determined in relation to food concentration and temperature, using pure suspensions of the unicellular alga Platymonas suecica in concentrations ranging from 3x10⁵ cells/l to 1.5x10⁸ cells/l. The rate of filtration (ml/h/mussel) generally decreased as cell concentrations increased, and dropped to low values when concentrations above 5x10⁷ cells/l were supplied. The amount of water swept clear varied continuously, and noticeable differences in the filtration activity of M. edulis were observed over short time intervals (5 min). Fluctuations of filtered volumes per unit time were greater with lower than with higher concentrations of algae. The influence of temperature on filtration activity was highest between 5°–15°C and 25°–30°C. A temperature increase from 15° to 25°C resulted in only a slight increase in filtration rate. At 5° and 30°C, filtration dropped to very low values, namely 350 and 100 ml/h, respectively. The temperature coefficients for the filtration rates of M. edulis were determined as: Q₁₀ (5° to 15°C)=4.96; Q₁₀ (10° to 20°C)=1.22. The amount of algae cells ingested per mussel per hour is directly related to food concentration. The maximum number of cells filtered/mussel/h in an algal suspension of 70x10⁶ cells/l was 21.5x10⁵ cells/h. Cell concentrations of up to 40x10⁶ cells/l were swept clear without producing pseudofaeces. The critical cell density for M. edulis was reached at algal concentrations of 70 to 80x10⁶ cells/l. Above these concentrations no normal filtration activity was observed.     

Leaf-cutter ants with worn mandibles cut half as fast,spend twice the energy,and tend to carry instead of cut

The importance of mechanical wear in the behavioral ecology and energetics of small organisms is an open question. We investigated wear in leaf-cutter ants, Atta cephalotes, because their cutting technique can be imitated and the leaves are the main energy source for the colony. We found that a razor-sharp (50-nm radius) “V-blade” that cuts leaves between the first and second mandibular teeth was dulled (∼10-μm radius) and often nearly worn away on foragers. We found that the force required to cut standard leaves, using mandibles removed from foragers cutting in the wild, varied by a factor of 2.5 with tooth wear, defined as the difference between pupal and actual tooth length. We also found that wear significantly reduced the cutting rate. From the distribution of wear among the cutting foragers, we estimated that the wild colony would have spent 44% less of both energy and time making the observed cuts if the cutters’ mandibles had all been pristine. Finally, wear correlated with behavioral differences—foragers with the most worn 10% of mandibles almost exclusively carried rather than cut. This previously unreported form of task partitioning suggests that eusociality may extend useful lifespans by making it possible to switch tasks as skills decline. We developed a model, assuming that ants do work at a constant rate proportional to their mass, to predict the cutting rate from head width, tooth wear, and force to cut leaves with a scalpel (R = 0.62), and we used this estimate to argue that the partitioning of cutting and carrying was sub-optimal but better than random. Wear’s strong effect on performance may promote wear-avoiding behavior and wear-resistant mandible composition; it may affect leaf selection and worker lifespan and it raises the possibility that wear is a similarly important constraint for many other small organisms.     

Modeling leaf dispersal in mixed hardwood forests using a ballistic approach

In mixed-species stands, modeling leaf litter dispersal is important to predict the physical and chemical characteristics of the forest floor, which plays a major role in nutrient cycling and in plant population dynamics. In this study, a spatially explicit model of leaf litterfall was developed and compared with two other models. These three models were calibrated for a mixed forest of oak and beech using litterfall data from mapped forest plots. All models assumed that an allometric equation described individual leaf litter production, but they strongly differed in the modeling of the probability density of leaf shedding with distance from source trees. Two models used a negative exponential function to account for leaf dispersal with distance, and this function was allowed to vary according to wind direction in one of them. In contrast, our approach was based on a simple ballistic equation considering release height, wind speed, wind direction, and leaf fall velocity; the distributions of wind speeds and wind directions were modeled according to a Weibull and a Von Mises distribution, respectively. Using an independent validation data set, all three models provided predictions well correlated to measurements (r > 0.83); however, the two models with a direction-dependent component were slightly more accurate. In addition, parameter estimates of the ballistic model were in close agreement with a foliar litter production equation derived from the literature for beech and with wind characteristics measured during leaf litterfall for both species. Because of its mechanistic background, such a spatially explicit model might be incorporated as a litterfall module in larger models (nutrient cycling, plant population dynamics) or used to determine the manner in which patch size in mixed-species stands influences litter mixture.     

House formation and feeding behaviour of<Emphasis Type="Italic"> Fritillaria borealis</Emphasis> (Appendicularia: Tunicata)

Numerous marine animals use elaborate filtration mechanisms to feed on particles suspended in the sea. Video-microscopy reveals that the common appendicularian Fritillaria borealis do so in a unique way. They live inside acellular, mucous houses, where their tail undulations act like a peristaltic pump in a close-fitting tail chamber to force water through a complex set of filters. Suspended particles, between ~0.2 and 30 µm in diameter, are retained in this process and propagated towards the mouth of the organism to be ingested. Coarser particles are back-washed out of the house during brief intervals of tail arrest, when the elastic recoil of the house walls reduces its volume to one-eighth of its inflated state. At the same time, the bifurcated tip of the animal's tail slides into a sleeve-like extension of the house. This sleeve acts like a flipper to propel the animal and collapsed house several millimetres to a few centimetres through the sea as soon as muscular activity of the tail is resumed. After a second tail arrest, a special muscular twist brings the tail out of the sleeve, whereupon undulatory pumping movements are resumed to reinflate and process particle-laden water through the house. The complete behavioural cycle lasts ~30 s. The repetitive inflation and deflation cycles of the house are probably of importance for unclogging of the coarse "inlet" filters downstream to the tail pump, as well as for agglutination of the smallest food particles trapped in the house to ease their recapture in the endostylar pharyngeal filter during the ingestion process. The unique tail sleeve and flipper at the same time enable the animal to move away from the previously filtered water and rejected coarse particles, before the house is reinflated for resumed food filtration.     

排水管道内不同粒径沉积颗粒物冲刷率的分析与计算

为探究和量化在水流冲刷下,排水管道中不同粒径颗粒物冲刷沉积的过程,本文模拟排水管道内沉积颗粒的冲刷过程.冲刷过程中,粒径较小的悬移质颗粒(小于0.1 mm),在管道沿线取样测得水流中悬浮固体质量浓度(SS);粒径较大的推移质颗粒(0.1—2 mm),测得管道沿线不同位置沉积的颗粒质量.并建立两个数学模型分别计算排水管道内两类颗粒物的冲刷率.研究发现,悬移质颗粒,以水流中SS为依据,可计算出输送通量和冲刷率;推移质颗粒,以管道不同位置沉积下来的颗粒质量为依据,拟合出了管道中的沉积分布函数,是以e为底数的指数函数,可计算出冲刷量和冲刷率.模型计算出的冲刷率呈现出规律性:悬移质颗粒被冲刷成悬浮状态,随水流迁移过程中部分会再次沉积,使得悬移质颗粒的冲刷率从管道前段至中后段逐步降低,如初始沉积质量为100 g的0.045 mm悬移质颗粒,在0.30 m·s^-1的冲刷流速下,计算出管道前段冲刷率为78.94%,最终在管道后段降至13.89%;对于两类颗粒而言,颗粒物粒径越小,冲刷流速越大,初始沉积质量越小,冲刷率越高.     

Suspension feeding in the brittle-star Ophiothrix fragilis : efficiency of particle retention and implications for the use of encounter-rate models

Dense beds of the suspension-feeding brittle-star Ophiothrix fragilis are common in European waters. Their potential importance in benthic–pelagic coupling has been highlighted, but little is known about the feeding dynamics of this species. Encounter-rate models provide a potential mechanism for the estimation of feeding rates on suspended material of varying sizes. This work investigates factors essential to the application of such models. Particle-retention efficiency (RE) converts encounter rate into capture, or clearance rate. Laboratory studies demonstrated that RE varied with the interactive effects of flow velocity and particle size. RE was lowest for large particles, particularly at high flow velocity where RE as low as 59% was observed. This indicates that if RE is not accounted for in encounter-rate models, significant overestimates of feeding rates on large particles may occur. Flow around feeding arms and tube feet was characterised by intermediate Reynolds numbers, precluding application of the most simple encounter-rate models. Complex secondary-flow patterns were observed, which carried particles along the downstream side of the feeding arms, but these did not appear to increase the area available for particle capture. Previously reported particle capture by arm spines was not observed. Evidence of active rejection of large particles by tube feet was recorded. Difficulties in the application of encounter-rate models for prediction of seston-removal rates are highlighted by these results. Predicted encounter rate may deviate from actual clearance rate due to the effects of retention dynamics, localised flow patterns and differential particle handling. Other methods of estimation of seston-removal rates are equally problematic however, so that encounter-rate models are likely to remain a useful tool for such estimates. Received: 23 January 1998 / Accepted: 24 June 1998