Restricted adaptive cluster sampling

Adaptive cluster sampling can be a useful design for sampling rare and patchy populations. With this design the initial sample size is fixed but the size of the final sample (and total sampling effort) cannot be predicted prior to sampling. For some populations the final sample size can be quite variable depending on the level of patchiness. Restricted adaptive cluster sampling is a proposed modification where a limit is placed on the sample size prior to sampling and quadrats are selected sequentially for the initial sample size. As a result there is less variation in the final sample size and the total sampling effort can be predicted with some certainty, which is impor- tant for many ecological studies. Estimates of density are biased with the restricted design but under some circumstances the bias can be estimated well by bootstrapping. © Rapid Science 1998     

Adaptive two-stage one-per-stratum sampling

We briefly describe adaptive cluster sampling designs in which the initial sample is taken according to a Markov chain one-per-stratum design (Breidt, 1995) and one or more secondary samples are taken within strata if units in the initial sample satisfy a given condition C. An empirical study of the behavior of the estimation procedure is conducted for three small artificial populations for which adaptive sampling is appropriate. The specific sampling strategy used in the empirical study was a single random-start systematic sample with predefined systematic samples within strata when the initially sampled unit in that stratum satisfies C. The bias of the Horvitz-Thompson estimator for this design is usually very small when adaptive sampling is conducted in a population for which it is suited. In addition, we compare the behavior of several alternative estimators of the standard error of the Horvitz-Thompson estimator of the population total. The best estimator of the standard error is population-dependent but it is not unreasonable to use the Horvitz-Thompson estimator of the variance. Unfortunately, the distribution of the estimator is highly skewed hence the usual approach of constructing confidence intervals assuming normality cannot be used here.     

Adaptive host choice and avoidance of superparasitism in the spawning decisions of bitterling (Rhodeus sericeus)

Choice of a site for oviposition can have fitness consequences. We investigated the consequences of female oviposition decisions for offspring survival using the bitterling, Rhodeus sericeus, a freshwater fish that spawns inside living unionid mussels. A field survey of nine bitterling populations in the Czech Republic revealed a significantly lower rate of release of juvenile bitterling from Anodonta cygnea compared to three other mussel species. A field experiment demonstrated that female bitterling show highly significant preferences for spawning in A. anatina, Unio pictorum, and U. tumidus. Within a species, female bitterling avoided mussels containing high numbers of bitterling embryos. Mortality rates of bitterling embryos in mussels were strongly density dependent and the strength of density dependence varied significantly among mussel species. Female preferences for mussels matched survival rates of embryos within mussels and females distributed their eggs among mussels such that embryo mortalities conformed to the predictions of an ideal free distribution model. Thus, female oviposition choice is adaptive and minimizes individual embryo mortality. Received: 6 October 1999 / Received in revised form: 7 January 2000 / Accepted: 13 March 2000     

Estimator bias and efficiency for adaptive cluster sampling with order statistics and a stopping rule

Practical problems facing adaptive cluster sampling with order statistics (acsord) are explored using Monte Carlo simulation for three simulated fish populations and two known waterfowl populations. First, properties of an unbiased Hansen-Hurwitz (HH) estimator and a biased alternative Horvitz-Thompson (HT) estimator are evaluated. An increase in the level of population aggregation or the initial sample size increases the efficiencies of the two acsord estimators. For less aggregated fish populations, the efficiencies decrease as the order statistic parameter r (the number of units about which adaptive sampling is carried out) increases; for the highly aggregated fish and waterfowl populations, they increase with r. Acsord is almost always more efficient than simple random sampling for the highly aggregated populations. Positive bias is observed for the HT estimator, with the maximum bias usually occurring at small values of r. Secondly, a stopping rule at the Sth iteration of adaptive sampling beyond the initial sampling unit was applied to the acsord design to limit the otherwise open-ended sampling effort. The stopping rule induces relatively high positive bias to the HH estimator if the level of the population aggregation is high, the stopping level S is small, and r is large. The bias of HT is not very sensitive to the stopping rule and its bias is often reduced by the stopping rule at smaller values of r. For more aggregated populations, the stopping rule often reduces the efficiencies of the estimators compared to the non-stopping-rule scheme, but acsord still remains more efficient than simple random sampling. Despite its bias and lack of theoretical grounding, the HT estimator is usually more efficient than the HH estimator. In the stopping rule case, the HT estimator is preferable, because its bias is less sensitive to the stopping level.     

Network inclusion probabilities and Horvitz-Thompson estimation for adaptive simple Latin square sampling

Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to determine which units to sample within the study region. Adaptive cluster sampling designs Thompson (1990) are sampling designs that are implemented by first selecting a sample of units according to some conventional probability sampling design. Then, whenever a specified criterion is satisfied upon measuring the variable of interest, additional units are adaptively sampled in neighborhoods of those units satisfying the criterion. The success of these adaptive designs depends on the probabilities of finding the rare clustered events, called networks. This research uses combinatorial generating functions to calculate network inclusion probabilities associated with a simple Latin square sample. It will be shown that, in general, adaptive simple Latin square sampling when compared to adaptive simple random sampling will (i) yield higher network inclusion probabilities and (ii) provide Horvitz-Thompson estimators with smaller variability.     

Performance of adaptive cluster sampling estimators in a multivariate setting

Adaptive cluster sampling has been proven to perform well in a univariate setting, but it may not perform well when there are several parameters of interest. The efficiency of adaptive sampling when there are several variables of interest depends on the relationship of the variables with one another.     

Operational sex ratio and alternative reproductive behaviours in the European bitterling,<Emphasis Type="Italic"> Rhodeus sericeus</Emphasis>

We investigated the effects of male population density and male-biased operational sex ratio (OSR) with constant and limited resource density on male mating tactics shown by a freshwater fish, the European bitterling, Rhodeus sericeus. This species spawns inside living unionid mussels. Large males defended territories and were aggressive towards conspecifics under equal sex ratios. They also monopolised pair spawnings with females, releasing 98% of all sperm clouds during mating. However, the mating tactic changed at high male density where large males ceased to be territorial and instead competed with groups of smaller males to release sperm when females spawned. Large, medium and small males now obtained 61%, 33%, and 6% of sperm releases respectively, thereby reducing the opportunity for sexual selection by half. Females spawned at equal rates in the two densities of males, despite lower courtship at high density. These results run counter to the usual expectation that an increasingly male-biased OSR should lead to higher variance in male mating success. Instead, the use of alternative reproductive behaviours by males can lead to lower resource competition and mating variance at high male densities.     

Distribution of a marginal population of <Emphasis Type="Italic">Mytilus edulis</Emphasis>: responses to biotic and abiotic processes at different spatial scales

Physical and biological processes interact to produce pattern in nature. Pattern is scale dependent as processes generating pattern are heterogeneous in time and space. We tested some causes of variation in abundance and distribution of three marginal populations of sublittoral blue mussels, Mytilus edulis, in the non-tidal northeastern Baltic Sea. We studied the role of substrate inclination, perennial algae and siltation along local wave exposure gradients on mussel distribution over a regional salinity gradient. We found marked differences on regional scales (p < 0.001) with lower densities and biomasses of mussels with declining salinity. Along local gradients, mussel densities increased with increasing exposure (p < 0.001) and declining slope and sedimentation (p < 0.01). Site specifically, densities of blue mussels and the perennial red algae, Furcellaria lumbricalis, were positively related, results supported by a colonisation experiment. Also, young post-recruits showed significant relations to adult biomass, wave exposure, algal biomass, bottom slope and sediment cover. Findings showed that the relative importance of the determinants affecting blue mussels at the edge of their range vary with scale and are affected by the density and size structure of mussel populations. The study provides an indication of the types of factors that may be invoked as causes of spatial variation in marginal blue mussel populations and reinforces the need to consider multiple aspects when distributional patterns are assessed.     

Adjusted two-stage adaptive cluster sampling

An adjusted two-stage sampling procedure is discussed for adaptive cluster sampling where some networks are large and others are small. A two-stage sample is drawn from the large networks and a single-stage sample is drawn from the rest. The simple random sampling (SRS) procedure without replacement is used at the initial stage. An estimator for the population mean along with its properties is discussed.     

The common mussel Mytilus edulis as an indicator of pollution by zinc,cadmium, lead and copper. I. Effects of environmental variables on uptake of metals

The net uptake of zinc, cadmium, lead and copper by the common mussel Mytilus edulis (L.) exposed to different conditions was investigated with a view to using this species as an indicator of contamination of the marine environment by these metals. The variables studied were season, position of the mussel in the water column, water salinity, water temperature, and the simultaneous presence of all four metals. Each of these 5 variables affected the net uptake of some or all of the metals studied under some conditions. Seasonal variation in concentrations of zinc, cadmium and copper was found in samples collected at three separate locations. The relationship of seasonal variation to tissue weight and absorption route of the metals is discussed. Near to freshwater inputs of trace metals, the concentrations of zinc, cadmium and lead in mussels were found to vary according to the depth at which the mussels were collected; in summer when freshwater run-off is less, this effect was absent. Low salinities did not affect the net uptake of zinc by mussels, but increased the net uptake of cadmium and decreased that of lead. Low temperatures had no effect on the net uptake of zinc or lead; the net uptake of cadmium was unaffected by low temperatures at high salinities but was decreased by low temperatures at low salinities. The presence of the other metals had no effect on the individual net uptake of either zinc, cadmium or lead. A sampling programme was devised to eliminate the effects of these environmental variables and to allow the use of M. edulis as an indicator of zinc, cadmium and lead in marine and estuarine environments. In contrast to the other metals, the net uptake of copper by the mussel was extremely erratic, and was affected by salinity and temperature changes and by the presence of the other metals and changes in their relative concentrations. The effects of other metals on the net uptake of copper cannot be easily eliminated or allowed for; it is, therefore, suggested that the mussel should not be used as an indicator of copper in the marine environment.     

Male-male interference competition decreases spawning rate in the European bitterling (<Emphasis Type="Italic">Rhodeus sericeus</Emphasis>)

We investigated the consequences of male-male interference competition associated with alternative male mating tactics in a freshwater fish, the European bitterling (Rhodeus sericeus). Male bitterling defend territories around living mussels and attract females to lay their eggs in the gill cavities of mussels. We experimentally manipulated spawning-site abundance and male density at two spatial scales. We showed that the total number of eggs spawned by females was constrained by the number of mussels available for oviposition. The effect was mediated by behavioral interactions among competing males because of variation in the Operational Sex Ratio (OSR) in close proximity to a mussel and not by a direct limitation in mussel capacity to accommodate the eggs. Both total and local male densities affected spawning behavior, and interacted in their effect on female spawning rate. Territorial male aggression caused courtship interruptions that prolonged the time until successful spawning and increased with male density. However, territoriality broke down at the highest male density, with a consequent stabilizing effect on spawning rate.Communicated by J. Krause     

Adaptive cluster sampling with clusters selected without replacement and stopping rule

Adaptive cluster sampling (ACS) has received much attention in recent years since it yields more precise estimates than conventional sampling designs when applied to rare and clustered populations. These results, however, are impacted by the availability of some prior knowledge about the spatial distribution and the absolute abundance of the population under study. This prior information helps the researcher to select a suitable critical value that triggers the adaptive search, the neighborhood definition and the initial sample size. A bad setting of the ACS design would worsen the performance of the adaptive estimators. In particular, one of the greatest weaknesses in ACS is the inability to control the final sampling effort if, for example, the critical value is set too low. To overcome this drawback one can introduce ACS with clusters selected without replacement where one can fix in advance the number of distinct clusters to be selected or ACS with a stopping rule which stops the adaptive sampling when a predetermined sample size limit is reached or when a given stopping rule is verified. However, the stopping rule breaks down the theoretical basis for the unbiasedness of the ACS estimators introducing an unknown amount of bias in the estimates. The current study improves the performance of ACS when applied to patchy and clustered but not rare populations and/or less clustered populations. This is done by combining the stopping rule with ACS without replacement of clusters so as to further limit the sampling effort in form of traveling expenses by avoiding repeat observations and by reducing the final sample size. The performance of the proposed design is investigated using simulated and real data.     

Allometric Scaling of Minimal Mammal Densities

Minimum viable densities have rarely been determined directly. Theoretical analyses, based on empirical relationships between average mammal densities, suggest that minimum densities of viable populations are lower for larger mammals. This suggestion has been cast into doubt by other field studies showing populations of small insects and birds at very low densities. We collected 143 of the closest approximations of minimum viable density available, those of minimal, rare, and endangered mammal populations. We found that minimal density decreases as the −0.68 power of body mass. Minimal densities of small mammals are 1000 times those of the largest species. The correlation between minimum viable population density and body mass is negative in the majority of the mammalian taxonomic orders. Although minimum density is, on average, 10% of mean population density, viable population densities of herbivores are 13 times those of carnivores and insectivores. Populations in the wet tropics can apparently sustain themselves at densities much lower than those in temperate climates.     

The application of adaptive cluster sampling for rare subtidal macroalgae

Adaptive cluster sampling (ACS) is a targeting sampling method that provides unbiased abundance estimators for populations of rare species that may be inadequately sampled with simple random sampling (SRS). ACS has been used successfully to estimate abundances of rockfish and sardine larvae from shipboard surveys. In this study, we describe the application of ACS for subtidal macroalgae. Using SCUBA, we measured abundances of Codium mamillosum, C. pomoides, and Halimeda cuneata at three islands and two levels of wave exposure. The three species were relatively patchy and could be sampled with ACS at one site per dive. Their distributions differed among islands and with exposure to wave energy, with H. cuneata found at only one island. ACS is a useful tool for understanding the spatial distribution and abundance of populations of rare benthic species, but, as was the case in this study, may not be as efficient as sampling with SRS with comparable replication.     

Interactive effects of wave exposure and tidal height on population structure of the mussel Perna perna

The influence of wave exposure and of tidal height on mussel (Perna perna Linnaeus) population structure (size, density, biomass and adult/juvenile correlations) was examined at 18 sites along the south coast of South Africa. Sites were classified as exposed or sheltered prior to sampling, without reference to the biota, on the basis of aspect, topography and wave regime. A single set of samples was collected from each site during three spring tide cycles. Adult mussels on these shores almost always attach directly to the rocks, and layering of mussels is virtually absent. Shore height always had a strong influence on population structure, but exposure had significant effects only lower on the shore, and almost exclusively on mussel sizes. Principal component analysis (PCA), based on size distribution data for each population, revealed a general upshore decrease in the modal size of the adult cohort. The effects of exposure on size distribution, however, varied with tidal height. PCA separated exposed zones, with larger mussels, from sheltered zones on the low-shore. Farther upshore the two shore types were increasingly confounded. The maximum size of mussels showed a similar pattern, with significant differences (ANOVA, p < 0.05) between exposed and sheltered sites only on the low- and mid-shores. Density was calculated from randomly placed quadrats (i.e. not necessarily from areas of 100% cover) and showed a different pattern. Adult (>15 mm) densities decreased up the shore, with low-, mid- and high-shore zones being significantly different from one another (ANOVA, p < 0.0001; followed by multiple range tests). However, exposure had no significant effect on density, nor was there a significant interaction with zone. Recruit (<15 mm) densities were positively correlated with adult (>15 mm) densities in all zones and for both exposure regimes ( p < 0.05 in all cases), but there was considerable variability and extremely low predictability in these relationships (r ² generally <0.2). Predictability tended to be greater towards the high-shore, where adults were more clumped. As with density, biomass was not affected by exposure, but decreased upshore as mean size and density decreased. A reduction in the influence of exposure farther upshore may be caused by greater emersion overriding the effects of exposure. The presence of free space within mussel beds and significant correlations between recruit and adult densities suggest that these mussel populations are recruit limited. Received: 7 January 2000 / Accepted: 6 July 2000     

The interplay between environmental conditions and allee effects during the recovery of stressed zooplankton communities

Many important ecological phenomena depend on the success or failure of small introduced populations. Several factors are thought to influence the fate of small populations, including resource and habitat availability, dispersal levels, interspecific interactions, mate limitation, and demographic stochasticity. Recent field studies suggest that Allee effects resulting from mate limitation can prevent the reestablishment of sexual zooplankton species following a disturbance. In this study, we explore the interplay between Allee effects and local environmental conditions in determining the population growth and establishment of two acid-sensitive zooplankton species that have been impacted by regional anthropogenic acidification. We conducted a factorial design field experiment to test the impact of pH and initial organism densities on the per capita population growth (r) of the sexual copepod Epischura lacustris and the seasonally parthenogenetic cladoceran Daphnia mendotae. In addition, we conducted computer simulations using r values obtained from our experiments to determine the probability of extinction for small populations of acid-sensitive colonists that are in the process of colonizing recovering lakes. The results of our field experiment demonstrated that local environmental conditions can moderate the impacts of Allee effects for E. lacustris: Populations introduced at low densities had a significantly lower r at pH 6 than at pH 7. In contrast, r did not differ between pH 6 and 7 environments when E. lacustris populations were introduced at high densities. D. mendotae was affected by pH levels, but not by initial organism densities. Results from our population growth simulations indicated that E. lacustris populations introduced at low densities to pH 6 conditions had a higher probability of extinction than those introduced at low densities to a pH 7 environment. Our study indicates that environmental conditions and mate limitation can interact to determine the fate of small populations of sexually reproducing zooplankton species. If a more rapid recovery of acid-damaged zooplankton communities is desired, augmentation of dispersal levels may be needed during the early phases of pH recovery in order to increase the probability of establishment for mate-limited zooplankton species.     

A review of adaptive cluster sampling: 1990–2003

Adaptive cluster sampling (ACS) is an adaptive sampling scheme which operates under the rule that when the observed value of an initially selected sampling unit satisfies some condition of interest, C, other additional units in some pre-defined accompanying neighborhood are also added to the sample. In turn, if any of these additional units satisfy C, then their corresponding unit neighborhoods are added to the sample as well, and so on. This process stops when no additional units satisfying C are encountered. This paper will provide a review of the major developments and issues in ACS since its introduction by Thompson (1990) [Journal of the American Statistical Association, 85, 1050–1059].     

Regulation of intertidal food webs by avian predators on New England rocky shores

Although there is a large body of research on food webs in rocky intertidal communities, most of the emphasis has been on the marine benthic components. Effects of avian predation on highly mobile predators such as crabs, remains practically unstudied in rocky shore ecosystems. The crab, Cancer borealis, is an important component of the diet of gulls (Larus marinus, L. argentatus) at the Isles of Shoals, Maine, USA. C. borealis prey include the predatory gastropod Nucella lapillus L., the herbivore Littorina littorea, and mussels Mytilus edulis L. We hypothesized that gulls reduce abundance of C. borealis in the low intertidal and shallow subtidal, thereby allowing C. borealis prey to persist in high numbers. A study of crab tidal migration showed that C. borealis density nearly doubled at high tide compared to low tide; thus, crabs from a large subtidal source population migrate into the intertidal zone during high tides and either emigrate or are removed by gulls during low tides. Results from a small-scale (1 m2) predator caging experiment in the low intertidal zone indicated that enclosed crabs significantly reduced L. littorea abundance when protected from gull predation. In a much larger-scale gull exclusion experiment, densities of C. borealis increased significantly during low and high tides in exclosures relative to the controls. C. borealis density was inversely correlated with changes in the abundance of two mesopredators Carcinus maenas and Nucella lapillus, and with the space-occupier M. edulis. There was a similar negative correlation between abundance of C. borealis and the change in abundance of the herbivore L. littorea, but the trend was not significant. Mortality of tethered L. littorea was associated with C. borealis density across sites. However, preferred algae did not change in response to L. littorea density during the experiment. Thus, we found suggestive, but not conclusive, evidence for a three-level cascade involving gulls, crabs, and L. littorea. Our studies strongly suggest that gulls, as apex predators, generate three-level trophic cascades in rocky intertidal food webs by preventing the highly mobile subtidal predator, C. borealis, from establishing substantial populations in the low-mid intertidal zone thereby indirectly enhancing densities of two key mesopredators (N. lapillus, Carcinus) and blue mussels (M. edulis).     

Shift in nutrient and plankton community in eutrophic lake following introduction of a freshwater bivalve

The impact of the freshwater bivalve Corbicula leana on plankton community dynamics was examined during a cyanobacterial bloom period. Nutrient and chlorophyll concentrations, primary productivity, and phytoplankton and zooplankton communities in the experimental enclosures were measured at 2-3 day intervals. The introduction of mussels reduced net primary productivity and phytoplankton and chlorophyll. Chlorophyll decreased immediately following addition of 100 mussels and then increased over time. After 600 mussels were added, chlorophyll decreased continuously from 87to 25 microg l(-1), approaching that in the mussel-free enclosure. Simultaneously, water transparency increased and concentrations of suspended solids and total phosphorus decreased. Mussel addition caused short-term increases in nutrient concentrations, especially following high-density treatment: phytoplankton density decreased, while cell density in the mussel-free enclosure increased. Zooplankton densities in the two enclosures were similar; however, carbon biomass in the mussel enclosure increased, associated with an increase in large zooplankton. The trophic relationship between phytoplankton and zooplankton was positive in the mussel-free enclosure and negative in the mussel-treatment enclosure, possibly reflecting effects of mussels on both consumer and resource control in the plankton community. Thus, filter feeding by Corbicula affects nutrient recycling and plankton community structure in a freshwater ecosystem through direct feeding and competition for food resources.     

Assessment of sperm competition by European bitterling,<Emphasis Type="Italic"> Rhodeus sericeus</Emphasis>

We investigated male assessment of sperm competition in the bitterling, Rhodeus sericeus, a freshwater fish that spawns on the gills of living unionid mussels. Field experiments showed males increased their inspection rate of mussels into which a testis solution containing sperm had been experimentally released. Males avoided leading females to mussels that contained high numbers of embryos, but did not alter their leading behavior in response to the presence of sperm. In laboratory experiments males also increased their inspection rate of mussels into which a testis solution had been released and also failed to alter their leading behavior in response to the presence of sperm in mussels. However, males avoided leading females to mussels in close proximity to other males, and thereby may avoid sperm competition. In a second field study, territorial males were shown to ejaculate into mussels at a low rate in the absence of competitors, increase the frequency of ejaculations in competition with a rival, then decrease relative ejaculate expenditure as the number of competing males increased. Observed data were shown to be significantly correlated with predicted estimates of ejaculate expenditure for a model of sperm competition intensity. We discuss our results in the context of adaptive responses of males to sperm competition.