Dispersion Of Patches Research Articles

Modeling urban expansion in Zahedan’s dry climate: insights from the SLEUTH model

IntroductionRapid uncontrolled growth of build-up areas has increasingly challenged the sustainable use of urban area simulating the growth patterns of fastest-growing cities is more necessary in dry climates, due to low ecological suitability for urban development and meeting the needs of citizens. Therefore, this research conducted aiming at predicting the expansion of urban land use in Zahedan City, Iran, which has a dry climate with an evenness landscape.MethodsUrban Expansion in Zahedan Modeled using SLEUTH (slope, land use, exclusion, urban extent, transportation and hillshade) in two historical and environmental scenarios until 2050. The input data were extracted from processing on DEM and remote sensing data and the SLEUTH model was calibrated in four stages from 1990 to 2020.Results and discussionThe results showed that the increase in Ahead extent in 2050 is more than twice as much as in 2020, and this increase was associated with a less dispersion of urban patches in the environmental scenario compared to the historical scenario. Also, the results clarified that the developable spaces are saturated in terms of slope in the east and there is the lack of urban green spaces. These results reveal the need for the attention of city managers in predicting the urban green space in the expected growth areas and compensating for the lack of vegetation cover in the former urban areas. Geographic extension of predicted urban land can be used in future environmental planning and urban developing strategies, as well as it is suggested to adopt this approach as a plan for urban planning in dry climates.

Quantifying F-actin patches in single melanoma cells using total-internal reflection fluorescence microscopy

Total-internal reflection fluorescence (TIRF) microscope is a unique technique for selective excitation of only those fluorophore molecules in a cellular environment, which are located at the sub-diffraction axial distance of a cell’s contact-area. Despite this prominent feature of the TIRF microscope, making quantitative use of this technique has been a challenge, since the excitation intensity strongly depends on the axial position of a fluorophore molecule. Here, we present an easy-implemented data analysis method to quantitatively characterize the fluorescent signal, without considering the intensity-value. We use F-actin patches in single-melanoma cells as an example and define two quantities of elongation and surface density for F-actin patches at the contact-area of a melanoma cell. The elongation parameter can evaluate the dispersion of F-actin patches at the contact-area of a cell and is useful to classify the attaching, spreading, and expanding stages of a cell. Following that, we present the profile of the surface density of F-actin patches as a quantity to probe the spatio-temporal distribution of the F-actin patches at the contact-area of a cell. The data analysis methods that are proposed here will also be applicable in the image analysis of the other advanced optical microscopic methods.

Effects of Spatial Boreal Forest Harvesting Practices on Efficiency through a Benchmarking Approach in Eastern Canada

In eastern Canada, harvesting practices and spatial organization of harvested sites are modulated according to ecosystem forest management objectives. We determined how spatial organization affects efficiency by evaluating wood procurement costs. A comparative analysis of efficiency was presented using a non-parametric technique, i.e., data envelopment analysis (DEA), which allows multiple variable analyses of different factors. A database of 50 harvested sites during the period 2015–2018, located along a north-south latitudinal gradient between 46° to 50°, was constructed with variables describing spatial organization (roads and dispersion of patches) and operational aspects (wood procurement costs). The evaluated financial efficiencies show high values greater than 70%. The causes of inefficiency were dispersion of the patches, distance to the mill, and the number of kilometers of built roads. When efficiency values were arranged by latitudinal location, northern sites exhibited a lower value of overall and scale efficiency due to the high values in the wood harvested, and developed road density of the zone.

Advection and diffusion in a chemically induced compressible flow

We study analytically the joint dispersion of Gaussian patches of salt and colloids in linear flows, and how salt gradients accelerate or delay colloid spreading by diffusiophoretic effects. Because these flows have constant gradients in space, the problem can be solved almost entirely for any set of parameters, leading to predictions of how the mixing time and the Batchelor scale are modified by diffusiophoresis. We observe that the evolution of global concentrations, defined as the inverse of the patches’ areas, are very similar to those obtained experimentally in chaotic advection. They are quantitatively explained by examining the area dilatation factor, in which diffusive and diffusiophoretic effects are shown to be additive and appear as the divergence of a diffusive contribution or of a drift velocity. An analysis based on compressibility is developed in the salt-attracting case, for which colloids are first compressed before dispersion, to predict the maximal colloid concentration as a function of the parameters. This maximum is found not to depend on the flow stretching rate nor on its topology (strain or shear flow), but only on the characteristics of salt and colloids (diffusion coefficients and diffusiophoretic constant) and the initial size of the patches.

The Impact of Tourism Development on the Structure of Agricultural Landscape of the Dongshan River Basin, Taiwan

After the establishment of the Dongshan River Scenic Area of Yilan County in Taiwan, land use in the Dongshan River Basin has undergone dramatic change, from agricultural areas to recreational settings, in the past decades. The pur- pose of this study was to apply the approach of landscape ecology to analyze the spatiotemporal transition of the agricultural landscape of the river basin after tour- ism development. The study aimed to examine the landscape change in the river ba- sin and its ecological implications. Maps of the Dongshan River Basin in 1987 and 2003 were generated, and the land uses were grouped into six land-use types: paddy fields, high-density vegetation, low-density vegetation, water, built-up land, and sandy beaches. A set of landscape metrics were computed with FRAGSTATS. The results showed that, from 1987 to 2003, the major change in the river basin land- scape was a rapid increase in the area of built-up patches, while the areas for all other land use decreased. As for paddy fields, although their overall structure was not significantly altered over the study period, 5% (301 hectares) of the paddy fields had been replaced by urban settlements, which included the river park, the booming country house scene, and a traditional arts center. These changes were the main con- tributors to the increasing patch edge of paddy fields. Viewing paddy fields in Dong- shan River Basin as unique man-made habitats, attention is then given to both the dispersion of built-up patches and the increase in size of the patch edge of paddy fields.

Interacting effects of discharge and channel morphology on transport of semibuoyant fish eggs in large, altered river systems.

Habitat fragmentation and flow regulation are significant factors related to the decline and extinction of freshwater biota. Pelagic-broadcast spawning cyprinids require moving water and some length of unfragmented stream to complete their life cycle. However, it is unknown how discharge and habitat features interact at multiple spatial scales to alter the transport of semi-buoyant fish eggs. Our objective was to assess the relationship between downstream drift of semi-buoyant egg surrogates (gellan beads) and discharge and habitat complexity. We quantified transport time of a known quantity of beads using 2–3 sampling devices at each of seven locations on the North Canadian and Canadian rivers. Transport time was assessed based on median capture time (time at which 50% of beads were captured) and sampling period (time period when 2.5% and 97.5% of beads were captured). Habitat complexity was assessed by calculating width∶depth ratios at each site, and several habitat metrics determined using analyses of aerial photographs. Median time of egg capture was negatively correlated to site discharge. The temporal extent of the sampling period at each site was negatively correlated to both site discharge and habitat-patch dispersion. Our results highlight the role of discharge in driving transport times, but also indicate that higher dispersion of habitat patches relates to increased retention of beads within the river. These results could be used to target restoration activities or prioritize water use to create and maintain habitat complexity within large, fragmented river systems.

Home range, activity and sociality of a top predator, the dingo: a test of the Resource Dispersion Hypothesis

The idea that groups of individuals may develop around resource patches led to the formulation of the Resource Dispersion Hypothesis (RDH). We tested the predictions of the RDH, within a quasi‐experimental framework, using Australia’s largest terrestrial predator, the dingo Canis lupus dingo. Average dingo group sizes were higher in areas with abundant focal food sources around two mine sites compared with those in more distant areas. This supports the notion that resource richness favours larger group size, consistent with the RDH. Irrespective of season or sex, average home range estimates and daily activity for dingoes around the mine sites were significantly less than for dingoes that lived well away. Assuming that a territory is the defended part of the home range and that territory size is correlated with home range size, consistent with the RDH, the spatial dispersion of food patches therefore determined territory size for dingoes in our study. However, although sample size was small, some dingoes that accessed the supplementary food resource at the mines also spent a large proportion of their time away, suggesting a breakdown of territorial defence around the focal food resource. This, in combination with the large variation in home range size among dingoes that accessed the same supplementary food resource, limits the predictive capabilities of the RDH for this species. We hypothesize that constraints on exclusive home range occupancy will arise if a surfeit of food resources (in excess of requirements for homeostasis) is available in a small area, and that this will have further effects on access to mates and social structure. We present a conceptual model of facultative territorial defence where focal resources are available to demonstrate our findings.

Habitat use and behaviour of cattle in a heterogeneous desert environment in central Australia

The activities of livestock in arid environments typically centre on watering points, with grazing impacts often predicted to decrease uniformly, as radial piospheres, with distance from water. In patchy desert environments, however, the spatial distribution of grazing impacts is more difficult to predict. In this study sightings and dung transects are used to identify preferred cattle habitats in the heterogeneous dune system of the Simpson Desert, central Australia. The importance of watering points as foci for cattle activity was confirmed and it was shown that patchily distributed gidgee woodland, which comprises only 16% of the desert environment, is the most heavily used habitat for cattle away from water and provides critical forage and shade resources. By contrast, dune swales and sides, which are dominated by shade- and forage-deficient spinifex grassland and comprise >70% of the available habitat, were less utilised. These results suggest that habitat use by cattle is influenced jointly by water point location and by the dispersion of woodland patches in a resource-poor matrix. The findings were used to build a modified conceptual model of cattle habitat use which was compared with an original piosphere model, and the consequences for wildlife in environments where the model applies are discussed.

Spatial organization of Molina’s hog-nosed skunk (Conepatus chinga) in two landscapes of the Pampas grassland of Argentina

We radio-tracked 16 individuals (6 males, 10 females) of the little known Molina’s hog-nosed skunk ( Conepatus chinga (Molina, 1782)) and compared home-range dynamics, movement rates, and densities between a protected area and a landscape fragmented by agriculture. The mean home-range size (95% fixed kernel) was 166.7 ha (SD = 107.5 ha), without significant differences between areas. Home-range size varied significantly between males (mean = 243.7 ha, SD = 76.5 ha) and females (mean = 120.4 ha, SD =77.6 ha). Overlap between home range and core area was extensive between and within sexes in the protected area and more limited in the cropland area. Mean distance traveled between two consecutive resting sites was 269.5 m (SD = 365 m) and did not differ between areas, although movements were greater for males than females. Distance moved was influenced by seasons, being greater during the cold period. Finally, density estimates were consistently greater at the protected area. We argue that home-range size in Molina’s hog-nosed skunks is an inherent species property, whereas population density and territoriality are more flexible parameters that could reflect how the ecosystem state was affected. In our study, the greater dispersion of food patches in the cropland area than in the protected area may be the major factor influencing these parameters.

Habitat patch size and mating system as determinants of social group size in coral-dwelling fishes

It is thought that the size and dispersion of habitat patches can determine the size and composition of animal social groups, however, this has rarely been tested. The relationship between group size, the mating system, and habitat patch size in six species of coral-dwelling gobies was examined. For all species, there was a positive correlation between coral colony size and social group size, however the strength of this relationship varied among species. Paragobiodon xanthosomus exhibited the strongest relationship and a manipulative field experiment confirmed that coral colony size limited group size in this species. For other species including Paragobiodon melanosomus and Eviota bifasciata, either a highly conservative mating system (P. melanosomus), or increased mobility (E. bifasciata) appeared to disrupt the relationship between habitat patch size and group size. There was no consistent relationship between the mating system exhibited and group size among the species investigated. These results demonstrate that habitat patch size, mobility, and mating systems can interact in complex ways to structure group size even among closely related species.

The effects of habitat fragmentation via forestry plantation establishment on spatial genotypic structure in the small marsupial carnivore, Antechinus agilis

Dispersal is an important influence on species' distributions, patch colonization and population persistence in fragmented habitat. We studied the impacts of habitat fragmentation resulting from establishment of an exotic pine plantation on dispersal of the marsupial carnivore, Antechinus agilis. We applied spatial analyses of individual multilocus microsatellite genotypes and mitochondrial haplotypes to study patterns of gene flow in fragmented habitat and natural habitat 'control' areas, and how this is affected by the spatial dispersion of habitat patches, the presence of corridors and a 'mainland' source of migrants. Spatial analysis of molecular variance and partial Mantel tests confirmed the absence of cryptic barriers to gene flow in continuous habitat, which if present would confound the comparison of genetic structures in fragmented vs. unfragmented habitats. Spatial genotypic structure suggested that although dispersal was male-biased in both habitat types, fragmentation restricted dispersal of males more than that of females and the degree of restriction of male dispersal was dependent on the geographical isolation of the patch. The scale of positive genotypic structure in fragmented habitat was restricted to the two closest patches for females and the three closest patches for males. Our results provide evidence for significantly increased gene flow through habitat corridors relative to that across the matrix and for significantly lower gene flow between 'mainland' unfragmented habitat and habitat patches relative to that within either habitat type, suggesting a behavioural barrier to crossing habitat interfaces.

Effect of a landfill on the home range and group size of coyotes (Canis latrans) in a tropical deciduous forest

AbstractThe effect of an open landfill and the seasonality of a tropical deciduous forest in Jalisco, Mexico, was tested on the home range and group size of coyotes Canis latrans under the Resource Dispersion Hypothesis (RDH), which proposes that in social carnivores dispersion of patches of limited resources determines home‐range size, whereas independently, abundance of resources affects group size. The predictions in this study were that coyotes using the landfill, where food is available all year due to the continuous arrival of food wastes that are concentrated in a single patch, would have smaller, seasonally constant home ranges than coyotes living outside, where food is distributed in several patches. In this area, coyotes would increase their home ranges during the dry season due to seasonal changes in resource availability. Also, a larger coyote group size should exist in the landfill, where food abundance is greater. Home‐range size and group size of coyotes living in and outside the landfill were estimated by radio‐tracking. Home ranges of coyotes in the landfill varied from 0.9 to 9.5 km2, whereas home‐range sizes of coyotes outside the landfill varied from 10.9 to 43.7 km2. Seasonality had no effect on the home‐range sizes. We identified a group of four adult coyotes in the landfill and no group formation in coyotes outside. These results support the predictions about home‐range and group size of coyotes in relation to landfill presence, and indicate that under the circumstances of our study, coyotes follow the postulates proposed by the RDH.

Linking landscape patterns of resource distribution with models of aggregation in ovipositing stream insects

Summary Many species rely on resources that are discrete patches distributed across the landscape. Models of the coexistence of such taxa often require intraspecific aggregation of individuals on some patches, brought about by choices made during dispersal and patch location. Few studies have employed spatially explicit measures of dispersion to detect whether and how any aggregation of individuals is influenced by the dispersion of resource patches in natural landscapes. For two species of caddisfly (Hydrobiosidae) in two stream stretches, we mapped the spatial distribution of potential and exploited patches (emergent rocks as oviposition sites) and enumerated egg masses on each rock. Point pattern analysis, based on Ripley's K‐function, was used to describe the spatial pattern of potential and exploited oviposition sites, and to test hypotheses about how site selection behaviour of females influences dispersion. Emergent rocks were clumped and equally abundant in both stretches. As documented previously, egg masses of Ulmerochorema and Apsilochorema were aggregated within oviposition sites. Both species showed velocity‐specific preferences for oviposition sites: oviposition of Ulmerochorema occurred most often on rocks in fast flowing water (>0·30 m s−1); oviposition of Apsilochorema on rocks in slow water (<0·30 m s−1). Our perception of how caddisflies deposit eggs into the stream landscape varied with the neutral landscape model. The most informative neutral landscape compared the spatial pattern of oviposition sites with the underlying pattern of emergent rocks, while constraining the random selection of rocks to reflect species’ velocity preferences. Oviposition sites of Ulmerochorema were clumped, over and above the background pattern of emergent rocks. Oviposition sites of Apsilochorema were over‐dispersed. Hydrobiosid caddisflies appear to make decisions about egg‐laying at different scales. We propose a mechanistic explanation, based on what we know about the oviposition behaviour of these species, to explain how the observed patterns might arise. Our results emphasize the importance of quantifying resource patch use in natural (rather than artificial) landscapes.

Long-term resource variation and group size: A large-sample field test of the Resource Dispersion Hypothesis

BackgroundThe Resource Dispersion Hypothesis (RDH) proposes a mechanism for the passive formation of social groups where resources are dispersed, even in the absence of any benefits of group living per se. Despite supportive modelling, it lacks empirical testing. The RDH predicts that, rather than Territory Size (TS) increasing monotonically with Group Size (GS) to account for increasing metabolic needs, TS is constrained by the dispersion of resource patches, whereas GS is independently limited by their richness. We conducted multiple-year tests of these predictions using data from the long-term study of badgers Meles meles in Wytham Woods, England. The study has long failed to identify direct benefits from group living and, consequently, alternative explanations for their large group sizes have been sought.ResultsTS was not consistently related to resource dispersion, nor was GS consistently related to resource richness. Results differed according to data groupings and whether territories were mapped using minimum convex polygons or traditional methods. Habitats differed significantly in resource availability, but there was also evidence that food resources may be spatially aggregated within habitat types as well as between them.ConclusionsThis is, we believe, the largest ever test of the RDH and builds on the long-term project that initiated part of the thinking behind the hypothesis. Support for predictions were mixed and depended on year and the method used to map territory borders. We suggest that within-habitat patchiness, as well as model assumptions, should be further investigated for improved tests of the RDH in the future.

The arrangement of resources in patchy landscapes: effects on distribution, survival, and resource acquisition of chironomids.

The spatial arrangement of resources in patchy habitats influences the distribution of individuals and their ability to acquire resources. We used Chironomus riparius, a ubiquitous aquatic insect that uses leaf particles as an important resource, to ask how the dispersion of resource patches influences the distribution and resource acquisition of mobile individuals in patchy landscapes. Two experiments were conducted in replicated laboratory landscapes (38×38 cm) created by arranging sand and leaf patches in a 5×5 grid so that the leaf patches were either aggregated or uniformly dispersed in the grid. One-day-old C. riparius larvae were introduced into the landscapes in one of three densities (low, medium, high). In experiment 1, we sampled larvae and pupae by coring each patch in each landscape 3, 6, 12, or 24 days after adding larvae. In experiment 2, emerging adults were collected daily for 42 days from each patch in each landscape. In aggregated landscapes, individuals were aggregated in one patch type or the other during a particular developmental stage, but the "preferred" type changed depending on developmental stage and initial density. Adult emergence was lower by about 30% in all aggregated landscapes. In dispersed landscapes, individuals used both types of patch throughout their life cycles at all initial densities. Thus, patch arrangement influences the distribution of mobile individuals in landscapes, and it influences resource acquisition even when average resource abundance is identical among landscapes. Regardless of patch arrangement, high initial density caused accumulation of early instars in edge patches, 75% mortality of early instars, a 25% increase in development time, and a 60% reduction in adult emergence. Because mortality was extremely high among early-instar larvae in high-density treatments, we do not have direct evidence that the mechanism by which patch arrangement operates is density dependent. However, the results of our experiments strongly suggest that dispersion of resource patches across a landscape reduces local densities by making non-resource patches available for use, thereby reducing intraspecific competition.

Slack in the Configuration of Habitat Patches for Northern Bobwhites

1 analyzed whether reported ranges in the optimal values of habitat features for northern bobwhites (Colinus virginianus) reflect incomplete knowledge or a property of the bobwhite-habitat interface. Simple landscape modeling revealed that patch configuration (the dispersion, quantity, and type of habitat patches) has a property defined as slack different patch configurations with ranges of values for habitat features may provide optimal habitat at the landscape level. Slack arises because different patel configurations lead to fully usable space through time. Slack also may arise because (1) bobwhites are broadly adapted to vegetation structures; (2) the interchange of time allocated to different behaviors has null effects on fitness, within ordinary limits; and (3) patch types have interchangeable functions. Ranges of optimality for habitat features seem to be a property of the bobwhite-habitat interface. Management should address limiting boundary conditions for patch configurations as opposed to secking a single, population-maximizing configuration. Slack permits flexibility in habitat management plans and thus fosters multiple-use management.

Ecological Factors Affecting Group and Territory Size in White-Throated Magpie-Jays

White-throated Magpie-Jays (Calocitta formosa) breed cooperatively and defend permanent, all-purpose group territories. We measured territory area, resource levels, resource dispersion, group size, and group reproductive success for 14 groups over a threeyear period in Costa Rica. Larger groups possessed larger territories containing more bullhorn acacia trees (Acacia cornigera and A. collinsii), which provided a critical food resource during the dry season. On a per capita basis, however, the number of acacia trees and territory area were the same for group members regardless of group size, and survivorship did not vary significantly with group size. Variation in reproductive success among groups was influenced by two factors:(1) larger groups produced more successful nests per year, and (2) territories with a higher density of acacia trees fledged more offspring per successfunel nest. Magpie-jays bred in pasture and foraged primarily in woodland; both habitat types were patchily distributed. The dispersion of woodland and pasture patches in the landscape appeared to constrain where jays could form territories, whereas the amount of acacia and other food resources determined the size that groups could attain. We conclude that ecological factors are critical to understanding the White-throated Magpie-Jay social system, along with other social and demographic constraints typically found in avianc ooperative breeders.

Distribution and Dispersion Patterns of Individuals in Nine Species of Lepanfhes (Orchidaceae)1

ABSTRACTDispersion of individuals and patches of individuals are presented for nine species of epiphytic and lithophytic orchids in the genus Lepanthes. Patches is defined as the group of individuals present on a phorophyte. Mean number and standard error of reproductive individuals per patch range from 11.6 ± 5.1 in Lepanthes eltoroensis, to 45.5 ± 5.2 in L. rupestris. Most patches contain few reproductive individuals and usually a few patches had most of the adults. The distribution of patch sizes is skewed and most patches carry fewer individuals than the mean. Therefore means may not be an appropriate measure of central tendency as shown by mean crowding and mean patchiness indices. Mean distances between patches range from 4.8 m in L. rupestris to 95.5 m in L. dodiana. Most species have a mean nearest neighbor patch further than 50 m. A regular pattern of dispersion is often observed at distances of 5 and 10 m while at larger distances patches are more frequently aggregated. Large distances between patches and few individuals per patch suggest that small effective population sizes may be common in these orchids if gene flow is restricted.RESUMENLa distribución espacial de los individuos y las poblaciones vegetales son componentes importantes que influyen en el proceso de apareamiento. La dispersión de los individuos y grupos de estos individuos (o parchos) se presentan para nueve especies de orquídeas epifíticas del género Lepanthes. El número promedio y error estandard de los individuos reproductivos por parcho fluctúa desde 11.6 ± 5.1 en Lepanthes eltoroensis hasta 45.5 ± 5.2 en L. rupestris. La mayoría de los parchos tenían pocos individuos reproductivos y usualmente algunos de los parchos tenían la mayoría de los individuos; la distribución de frecuencias del tamaño de los parchos es desviada y la mayoría de los parchos tenía menos adultos que el promedio. Parece que el promedio no es una medida mas apropriada de la tendencia central como suguiera los indices de “mean crowding and mean patchiness”. Las distancias promedio entre los forofitos fluctúa desde 4.8 m en L. rupestris hasta 95.5 m en L. dodiana; la mayoría de las especies tenían un forofito vecino más cercano con un promedio mayor de 50 m. El patrón de dispersión observado a distancias de 5 y 10 metros es mayormente regular mientras que a distancias mayores los parchos se encuentran más agregados. Distancias grandes entre los forofitos y pocos individuos por forofito sugieren que tamaños de población efectiva pequeños pueden ser comunes en estas orquídeas.

Habitat richness affects home range size in the red fox Vulpes vulpes

The spatial behaviour of the red fox Vulpes vulpes shows a great flexibility (Voigt and Macdonald, 1984). Home range size varies from 10 to over 5000 ha (Macdonald, 1987; Voigt, 1987). In carnivores, variations in home range size, weighed for body mass (Gittleman and Harvey, 1982), are largely related to differences in habitat productivity, but the intraspecific local variation in home range size can be only partially explained by differences in productivity. Macdonald (1981, 1983) suggested that home range size and configuration were determined in the red fox by the dispersion of food-rich patches. More recently, it has been found that 1. (a) where humans are intolerant to the fox, resting sites are often located peripherally in home ranges (Meia and Weber, 1993); 2. (b) the ranging behaviour of red foxes cannot be explained only by the features of food resources in a complex environment with several productive habitats (Cavallini and Lovari, 1994); and 3. (c) home range size may be directly related to the distance between rest sites and food patches (Lucherini et al., in press). In this paper, evidence for a strong influence of habitat richness on home range size of the red fox is reported.

The sociality of solitary foragers: a model based on resource dispersion

Under versions of what may broadly be called the Resource Dispersion Hypothesis (RDH) several authors have concluded that territory size and group size are limited respectively, and independently, by the dispersion and richness of patches of food. This paper presents a model that shows how the frequency distribution of resources available per unit time within a territory may permit the formation of groups even in the absence of any functional advantage to any individual from the presence of another. In this model, animals (called primary occupants) occupy territories containing sufficient resources to meet or exceed their requirements for a critical proportion of feeding periods. The availability of these resources is described in terms of their mean richness and their heterogeneity, and plots of these parameters indicate the circumstances within which individuals may share the minimum territory with the primary occupants. The model shows how, under plausible conditions of resource dispersion, a territory that provides almost total food security for two occupants could also provide, at no cost to the original occupants, substantial food security for an additional group member, even if it never used the same food patches as the originals. It is not therefore, as is sometimes supposed, a necessary condition for the RDH that members of a group often forage simultaneously in the same patch. Thus the model describes ecological circumstances whereby groups could evolve amongst species whose members neither forage communally, nor even meet frequently.