Small-scale Spatial Structure Research Articles

Special IASS 2024_Wooden Spatial Structures Assembled with Disassembly Possible Simple Joints

The authors propose a new clamped joint system that can easily join wooden members together and a wooden spatial structure that utilizes this system. The proposed system can clamp wood members without the need for special joint machining. Two types of clamps have been developed: orthogonal clamps and rotational clamps. A frame system is developed using this joint system, which is a hemispherical dome with a diameter of 5.4 m and its membrane is tensioned using struts. A preliminary framework structural analysis and membrane stress analysis are conducted to confirm the safety of the frame. Full-scale mock-up was constructed to validate the frame model analysis and verify construction procedures. As a combinable structure, a small-scale spatial structure assembled using orthogonal clamps is also described. It is the cube-shaped structure 2.8 m wide consisting of wall and ceiling planes. Single clamp loading tests are conducted to understand the load-deformation relationship of the proposed clamped joint system and full-scale horizontal loading tests have been conducted.

Small-Scale Variability in Bacterial Community Structure in Different Soil Types.

Microbial spatial distribution has mostly been studied at field to global scales (i.e., ecosystem scales). However, the spatial organization at small scales (i.e., centimeter to millimeter scales), which can help improve our understanding of the impacts of spatial communities structure on microbial functioning, has received comparatively little attention. Previous work has shown that small-scale spatial structure exists in soil microbial communities, but these studies have not compared soils from geographically distant locations, nor have they utilized community ecology approaches, such as the core and satellite hypothesis and/or abundance-occupancy relationships, often used in macro-ecology, to improve the description of the spatial organization of communities. In the present work, we focused on bacterial diversity (i.e., 16S rRNA gene sequencing) occurring in micro-samples from a variety of locations with different pedo-climatic histories (i.e., from semi-arid, alpine, and temperate climates) and physicochemical properties. The forms of ecological spatial relationships in bacterial communities (i.e., occupancy-frequency and abundance-occupancy) and taxa distributions (i.e., habitat generalists and specialists) were investigated. The results showed that bacterial composition differed in the four soils at the small scale. Moreover, one soil presented a satellite mode distribution, whereas the three others presented bimodal distributions. Interestingly, numerous core taxa were present in the four soils among which 8 OTUs were common to the four sites. These results confirm that analyses of the small-scale spatial distribution are necessary to understand consequent functional processes taking place in soils, affecting thus ecosystem functioning.

Gaussian processes, median statistics, Milky Way rotation curves

We use the Iocco et al. (2015) compilation of 2,780 circular velocity measurements to analyze the Milky Way rotation curve. We find that the error bars for individual measurements are non-gaussian, and hence instead derive median statistics binned central circular velocity values and error bars from these data. We use these median statistics central values and error bars to fit the data to simple, few parameter, rotation curve functions. These simple functions are unable to adequately capture the significant small scale spatial structure in these data and so provide poor fits. We introduce and use the Gaussian Processes (GP) method to capture this small scale structure and use it to derive Milky Way rotation curves from the binned median statistics circular velocity data. The GP method rotation curves have significant small-scale spatial structure superimposed on a broad rise to galactocentric radius $R\approx7$ kpc and a decline at larger $R$. We use the GP method median statistics rotation curve to measure the Oort $A$ and $B$ constants and other characteristic rotation curve quantities. We study correlations in the residual circular velocities (relative to the GP method rotation curve). Along with other evidence for azimuthal asymmetry of the Milky Way circular rotation velocity field, we find that larger residual circular velocities seem to favor parts of spiral arms.

Small-scale spatial structure affects predator-prey dynamics and coexistence

Small-scale spatial variability can affect community dynamics in many ecological and biological processes, such as predator-prey dynamics and immune responses. Spatial variability includes short-range neighbour-dependent interactions and small-scale spatial structure, such as clustering where individuals aggregate together, and segregation where individuals are spaced apart from one another. Yet, a large class of mathematical models aimed at representing these processes ignores these factors by making a classical mean-field approximation, where interactions between individuals are assumed to occur in proportion to their average density. Such mean-field approximations amount to ignoring spatial structure. In this work, we consider an individual-based model of a two-species community that is composed of consumers and resources. The model describes migration, predation, competition and dispersal of offspring, and explicitly gives rise to varying degrees of spatial structure. We compare simulation results from the individual-based model with the solution of a classical mean-field approximation, and this comparison provides insight into how spatial structure can drive the system away from mean-field dynamics. Our analysis reveals that mechanisms leading to intraspecific clustering and interspecific segregation, such as short-range predation and short-range dispersal, tend to increase the size of the resource species relative to the mean-field prediction. We show that under certain parameter regimes these mechanisms lead to the extinction of consumers whereas the classical mean-field model predicts the coexistence of both species.

Small-scale spatial structure influences large-scale invasion rates

Local interactions among individual members of a population can generate intricate small-scale spatial structure, which can strongly influence population dynamics. The two-way interplay between local interactions and population dynamics is well understood in the relatively simple case where the population occupies a fixed domain with a uniform average density. However, the situation where the average population density is spatially varying is less well understood. This situation includes ecologically important scenarios such as species invasions, range shifts, and moving population fronts. Here, we investigate the dynamics of the spatial stochastic logistic model in a scenario where an initially confined population subsequently invades new, previously unoccupied territory. This simple model combines density-independent proliferation with dispersal, and density-dependent mortality via competition with other members of the population. We show that, depending on the spatial scales of dispersal and competition, either a clustered or a regular spatial structure develops over time within the invading population. In the short-range dispersal case, the invasion speed is significantly lower than standard predictions of the mean-field model. We conclude that mean-field models, even when they account for non-local processes such as dispersal and competition, can give misleading predictions for the speed of a moving invasion front.

Wind pollination over 70 years reduces the negative genetic effects of severe forest fragmentation in the tropical oak Quercus bambusifolia.

Whether wind pollination in trees can offset the negative genetic consequences of anthropogenic forest fragmentation is not clearly established. To answer this question, we examined the demographic genetics of Quercus bambusifolia over a 70-year recovery period in highly fragmented forests in Hong Kong. We sampled 1138 individuals from 37 locations, and genetically analysed the chronosequence through the classification of tree diameters from the same populations using 13 microsatellite markers. Our study reveals that severe fragmentation caused a significant genetic bottleneck with very few remaining but genetically diverse individuals. We observed an enhanced genetic diversity during demographic recovery. We found full-sibs within populations and half-sibs across the study range. This reflects a limited seed dispersal and extensive pollen flow. Despite reduced genetic structure both among and within populations, overall a strong persisting genetic differentiation (F'ST = 0.240, P < 0.01) and significant small-scale spatial genetic structure (F(1) = 0.13, Sp = 0.024, P < 0.01) were observed. Existing bottlenecks and low effective population sizes within the temporal chronosequence suggest that the long-term effect of severe fragmentation cannot be entirely eliminated by wind pollination with demographic recovery in the absence of effective seed dispersal. Our results lead to recommendations for forest management.

Effects of solid particles and wall roughness on turbulent boundary layer in a two-phase horizontal channel flow

Particle image velocimetry (PIV) is used to study the horizontal channel flow over smooth and rough walls, and single-phase flow and particle-laden flow with two different particle diameters are analyzed. The phenomenon of the wall similarity hypothesis is verified in single-phase flow but affected by particles in particle-laden flows. The pyramidal rough elements cause a velocity defect and shorten the length scale of the vortex. The streamwise and wall-normal Reynolds stresses are enhanced by wall roughness. Two kinds of particles with different Stokes numbers are used in the experiment. The collision and dragging effect of particles in the inner region offset the influence of wall roughness on the mean velocity profile. The introduction of particles tends to increase the streamwise Reynolds stress, but makes the peak of the Reynolds stress move to a higher wall-normal position, which is contrary to the effect of wall roughness. Quadrant analysis indicates that the wall roughness and particles show a converse effect on quadrant events, but neither change the majority of second and forth quadrant events. The analysis of the spatial structure of turbulence shows that the small scale spatial structure is less affected by wall roughness compared with the large scale spatial structure. According to the statistics of the particle distribution, it was found that particles with a larger Stokes number show a more distinct prioritized deposition in the inner region, and the deposition of particles is reduced due to the existence of rough elements.

Comparison of the Arctic upper-air temperatures from radiosonde and radio occultation observations

The air temperature is one of the most important parameters used for monitoring the Arctic climate change. The constellation observing system for meteorology, ionosphere, and climate and Formosa Satellite Mission 3 (COSMIC/FORMOSAT-3) radio occultation (RO) “wet” temperature product (i.e., “wetPrf”) is used to analyze the Arctic air temperature profiles at 925–200 hPa in 2007–2012. The “wet” temperatures are further compared with radiosonde (RS) observations. The results from the spatially and temporally synchronized RS and COSMIC observations show that their temperatures agree well with each other, especially at 400 hPa. Comparisons of seasonal temperatures and anomalies from the COSMIC and homogenized RS observations suggest that the limited number of COSMIC observations during the spatial matchup may be insufficient to describe the smallscale spatial structure of temperature variations. Furthermore, comparisons of the seasonal temperature anomalies from the RS and 5°×5° gridded COSMIC observations at 400 hPa during the sea ice minimum (SIM) of 2007 and 2012 are also made. The results reveal that similar Arctic temperature variation patterns can be obtained from both RS and COSMIC observations over the land area, while extra information can be further provided from the densely distributed COSMIC observations. Therefore, despite COSMIC observations being unsuitable to describe the Arctic temperatures in the lowest level, they provide a complementary data source to study the Arctic upper-air temperature variations and related climate change.

Breeding system, gene dispersal and small-scale spatial genetic structure of a threatened food tree species, Pentadesma butyracea (Clusiaceae) in Benin

Pentadesma butyracea Sabine, a rain forest food tree species, plays a vital role in the socio-economic livelihood of some West African rural communities due to its various products. However, its scattered populations are threatened in Benin. Defining appropriate conservation strategies requires a good knowledge of mating patterns and their consequences for population genetics. The outcrossing rate, levels of correlated paternity and fine-scale spatial genetic structure of adults and maternal sibships were estimated for one small population and three large populations in Benin using microsatellite markers. Similar outcrossing rates (88–95%) were found in all populations, showing that P. butyracea is mainly an outbreeding species. We found no evidence of inbreeding depression from a decay of inbreeding with age. The spatial genetic structure within the large populations (Sp statistic = 0.003–0.038) was consistent with isolation-by-distance expectations, showing that gene dispersal is spatially limited. Limited pollen dispersal is highlighted by the decay of the degree of correlated paternity between sibships with spatial distance. The mean pollen dispersal distance was estimated between 50 m and 450 m, but up to 21% pollen may migrate from external sources. The smallest population displayed slightly higher correlated paternity than the large populations (r p = 0.37 vs. r p = 0.17–0.30). In conclusion, our results suggest that small populations may show a reduction in sire numbers in seed, while the fragmented populations, large and small, are connected through gene flow. There is little inbreeding and no evidence of inbreeding depression.

Euglossine bees mediate only limited long-distance gene flow in a tropical vine.

Euglossine bees (Apidae: Euglossini) have long been hypothesized to act as long-distance pollinators of many low-density tropical plants. We tested this hypothesis by the analysis of gene flow and genetic structure within and among populations of the euglossine bee-pollinated vine Dalechampia scandens. Using microsatellite markers, we assessed historical gene flow by the quantification of regional-scale genetic structure and isolation by distance among 18 populations, and contemporary gene flow by the estimation of recent migration rates among populations. To assess bee-mediated pollen dispersal on a smaller scale, we conducted paternity analyses within a focal population, and quantified within-population spatial genetic structure in four populations. Gene flow was limited to certain nearby populations within continuous forest blocks, whereas drift appeared to dominate on larger scales. Limited long-distance gene flow was supported by within-population patterns; gene flow was biased towards nearby plants, and significant small-scale spatial genetic structure was detected within populations. These findings suggest that, although female euglossine bees might be effective at moving pollen within populations, and perhaps within forest blocks, their contribution to gene flow on the regional scale seems too limited to counteract genetic drift in patchily distributed tropical plants. Among-population gene flow might have been reduced following habitat fragmentation.

Clonality increases with snow depth in the arctic dwarf shrub Empetrum hermaphroditum.

Vegetative reproduction and spread through clonal growth plays an important role in arctic-alpine ecosystems with short cool growing seasons. Local variation in winter snow accumulation leads to discrete habitat types that may provide divergent conditions for sexual and vegetative reproduction. Therefore, we studied variation in clonal structure of a dominant, evergreen, dwarf shrub (Empetrum nigrum s.l. with the two taxa E. nigrum L. s.s. and E. hermaphroditum Hagerup) along a snow cover gradient and compared clonal diversity and spatial genetic structure between habitats. We studied 374 individual shoots using 105 polymorphic AFLP markers and analyses based on hierarchical clustering, clonal diversity indices, and small-scale spatial genetic structure with pairwise kinship coefficient. We used two approaches to define a threshold of genotypic distance between two samples that are considered the same clone. Clonality was examined among three habitats (exposed ridges, sheltered depressions, birch forest) differing in snow conditions replicated in four study regions in Norway and Sweden. Clonality of E. hermaphroditum differed between habitats with an increase in clonal diversity with decreasing snow depth. Small-scale spatial genetic structure increased with decreasing clonal diversity and increasing clone size. In three study regions, E. hermaphroditum was the only species, whereas in one region E. nigrum also occurred, largely confined to exposed ridges. Our results demonstrated that snow cover in conjunction with associated habitat conditions plays an important role for the mode of propagation of the dwarf shrub E. hermaphroditum.

Consequences of natal philopatry for reproductive success and mate choice in an Alpine rodent

Quantifying the interaction between dispersal, kinship, and genetic structure can provide insights into the factors that shape kin-structured mammal societies. Here, we first employ a combination of 8 years of capture–mark–recapture and molecular data to characterize the spatial and genetic relationships among female snow voles (Chionomys nivalis) in a population located in the Swiss Alps. Subsequently, we examine the individual-level consequences of kin structure in terms of fitness and mating patterns. Behavioral data, relatedness estimates, and spatial autocorrelation analyses indicate that females show strong philopatry, with spatially clustered females being characterized by high levels of genetic relatedness, leading to significant small-scale (<30 m) spatial genetic structure (SGS). In line with selection favoring female philopatry, dispersing females had a lower fitness compared with philopatric individuals. However, we found a negative association between female reproductive success and the number of neighboring females. This suggests that female kin clustering does not constitute an adaptive strategy in this species, but rather that site tenacity is a by-product of the costs of dispersal. Although dispersal is frequently invoked as a means to avoid inbreeding, our results provide no evidence for premating inbreeding avoidance, which is in line with previous studies on mammals. Instead, in the majority of years, we observed that pairs were more-closely related than expected by chance. However, we found that both males and females with related partners had reduced reproductive success, suggesting the existence of inbreeding depression and/or postmating inbreeding avoidance mechanisms. On the whole, our results show how quantification of SGS within populations can provide insights into individual dispersal behavior and its fitness consequences, and into the ways in which social and genetic structure interacts to shape the evolution of free-living populations.

Optimizing Posidonia oceanica (L.) Delile shoot density: Lessons learned from a shallow meadow

Although shoot patchiness has long been studied in Posidonia oceanica meadows, small scale spatial structure of P. oceanica meadows is poorly known, as very few studies focused on this feature. In order to analyze spatial patterns within P. oceanica meadows that appear uniformly dense and undisturbed, we collected shoot density data at the Capo Rizzuto Marine Protected Area (NE Mediterranean, Italy). Intensive sampling was carried out within a square lattice at small spatial scale (i.e. in the 10−1–102m2 range) and shoot counts were obtained from sample quadrats of different size (60, 40, 20cm). Spatial data analysis highlighted high irregularity in shoot density from centimetric to larger spatial scales. Therefore the deviation between shoot density estimates obtained using conventional methods and the overall average of quadrat counts (assumed as the best estimate for true density) was never negligible even when larger counting quadrats or higher numbers of replicates were adopted. While shoot density is regarded as the most important property of a P. oceanica meadow and as an indicator of ecosystem health, uncertainty in density estimates and unknown expected errors impair the effectivity of this approach. However, we showed that error could be predicted based on sampling intensity and design in an apparently uniform meadow. Although results from a single case study cannot be generalized, our work is the first attempt at analyzing the problems related to density estimates obtained from shoot counts and it shows how sampling can be optimized to achieve any desired level of accuracy.

Hidden founder effects: small-scale spatial genetic structure in recently established populations of the grassland specialist plant Anthyllisvulneraria.

The long-term establishment success of founder plant populations has been commonly assessed based on the measures of population genetic diversity and among population genetic differentiation, with founder populations expected to carry sufficient genetic diversity when population establishment is the result of many colonists from multiple source populations (the 'migrant pool' colonization model). Theory, however, predicts that, after initial colonization, rapid population expansion may result in a fast increase in the extent of spatial genetic structure (SGS), independent of extant genetic diversity. This SGS can reduce long-term population viability by increasing inbreeding. Using 12 microsatellite markers, we inferred colonization patterns in four recent populations of the grassland specialist plant Anthyllisvulneraria and compared the extent of SGS between recently established and old populations. Assignment analyses of the individuals of recent population based on the genetic composition of nine adjacent putative source populations suggested the occurrence of the 'migrant pool' colonization model, further confirmed by high genetic diversity within and low genetic differentiation among recent populations. Population establishment, however, resulted in the build-up of strong SGS, most likely as a result of spatially restricted recruitment of the progeny of initial colonists. Although reduced, significant SGS was nonetheless observed to persist in old populations. The presence of SGS was in all populations associated with elevated inbreeding coefficients, potentially affecting the long-term viability of these populations. In conclusion, this study illustrates the importance of studying SGS next to population genetic diversity and differentiation to adequately infer colonization patterns and long-term establishment success of plant species.

Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias.

The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and ignores the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual-based model (IBM) that uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. We incorporate a mechanism for local directional bias such that an individual's direction of movement is dependent on the degree of cell crowding in its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias, and we solve this numerically for a spatially homogeneous case.

Changes in small scale spatial structure of cockle Cerastoderma edule (L.) post-larvae

Understanding the spatial structure of populations at different scales can help reveal the processes controlling abundance and distribution. For species with a pelagic larval stage and highly mobile post-larval stage, such as the common cockle Cerastoderma edule, changes in spatial distribution from settlement through to juvenile stages are likely to occur at various spatial scales influencing the population dynamics observed. To record such changes a small-scale high-resolution field survey of C. edule post-larvae over an area of 60×80m of tidal mudflat was conducted using a spatially-explicit staggered-nested design in the Dee estuary, UK. The survey was repeated three times from initial settlement in June 2011 to March 2012. Changes in spatial structure were described using Moran's I correlograms and prediction mapping, with analysis of correlations between cockle cohorts and sediment composition. At the first sampling event in June 2011, when initial settlement was occurring, post-larvae were highly aggregated in patches of 10–14m in size and 16–20m apart. By October 2011 the post-larvae had become more evenly dispersed with some small scale (<4m) random patchiness and a gradient in post-larval density. This spatial structure was maintained into March 2012 but with increased patchiness. At settlement post-larval density showed no correlation with adult abundance or sediment mud content, but by October 2011 and March 2012 there was a strong positive correlation with adult abundance. Such changes in spatial structure, abundance and adult association after settlement show the likely importance of small scale (metres to tens of metres) processes on post-larval survival from predation and adult interactions, thereby potentially shaping adult distributions. Small scale patchiness in post-larvae can be created at settlement; however the distribution and association with adults change over time.

Isotopic evidence for the spatial heterogeneity of the planktonic food webs in the transition zone between river and lake ecosystems

Resources and organisms in food webs are distributed patchily. The spatial structure of food webs is important and critical to understanding their overall structure. However, there is little available information about the small-scale spatial structure of food webs. We investigated the spatial structure of food webs in a lake ecosystem at the littoral transition zone between an inflowing river and a lake. We measured the carbon isotope ratios of zooplankton and particulate organic matter (POM; predominantly phytoplankton) in the littoral zone of a saline lake. Parallel changes in the δ 13C values of zooplankton and their respective POMs indicated that there is spatial heterogeneity of the food web in this study area. Lake ecosystems are usually classified at the landscape level as either pelagic or littoral habitats. However, we showed small-scale spatial heterogeneity among planktonic food webs along an environmental gradient. Stable isotope data is useful for detecting spatial heterogeneity of habitats, populations, communities, and ecosystems.

Transmission of genetic variation from the adult generation to naturally established seedling cohorts in small forest stands of pedunculate oak (Quercus robur L.)

Natural regeneration is being increasingly encouraged as the preferred regeneration method for sustainable forestry. However, the benefits of natural regeneration may be jeopardized in small and low-density forest stands, as genetic drift and inbreeding may reduce genetic diversity of the standing population. Particularly in light-demanding tree species, which are characterized by a narrow recruitment window, conservation of genetic diversity during natural regeneration can be challenging in small forest stands. In this study, we investigated intergenerational transmission of genetic diversity in stands of the light-demanding tree species pedunculate oak (Quercus robur L.) by comparing genetic diversity between adults and recently established seedling cohorts (1–3years old) in four small stands of Q. robur (<4.5ha) that varied in tree density. We also quantified mating patterns and gene flow to investigate their role in shaping the genetic diversity and spatial genetic structure of the studied populations. Because of the closed canopy cover of these stands, only recently established seedlings could be studied. When all stands were pooled, the adult cohort showed a slightly but significantly higher allelic richness (Ar) than the established seedling cohort (Ar=11.5 and 10.7, respectively). However, at the stand level, no significant negative effects on the genetic diversity of the offspring generation were found in the small and low-density forest stands. As expected, acorn dispersal resulting in seedling recruitment was restricted to a few meters from mother trees, resulting in significant small-scale spatial genetic structure in the offspring. Pollen inflow from outside the study plots varied strongly among stands, but all plots showed a significant correlated paternity (rp), with higher estimates of rp found in forest stands with lower effective population sizes (Ne) and lower tree densities. Given that high rp-levels may increase the probability of biparental inbreeding and may incur fitness costs in subsequent offspring generations, we recommend that in small scale forestry of Q. robur, Ne and tree densities should be sufficiently large to maintain genetic diversity over the long term. Allowing gene flow between tree populations through reducing the spatial isolation among forest stands, can help to increase Ne in small-scale silvicultural systems of pedunculate oak, but also in other species characterized by narrow recruitment windows.

Habitat fragmentation and the genetic structure of the Amazonian palm Mauritia flexuosa L.f. (Arecaceae) on the island of Trinidad

Studies of small-scale spatial genetic structure are crucial to the implementation of successful conservation and management plans as they enhance our capacity to predict a population’s ability to respond to changing environments. This is especially true in the tropics, where high rates of fragmentation and land-use change create potential barriers to genetic exchanges. We examined the genetic diversity of wild populations of the economically important palm, Mauritia flexuosa (Arecaceae) in and around the Aripo Savannas Scientific Reserve on the island of Trinidad. In a small collection area of roughly 10 km2 we found both spatial and inter-generational, or temporal, genetic differentiation and high levels of nearest-neighbor mating. We suggest that these spatial and temporal genetic patterns reflect recent and drastic anthropogenic changes to the landscape of the study sites, which we hypothesize have created barriers to gene flow through animal-mediated seed dispersal and pollination.

Small-scale spatial and temporal genetic structure of the Atlantic sea scallop (Placopecten magellanicus) in the inshore Gulf of Maine revealed using AFLPs

Despite long planktonic durations, many species of broadcast spawning invertebrates exhibit genetic structure at small spatial and temporal scales. Amplified fragment length polymorphisms were used to assess genetic variation in the sea scallop, Placopecten magellanicus, among four inshore and one offshore location in the Gulf of Maine and temporal genetic variation among age classes of sea scallops at one site. Our results indicated that genetic structure for P. magellanicus exists on smaller spatial scales (tens to hundreds of kilometers) than expected given the 40-day planktonic larval period. In addition, genetic differences among age classes may be influenced by inter-annual differences in larval supply or reproductive success. Future genetic studies should sample multiple age classes prior to comparison among locations.