Brownian Bridge Research Articles

Construction and sample path properties of Brownian house-moving between two curves

AbstractThis study aims to construct a stochastic process called “Brownian house-moving,” which is a Brownian bridge conditioned to stay between two curves. To construct this process, statements are prepared on the weak convergence of conditioned Brownian motions, conditioned Brownian bridges, and conditioned three-dimensional Bessel bridges. Moreover, the sample path properties of Brownian house-moving are studied as well.

Unveiling change-point detection with ARMA-GARCH models

This paper delves into the asymptotic theory of a Cumulative Sum (CUSUM)-type test designed for detecting change points in the variance of AutoRegressive Moving-Average models, incorporating Generalized Autoregressive Conditional Heteroscedasticity innovations. We demonstrate that, under the null hypothesis (no change), the CUSUM test statistic converges to the supremum of a standard Brownian bridge. Using a Monte Carlo simulation, we highlight the strong performance of our proposed test when compared to the methods by Song and Kang. Furthermore, we provide a real-world data analysis to showcase the practical application of our test.

Modeling regional occupancy of fishes using acoustic telemetry: a model comparison framework applied to lake trout

Acoustic telemetry is a common tool used in fisheries management to estimate fish space use (i.e., occupancy) from a local habitat scale to entire systems. Numerous analytical models have been developed to estimate different aspects of fish movement from telemetry datasets, yet evaluations of model performance and comparisons among models are limited. Here, we develop a framework to evaluate model estimates of regional occupancy in large and fragmented systems using an acoustic receiver array in Lake Champlain. We simulated the tracks of 100 acoustically tagged fish using a random walk function and created detection events based on receiver positions and distance-based detection probability. Regional occupancy for the simulated data was estimated by six movement models that ranged in analytical complexity, and results were compared to the true distributions for each simulated track to evaluate model error. The six movement models included: (1) a basic residency index using detections alone; (2) a residency index using last-observation-carried-forward; (3) a centers of activity model; (4) linear and non-linear interpolations (i.e., least-cost paths); and (5 and 6) two dynamic Brownian bridge movement models generated using separate packages in R. We developed a model selection process to compare model performance and select the optimal analysis based on simulation error. This process showed significant differences in model performance among the six movement models based on model error. Overall, the model generating least-cost paths using linear and non-linear interpolations consistently provided the most accurate regional occupancy estimates. Based on these simulation results, we applied this model to a case study that evaluated patterns in the regional distribution of stocked lake trout (Salvelinus namaycush) in Lake Champlain, which demonstrated distinct regional occupancy of two stocked lake trout groups. These results demonstrate potential for large variability in interpretation of acoustic telemetry data for describing regional fish distribution dependent on the analytical method used.

Estimating ungulate migration corridors from sparse movement data

AbstractMany ungulates migrate between distinct summer and winter ranges, and identifying, mapping, and conserving these migration corridors have become a focus of local, regional, and global conservation efforts. Brownian bridge movement models (BBMMs) are commonly used to empirically identify these seasonal migration corridors; however, they require location data sampled at relatively frequent intervals to obtain a robust estimate of an animal's movement path. Fitting BBMMs to sparse location data violates the assumption of conditional random movement between successive locations, overestimating the area (and width) of a migration corridor when creating individual and population‐level occurrence distributions and precluding the use of low‐frequency, or sparse, data in mapping migration corridors. In an effort to expand the utility of BBMMs to include sparse GPS data, we propose an alternative approach to model migration corridors from sparse GPS data. We demonstrate this method using GPS data collected every 2 h from four mule deer (Odocoileus hemionus) and four elk (Cervus canadensis) herds within Wyoming and Idaho. First, we used BBMMs to estimate a baseline corridor for the 2‐h data. We then subsampled the 2‐h data to one location every 12 h (a proxy for sparse data) and fitted BBMMs to the 12‐h data using a fixed motion variance (FMV) value, instead of estimating the Brownian motion variance empirically. A range of FMV values was tested to identify the value that best approximated the baseline migration corridor. FMV values within a species‐specific range (mule deer: 400–1200 m2; elk: 600–1600 m2) successfully delineated migration corridors similar to the 2‐h baseline corridors; overall, lower values delineated narrower corridors and higher values delineated wider corridors. Optimal FMV values of 800 m2 (mule deer) and 1000 m2 (elk) decreased the inflation of the 12‐h corridors relative to the 2‐h corridors from traditional BBMMs. This FMV approach thus enables using sparse movement data to approximate realistic migration corridor dimensions, providing an important alternative when movement data are collected infrequently. This approach greatly expands the number of datasets that can be used for migration corridor mapping—a useful tool for management and conservation across the globe.

Efficient generation of barrier crossing trajectories using approximate Brownian bridges.

We examine continuous random walks that are conditioned to reach one region before another. These conditioned processes are used to generate stochastic trajectories for barrier crossing events, which are generally rare and difficult to sample. The processes are generated using a Brownian bridge technique, resulting in near perfect sampling efficiency without accruing error in the conditional statistics of the process. The construction requires the hitting probability or committer function, which is a solution to the backward Fokker-Planck equation, a partial-differential equationthat can be difficult to solve through general means. Therefore, we derive a one-dimensional approximation through asymptotic methods for barrier crossing trajectories. We show that this approximation has a simple analytical representation and approaches the true solution as the barrier height increases. Brownian bridge trajectories generated with this approximate solution are then shown to result in accurate conditional statistics when used in conjunction with importance sampling, even in the case when potential energy barriers are not large. We show this idea's effectiveness by simulating rare events in a stochastic chemical reaction network (Schögl reaction) with multiple steady states. This methodology shows great promise for future implementation to simulate rare barrier crossing events for a wide variety of physical processes.

Statistical inference for innovation distribution in ARMA and multi-step-ahead prediction via empirical process

The kernel distribution estimator (KDE) is proposed based on residuals of the innovation distribution in the autoregressive moving-average (ARMA) time series. The deviation between KDE and the innovation distribution function is shown to converge to the Brownian bridge, leading to the construction of a Kolmogorov–Smirnov smooth simultaneous confidence band for the innovation distribution function. Additionally, an empirical cumulative distribution function (CDF) based on prediction residuals is introduced for the multi-step-ahead prediction error distribution function. This empirical process weakly converges to a Gaussian process with a specific covariance function. Furthermore, a quantile estimator is derived from the empirical CDF of prediction residuals, and multi-step-ahead prediction intervals (PIs) for future observations are established using these estimated quantiles. The PIs achieve the nominal prediction level asymptotically for the finite variance ARMA model. Simulation studies and analysis of crude oil price data confirm the validity of the asymptotic theory.

Many-sided Poisson–Voronoi cells with only Gabriel neighbors

Let pnG be the probability for a planar Poisson–Voronoi cell to be n-sided and have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of log⁡pnG in the limit of large n. We determine several statistical properties of a many-sided cell obeying this ‘Gabriel condition.’ In particular, the cell perimeter, when parametrized as a function τ(θ) of the polar angle θ, behaves as a Brownian bridge on the interval 0⩽θ⩽2π . We point out similarities and differences with related problems in random geometry.

Non-parametric inference on calibration of predicted risks.

Moderate calibration, the expected event probability among observations with predicted probability z being equal to z, is a desired property of risk prediction models. Current graphical and numerical techniques for evaluating moderate calibration of risk prediction models are mostly based on smoothing or grouping the data. As well, there is no widely accepted inferential method for the null hypothesis that a model is moderately calibrated. In this work, we discuss recently-developed, and propose novel, methods for the assessment of moderate calibration for binary responses. The methods are based on the limiting distributions of functions of standardized partial sums of prediction errors converging to the corresponding laws of Brownian motion. The novel method relies on well-known properties of the Brownian bridge which enables joint inference on mean and moderate calibration, leading to a unified "bridge" test for detecting miscalibration. Simulation studies indicate that the bridge test is more powerful, often substantially, than the alternative test. As a case study we consider a prediction model for short-term mortality after a heart attack, where we provide suggestions on graphical presentation and the interpretation of results. Moderate calibration can be assessed without requiring arbitrary grouping of data or using methods that require tuning of parameters.

Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge

This paper discusses pricing barrier options and autocallable structured products on underlying three assets via exit probabilities of a three-dimensional Brownian bridge. We derive the marginal exit and bivariate co-exit probabilities. Despite the impossibility of finding an analytical trivariate co-exit probability, its calculation is not regarded as a purely numerical problem. After specifying the component to be numerically evaluated, logistic regression with the Monte Carlo method is adopted to predict it. Extensive numerical experiments of calculating the exit probability of the three-dimensional Brownian motion and pricing complex products demonstrate the effectiveness and efficiency of our approach.

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

Female black bear home range and resource selection in relation to West Virginia mine lands

AbstractWildlife space use is a key component to consider when determining effective conservation and management practices. Home range size and resource selection can vary temporally and spatially depending on mortality risks in the area and access to food resources. Lands used for coal mining can contain unique mortality risks and food resources for American black bears (Ursus americanus) relative to surrounding landscapes due to hunting restrictions, vegetation planted as part of the reclamation process, and other anthropogenic food resources. We examined the influence of mining sites on resource selection of female black bears in southern West Virginia, USA, during 2007 and 2008. We estimated home range size with Brownian Bridge movement models and evaluated variation in home range size with mixed‐effects linear regression. We evaluated resource selection with mixed‐effects Poisson point process models. On mine sites, the mean 95% home range for female black bears was 1,996 ha (SD = 2,633) in the autumn and 544 ha (SD = 238) in the spring, and on non‐mine sites the mean 95% home range was 1,926 ha (SD = 1,292) in the autumn and 771 ha (SD = 480) in the spring. Our resource selection model indicated black bears used nonhunting zones associated with mine lands much greater than their availability, and that black bears in mine areas disproportionately used areas in close proximity to anthropogenic structures. Interannual variation in home range size was likely associated with mast conditions, because smaller home ranges were associated with high mast production. Our data suggest black bears may be using non‐hunting zones within mine lands as refugia from hunting and are using anthropogenic food sources available at refuse sites on mine lands, which may increase regional black bear populations in areas with high mining activity.

Spatial overlap of gray wolves and ungulate prey changes seasonally corresponding to prey migration

BackgroundPrey are more vulnerable during migration due to decreased familiarity with their surroundings and spatially concentrated movements. Predators may respond to increased prey vulnerability by shifting their ranges to match prey. Moose (Alces alces) and white-tailed deer (Odocoileus virginianus) are primary gray wolf (Canis lupus) prey and important subsistence species for Indigenous communities. We hypothesized wolves would increase use of ungulate migration corridors during migrations and predicted wolf distributions would overlap primary available prey.MethodsWe examined seasonal gray wolf, moose, and white-tailed deer movements on and near the Grand Portage Indian Reservation, Minnesota, USA. We analyzed GPS collar data during 2012–2021 using Brownian bridge movement models (BBMM) in Migration Mapper and mechanistic range shift analysis (MRSA) to estimate individual- and population-level occurrence distributions and determine the status and timing of range shifts. We estimated proportional overlap of wolf distributions with moose and deer distributions and tested for differences among seasons, prey populations, and wolf sex and pack affiliations.ResultsWe identified a single migration corridor through which white-tailed deer synchronously departed in April and returned in October–November. Gray wolf distributions overlapped the deer migration corridor similarly year-round, but wolves altered within-range distributions seasonally corresponding to prey distributions. Seasonal wolf distributions had the greatest overlap with deer during fall migration (10 October–28 November) and greatest overlap with moose during summer (3 May–9 October).ConclusionsGray wolves did not increase their use of the white-tailed deer migration corridor but altered distributions within their territories in response to seasonal prey distributions. Greater overlap of wolves and white-tailed deer in fall may be due to greater predation success facilitated by asynchronous deer migration movements. Greater summer overlap between wolves and moose may be linked to moose calf vulnerability, American beaver (Castor canadensis) co-occurrence, and reduced deer abundance associated with migration. Our results suggest increases in predation pressure on deer in fall and moose in summer, which can inform Indigenous conservation efforts. We observed seasonal plasticity of wolf distributions suggestive of prey switching; that wolves did not exhibit migratory coupling was likely due to spatial constraints resulting from territoriality.

Female Deer Movements Relative to Firearms Hunting in Northern Georgia, USA.

Perceived risk associated with hunters can cause white-tailed deer (Odocoileus virginianus) to shift their activity away from key foraging areas or alter normal movements, which are important considerations in managing hunting and its effects on a population. We studied the effects of seven firearms hunts on the movements of 20 female deer in two Wildlife Management Areas within the Chattahoochee National Forest of northern Georgia, USA, during the 2018-2019 and 2019-2020 hunting seasons. Deer populations and the number of hunters in our study area have declined significantly since the 1980s. In response, hunting regulations for the 2019-2020 hunting season eliminated opportunities for harvesting female deer. To evaluate the indirect effects of antlered deer hunting on non-target female deer, we calculated 90% utilization distributions (UDs), 50% UDs, and step lengths for pre-hunt, hunt, and post-hunt periods using the dynamic Brownian bridge movement model. Data included 30 min GPS locations for 44 deer-hunt combinations. Pre-hunt 50% UDs (x- = 7.0 ha, SE = 0.4 ha) were slightly greater than both hunt (x- = 6.0 ha, SE = 0.3 ha) and post-hunt (x- = 6.0 ha, SE = 0.2 ha) 50% UDs (F = 3.84, p = 0.03). We did not detect differences in step length, nor did we detect differences in size or composition of 90% UDs, among the periods. Overall, our results suggest that the low level of hunting pressure in our study area and lack of exposure to hunters led to no biologically significant changes in female deer movements. To the extent of the findings presented in this paper, adjustments to the management of hunting in our study area do not appear to be necessary to minimize hunting-related disturbances for female deer. However, managers should continue to consider female deer behavior when evaluating future changes to hunting regulations.

BBScore: A Brownian Bridge Based Metric for Assessing Text Coherence

Measuring the coherence of text is a vital aspect of evaluating the quality of written content. Recent advancements in neural coherence modeling have demonstrated their efficacy in capturing entity coreference and discourse relations, thereby enhancing coherence evaluation. However, many existing methods heavily depend on static embeddings or focus narrowly on nearby context, constraining their capacity to measure the overarching coherence of long texts. In this paper, we posit that coherent texts inherently manifest a sequential and cohesive interplay among sentences, effectively conveying the central theme, purpose, or standpoint. To explore this abstract relationship, we introduce the "BB Score," a novel reference-free metric grounded in Brownian bridge theory for assessing text coherence. Our findings showcase that when synergized with a simple additional classification component, this metric attains a performance level comparable to state-of-the-art techniques on standard artificial discrimination tasks. We also establish in downstream tasks that this metric effectively differentiates between human-written documents and text generated by large language models within specific domains. Furthermore, we illustrate the efficacy of this approach in detecting written styles attributed to various large language models, underscoring its potential for generalizability. In summary, we present a novel Brownian bridge coherence metric capable of measuring both local and global text coherence, while circumventing the need for end-to-end model training. This flexibility allows for its application in various downstream tasks.

Optimal stopping of an Ornstein–Uhlenbeck bridge

In this paper we make a rigorous analysis of the existence and characterization of the free boundary related to the optimal stopping problem that maximizes the mean of an Ornstein–Uhlenbeck bridge. The result includes the Brownian bridge problem as a limit case. The methodology hereby presented relies on a time–space transformation that casts the original problem into a more tractable one with an infinite horizon and a Brownian motion underneath. We comment on two different numerical algorithms to compute the free-boundary equation and discuss illustrative cases that shed light on the boundary’s shape. In particular, the free boundary generally does not share the monotonicity of the Brownian bridge case.

Home range in genus Capra: from polygons to Brownian bridges of scabietic and healthy Iberian ibexes (Capra pyrenaica).

Home range and home range overlap can be used to describe use of space and movement of wildlife. During the last years, advancements in technology have greatly improved our understanding of animal movement, especially among large herbivores. Wild ungulate abundance and distribution have increased in temperate areas. Moreover, their diseases-including sarcoptic mange in the Iberian Ibex (Capra pyrenaica)-have become a cause of concern for livestock, public health, and wildlife conservation. In this study, we first reviewed existing literature on the home range of species in the genus Capra. We then analyzed data from 52 GPS-GSM-collared Iberian ibexes, of which 33 were healthy and 19 were affected by sarcoptic mange from 3 different populations in the southeastern Iberian Peninsula to analyze: (1) differences in size and characteristics of home ranges obtained by the 3 most commonly used methodologies-minimum convex polygon, kernel density estimation, and Brownian bridges movement models (BBMMs); and (2) the impact of endemic sarcoptic mange on Iberian Ibex home range. The literature review revealed that available information on spatial behavior of Capra spp. was based only on 3 species, including the Iberian Ibex, estimated through a diversity of methods which made it difficult to compare results. We found positive correlations among the different home range estimation methods in the Iberian Ibex, with BBMMs proving to be the most accurate. This study is the first to use BBMMs for estimating home range in this species, and it revealed a marked seasonal behavior in spatial use, although sarcoptic mange smoothed such seasonal pattern. The seasonal overlaps obtained suggest that core areas of the Iberian Ibex change within wider home range areas, which are ecological parameters relevant to identifying key areas for species management and conservation.

GPS Tracking Reveals the White-Tailed Eagle Haliaeetus albicilla as an Ambassador for the Natura 2000 Network

The Natura 2000 network of protected areas is the backbone of species conservation in the European Union. We investigated whether Austrian-hatched white-tailed eagles (Haliaeetus albicilla) make particular use of this multinational network during their natal dispersal, and what habitats were of importance to the eagles. We analyzed the utilization distribution of 907,466 GPS locations from 38 dispersing white-tailed eagles using a dynamic Brownian Bridge Movement Model. Eagles ranged over a huge area of central-eastern Europe. Natura 2000 sites overlapped with 67% of the resulting 50% isopleth; i.e., a high probability of utilization of Natura 2000 areas by white-tailed eagles was found. White-tailed eagles used wetlands, waterbodies, and deciduous forests adjacent to wet habitats disproportionately often. Coniferous forests and settlements were avoided. Anthropogenically caused mortalities hardly occurred within Natura 2000 sites. Our study suggests that the Natura 2000 network is a crucial tool for conserving the white-tailed eagle. This top predator is an ambassador for the Natura 2000 idea during all life stages, and should continue to be a conservation priority of the network.

Weierstrass bridges

We introduce a new class of stochastic processes called fractional Wiener–Weierstraß bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstraß functions to an underlying fractional Brownian bridge. By analyzing the p p -th variation of the fractional Wiener–Weierstraß bridge along the sequence of b b -adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener–Weierstraß bridges that are based on a standard Brownian bridge. We furthermore prove that fractional Wiener–Weierstraß bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstraß bridges based on bridges derived from a general continuous Gaussian martingale.

Survival and behavior of Mojave desert tortoises head‐started with and without outdoor rearing

AbstractMojave desert tortoise (Gopherus agassizii) populations in some regions have declined by >50% since 2004, prompting the need for more research on ways to recover populations. One possible recovery tool is head‐starting (i.e., the act of protecting and raising juvenile tortoises to sizes that increase survival upon release); however, head‐starting can have high start‐up and maintenance costs that can limit its feasibility. Strategies that reduce cost and rearing duration may foster broader and more effective use. We released and radio‐tracked 60 juvenile tortoises in the Mojave National Preserve in California, USA, that had been reared under 2 treatments: those reared 1 year indoors after hatching, then 1 year outdoors (combo) and those reared just 1 year indoors (indoor‐only). We tested whether indoor‐only rearing alone could be a more efficient means of producing robust head‐started tortoises. We examined the behavior, movement, and survival of tortoises after release into the wild from 2020 to 2021 to determine whether these outcomes differed between husbandry treatments. Combo tortoises tended to perform settling behaviors (mean ± 1 SE days to building first burrow = 6.7 ± 0.8, entering dormancy = 23.3 ± 2.1, and emerging from dormancy = 189.6 ± 4.4) earlier than indoor‐only tortoises (7.4 ± 0.9, 31.5 ± 2.6, and 193.9 ± 5.9, respectively), but this difference was not significant, suggesting the rearing method did not greatly alter settling behavior. Indoor‐only tortoises dispersed at least twice as far from their release site (156.2 ± 26.3 m compared with 77.3 ± 20.6 m for combo tortoises), had larger mean use areas (3.7 ± 0.1 ha compared with 2.8 ± 0.3 ha for combo tortoises for 95% Brownian bridge movement model estimates), and greater variability in their movements than combo tortoises (daily average step length post‐emergence: 4.3 ± 0.2 m compared with 2.8 ± 0.1 m for combo tortoises). Despite differences in their movements, indoor‐only and combo tortoises had similar survival rates over the study, 51% versus 42%, respectively, during a period of extreme drought in 2021. The similarity in survival between groups gives head‐starting practitioners freedom in their rearing methods. The indoor‐only group had lower site fidelity, which should be considered when this is an undesirable trait for released tortoises.

Home range, site fidelity, and movements of timber rattlesnakes (Crotalus horridus) in west-central Illinois

Understanding the home range of imperiled reptiles is important to the design of conservation and recovery efforts. Despite numerous home range studies for the Threatened timber rattlesnake (Crotalus horridus), many have limited sample sizes or outdated analytical methods and only a single study has been undertaken in the central midwestern United States. We report on the home range size, site fidelity, and movements of C. horridus in west-central Illinois. Using VHF telemetry, we located 29 C. horridus (13 female, 16 male) over a 5-year period for a total of 51 annual records of the species' locations and movements. We calculated annual home ranges for each snake per year using 99%, 95%, and 50% isopleths derived from Brownian Bridge utilization distributions (BBMM), and we also report 100% minimum convex polygons to be consistent with older studies. We examined the effects of sex, mass, SVL, and year on home range sizes and reported on movement metrics as well as home range fidelity using both Utilization Distribution Overlap Index (UDOI) and Bhattacharyya's affinity (BA) statistics. The home range sizes for male and non-gravid C. horridus were 88.72 Ha (CI 63.41–110.03) and 28.06 Ha (CI 17.17–38.96) for 99% BBMM; 55.65 Ha (CI 39.36–71.93) and 17.98 (CI 10.69–25.28) for 95% BBMM; 7.36 Ha (CI 5.08–9.64) and 2.06 Ha (CI 1.26–2.87) for 50% BBMM; and 78.54Ha (CI 47.78–109.30) and 27.96 Ha (CI 7.41–48.51) for MCP. The estimated daily distance traveled was significantly greater for males (mean = 57.25 m/day, CI 49.06–65.43) than females (mean = 27.55 m/day, CI 18.99–36.12), particularly during the summer mating season. Similarly, maximum displacement distances (i.e., maximum straight-line distance) from hibernacula were significantly greater for males (mean = 2.03 km, CI 1.57–2.48) than females (mean = 1.29 km, CI 0.85–1.73], and on average, males were located further from their hibernacula throughout the entirety of their active season. We calculated fidelity to high-use areas using 11 snakes that were tracked over multiple years. The mean BBMM overlap using Bhattacharyya's affinity (BA) for all snakes at the 99%, 95%, and 50% isopleths was 0.48 (CI 0.40–0.57), 0.40 (0.32–0.49), and 0.07 (0.05–0.10), respectively. The mean BBMM overlap for all snakes using the Utilization Distribution Overlap Index (UDOI) at the 99%, 95%, and 50% isopleths was 0.64 (CI 0.49–0.77), 0.32 (CI 0.21–0.47), and 0.02 (CI 0.01–0.05)), respectively. Our results are largely consistent with those of other studies in terms of the influence of sex on home range size and movements. The species also exhibits strong site fidelity with snakes generally using the same areas each summer, though there is far less overlap in specific (e.g., 50% UDOI) high-use areas, suggesting some plasticity in hunting areas. Particularly interesting was the tendency for snakes to disperse from specific hibernacula in the same general direction to the same general areas. We propose some possible reasons for this dispersal pattern.