Dispersal connectivity and reserve selection for marine conservation

Although larval dispersal is crucial for the persistence of most marine populations, dispersal connectivity between sites is rarely considered in designing marine protected area networks. In particular the role of structural characteristics (known as topology) for the network of larval dispersal routes in the conservation of metapopulations has not been addressed. To determine reserve site configurations that provide highest persistence values with respect to their connectivity characteristics, we model nine connectivity topological models derived from graph theory in a demographic metapopulation model. We identify reserve site configurations that provide the highest persistence values for each of the metapopulation connectivity models. Except for the minimally connected and fully connected populations, we observed two general ‘rules of thumb’ for optimising the mean life time for all topological models: firstly place the majority of reserves, so that they are neighbours of each other, on the sites where the number of connections between the populations is highest (hub), secondly when the reserves have occupied the majority of the vertices in the hub, then select another area of high connectivity and repeat. If there are no suitable hubs remaining then distribute the remaining reserves to isolated locations optimising contact with non-reserved sites.     

Habitat-quality effects on metapopulation dynamics in greater white-toothed shrews, Crocidura russula

The effects of patch size and isolation on metapopulation dynamics have received wide empirical support and theoretical formalization. By contrast, the effects of patch quality seem largely underinvestigated, partly due to technical difficulties in properly assessing quality. Here we combine habitat-quality modeling with four years of demographic monitoring in a metapopulation of greater white-toothed shrews (Crocidura russula) to investigate the role of patch quality on metapopulation processes. Together, local patch quality and connectivity significantly enhanced local population sizes and occupancy rates (R2 = 14% and 19%, respectively). Accounting for the quality of patches connected to the focal one and acting as potential sources improved slightly the model explanatory power for local population sizes, pointing to significant source-sink dynamics. Local habitat quality, in interaction with connectivity, also increased colonization rate (R2 = 28%), suggesting the ability of immigrants to target high-quality patches. Overall, patterns were best explained when assuming a mean dispersal distance of 800 m, a realistic value for the species under study. Our results thus provide evidence that patch quality, in interaction with connectivity, may affect major demographic processes.     

Connectivity or demography: Defining sources and sinks in coral reef fish metapopulations

The identity of an individual patch as a source or a sink within a metapopulation is a function of its ability to produce individuals and to disperse them to other patches. In marine systems patch identity is very often defined by dispersal ability alone—upstream patches are sources—while issues of variable habitat quality (which affects local production) are ignored. This can have important ramifications for the science of marine reserve siting. This study develops a spatially explicit source–sink metapopulation model for reef fish and uses it to evaluate the relative importance of connectivity versus demography and how this depends upon the level of local larval retention and the strength of density-dependent recruitment. Elasticity analyses indicated that patch contribution (source or sink) was more sensitive to demographic parameters (particularly survival) than connectivity and this effect was conserved even under strong levels of density-dependence and was generally strengthened as local retention increased. Variability in the relationship between parameter elasticity and local retention was shown to be dependent upon the magnitude of connectivity for an individual patch relative to a critical connectivity value. The proportion of larvae lost due to transport processes was an important parameter which directly affected the magnitude of this critical connectivity value. Patches with connectivity values less than the critical value contributed to the metapopulation largely via production (i.e., local demographics most important). As local retention increased, so did the importance of demographic parameters in these patches. Patches with connectivity values greater than the critical value contributed largely via dispersal of larvae and thus the importance of local demographics decreased as local retention increased.     

Developing Metapopulation Connectivity Criteria from Genetic and Habitat Data to Recover the Endangered Mexican Wolf

Restoring connectivity between fragmented populations is an important tool for alleviating genetic threats to endangered species. Yet recovery plans typically lack quantitative criteria for ensuring such population connectivity. We demonstrate how models that integrate habitat, genetic, and demographic data can be used to develop connectivity criteria for the endangered Mexican wolf (Canis lupus baileyi), which is currently being restored to the wild from a captive population descended from 7 founders. We used population viability analysis that incorporated pedigree data to evaluate the relation between connectivity and persistence for a restored Mexican wolf metapopulation of 3 populations of equal size. Decreasing dispersal rates greatly increased extinction risk for small populations (<150–200), especially as dispersal rates dropped below 0.5 genetically effective migrants per generation. We compared observed migration rates in the Northern Rocky Mountains (NRM) wolf metapopulation to 2 habitat‐based effective distance metrics, least‐cost and resistance distance. We then used effective distance between potential primary core populations in a restored Mexican wolf metapopulation to evaluate potential dispersal rates. Although potential connectivity was lower in the Mexican wolf versus the NRM wolf metapopulation, a connectivity rate of >0.5 genetically effective migrants per generation may be achievable via natural dispersal under current landscape conditions. When sufficient data are available, these methods allow planners to move beyond general aspirational connectivity goals or rules of thumb to develop objective and measurable connectivity criteria that more effectively support species recovery. The shift from simple connectivity rules of thumb to species‐specific analyses parallels the previous shift from general minimum‐viable‐population thresholds to detailed viability modeling in endangered species recovery planning. Desarrollo de Criterios de Conectividad Metapoblacional a Partir de Datos Genéticos y de Hábitat para Recuperar al Lobo Mexicano en Peligro de Extinción     

A model for behavioral regulation of metapopulation dynamics

Habitat fragmentation can decrease local population persistence by reducing connectivity, which is a function of dispersal of individuals among habitat fragments. Dispersal is often treated as diffusion in population models, even though for many species it is a result of a series of behavioral decisions. We developed a metapopulation model to explore the potential importance of dispersal behaviors in driving metapopulation dynamics. We incorporated types of behavior that affect dispersal—colonization inhibiting, colonization enhancing, extinction inhibiting, extinction enhancing, rescue enhancing, rescue inhibiting—into Levins’ (1969) metapopulation model and projected occupancy rates for a variety of parameter values. Examples from the literature of behaviors associated with each of these parameters are provided. Our model simplifies into previously published metapopulation models that incorporate only a single behavior, and we present a density-dependent rescue function that leads to multiple non-zero equilibria. We found a variety of behavioral effects on metapopulations. Rescue enhancement fills patches faster than does colonization enhancement or extinction inhibition, and declines in patch occupancy are moderate with extinction enhancement, but colonization inhibition causes metapopulation extinction. We also found that with colonization and extinction inhibitions, equilibrium patch occupancy is inversely related to patch turnover rate. With density-dependent rescue, persistence depends not only on the strength of the strong rescue effect, but also on having a sufficient initial fraction of patches occupied; the stronger the rescue effect, the lower this fraction can be. This study suggests that dispersal behavior can have strong influences on metapopulation dynamics. It confirms the importance of understanding the relationship between landscape structure and dispersal behavior in understanding population persistence.     

Evaluating the importance of demographic connectivity in a marine metapopulation

Recently researchers have gone to great lengths to measure marine metapopulation connectivity via tagging, genetic, and trace-elemental fingerprinting studies. These empirical estimates of larval dispersal are key to assessing the significance of metapopulation connectivity within a demographic context, but the life-history data required to do this are rarely available. To evaluate the demographic consequences of connectivity we constructed seasonal, size-structured metapopulation matrix models for two species of mytilid mussel in San Diego County, California, USA. The self-recruitment and larval exchange terms were produced from a time series of realized connectivities derived from trace-elemental fingerprinting of larval shells during spring and fall from 2003 to 2008. Both species exhibited a strong seasonal pattern of southward movement of recruits in spring and northward movement in fall. Growth and mortality terms were estimated using mark-recapture data from representative sites for each species and subpopulation, and literature estimates of juvenile mortality. Fecundity terms were estimated using county-wide settlement data from 2006-2008; these data reveal peak reproduction and recruitment in fall for Mytilus californianus, and spring for M. galloprovincialis. Elasticity and life-stage simulation analyses were employed to identify the season- and subpopulation-specific vital rates and connectivity terms to which the metapopulation growth rate (lambda) was most sensitive. For both species, metapopulation growth was most sensitive to proportional changes in adult fecundity, survival and growth of juvenile stages, and population connectivity, in order of importance, but relatively insensitive to adult growth or survival. The metapopulation concept was deemed appropriate for both Mytilus species as exchange between the subpopulations was necessary for subpopulation persistence. However, highest metapopulation growth occurred in years when a greater proportion of recruits was retained within the predominant source subpopulation. Despite differences in habitat and planktonic duration, both species exhibited similar overall metapopulation dynamics with respect to key life stages and processes. However, different peak reproductive periods in an environment of seasonal current reversals led to different regional (subpopulation) contributions to metapopulation maintenance; this result emphasizes the importance of connectivity analysis for spatial management of coastal resources.     

Consistent scaling of persistence time in metapopulations

Recent theory and experimental work in metapopulations and metacommunities demonstrates that long-term persistence is maximized when the rate at which individuals disperse among patches within the system is intermediate; if too low, local extinctions are more frequent than recolonizations, increasing the chance of regional-scale extinctions, and if too high, dynamics exhibit region-wide synchrony, and local extinctions occur in near unison across the region. Although common, little is known about how the size and topology of the metapopulation (metacommunity) affect this bell-shaped relationship between dispersal rate and regional persistence time. Using a suite of mathematical models, we examined the effects of dispersal, patch number, and topology on the regional persistence time when local populations are subject to demographic stochasticity. We found that the form of the relationship between regional persistence time and the number of patches is consistent across all models studied; however, the form of the relationship is distinctly different among low, intermediate, and high dispersal rates. Under low and intermediate dispersal rates, regional persistence times increase logarithmically and exponentially (respectively) with increasing numbers of patches, whereas under high dispersal, the form of the relationship depends on local dynamics. Furthermore, we demonstrate that the forms of these relationships, which give rise to the bell-shaped relationship between dispersal rate and persistence time, are a product of recolonization and the region-wide synchronization (or lack thereof) of population dynamics. Identifying such metapopulation attributes that impact extinction risk is of utmost importance for managing and conserving the earth's evermore fragmented populations.     

Consequences of ignoring dispersal variation in network models for landscape connectivity

Habitat loss and fragmentation can negatively influence population persistence and biodiversity, but the effects can be mitigated if species successfully disperse between isolated habitat patches. Network models are the primary tool for quantifying landscape connectivity, yet in practice, an overly simplistic view of species dispersal is applied. These models often ignore individual variation in dispersal ability under the assumption that all individuals move the same fixed distance with equal probability. We developed a modeling approach to address this problem. We incorporated dispersal kernels into network models to determine how individual variation in dispersal alters understanding of landscape-level connectivity and implemented our approach on a fragmented grassland landscape in Minnesota. Ignoring dispersal variation consistently overestimated a population's robustness to local extinctions and underestimated its robustness to local habitat loss. Furthermore, a simplified view of dispersal underestimated the amount of habitat substructure for small populations but overestimated habitat substructure for large populations. Our results demonstrate that considering biologically realistic dispersal alters understanding of landscape connectivity in ecological theory and conservation practice.     

Comparing spatial conservation prioritization methods with site- versus spatial dependency-based connectivity

Larval dispersal is an important component of marine reserve networks. Two conceptually different approaches to incorporate dispersal connectivity into spatial planning of these networks exist, and it is an open question as to when either is most appropriate. Candidate reserve sites can be selected individually based on local properties of connectivity or on a spatial dependency-based approach of selecting clusters of strongly connected habitat patches. The first acts on individual sites, whereas the second acts on linked pairs of sites. We used a combination of larval dispersal simulations representing different seascapes and case studies of biophysical larval dispersal models in the Coral Triangle region and the province of Southeast Sulawesi, Indonesia, to compare the performance of these 2 methods in the spatial planning software Marxan. We explored the reserve design performance implications of different dispersal distances and patterns based on the equilibrium settlement of larvae in protected and unprotected areas. We further assessed different assumptions about metapopulation contributions from unprotected areas, including the case of 100% depletion and more moderate scenarios. The spatial dependency method was suitable when dispersal was limited, a high proportion of the area of interest was substantially degraded, or the target amount of habitat protected was low. Conversely, when subpopulations were well connected, the 100% depletion was relaxed, or more habitat was protected, protecting individual sites with high scores in metrics of connectivity was a better strategy. Spatial dependency methods generally produced more spatially clustered solutions with more benefits inside than outside reserves compared with site-based methods. Therefore, spatial dependency methods potentially provide better results for ecological persistence objectives over enhancing fisheries objectives, and vice versa. Different spatial prioritization methods of using connectivity are appropriate for different contexts, depending on dispersal characteristics, unprotected area contributions, habitat protection targets, and specific management objectives. Comparación entre los métodos de priorización de la conservación espacial con sitio y la conectividad espacial basada en la dependencia     

Epiphyte metapopulation dynamics are explained by species traits, connectivity, and patch dynamics

The colonization-extinction dynamics of many species are affected by the dynamics of their patches. For increasing our understanding of the metapopulation dynamics of sessile species confined to dynamic patches, we fitted a Bayesian incidence function model extended for dynamic landscapes to snapshot data on five epiphytic lichens among 2083 mapped oaks (dynamic patches). We estimate the age at which trees become suitable patches for different species, which defines their niche breadth (number of suitable trees). We show that the colonization rates were generally low, but increased with increasing connectivity in accordance with metapopulation theory. The rates were related to species traits, and we show, for the first time, that they are higher for species with wide niches and small dispersal propagules than for species with narrow niches or large propagules. We also show frequent long-distance dispersal in epiphytes by quantifying the relative importance of local dispersal and background deposition of dispersal propagules. Local stochastic extinctions from intact trees were negligible in all study species, and thus, the extinction rate is set by the rate of patch destruction (tree fall). These findings mean that epiphyte metapopulations may have slow colonization-extinction dynamics that are explained by connectivity, species traits, and patch dynamics.     

Populations in small, ephemeral habitat patches may drive dynamics in a Daphnia magna metapopulation

Migration is the key process to understand the dynamics and persistence of a metapopulation. Many metapopulation models assume a positive correlation between habitat patch size or stability and the number of emigrants. However, few empirical data exist, and habitat patch size and habitat stability may affect dispersal differently than they affect local persistence. Here, we studied the production of the migration stage (i.e., resting eggs called ephippia) of the cladoceran Daphnia magna in a metapopulation consisting of 530 rock pool habitat patches over 25 years. Earlier, the functioning of this metapopulation was explained with a Levins-type metapopulation model or with a mainland-island metapopulation model, based on local extinction and colonization data or time series data, respectively. We used pool volume, hydroperiod length, and number of desiccation events to calculate per-pool production of ephippia (i.e., migration stages). We estimated that populations in small and ephemeral habitat patches produced more than half of the 250 000 to 1 million ephippia that were produced in the metapopulation as a whole per year between 1982 and 2006. Furthermore, these small populations contributed approximately 90% of the ephippia exposed during desiccation events, while the contribution of the long-lived populations in large pools was minimal. We term this an "inverse mainland-island" type metapopulation and propose that populations in small, ephemeral habitat patches may also be the driving force for metapopulation dynamics in other systems.     

Dispersal depression with habitat fragmentation in the bog fritillary butterfly

Habitat fragmentation is expected to impose strong selective pressures on dispersal rates. However, evolutionary responses of dispersal are not self-evident, since various selection pressures act in opposite directions. Here we disentangled the components of dispersal behavior in a metapopulation context using the Virtual Migration model, and we linked their variation to habitat fragmentation in the specialist butterfly Proclossiana eunomia. Our study provided a nearly unique opportunity to study how habitat fragmentation modifies dispersal at the landscape scale, as opposed to microlandscapes or simulation studies. Indeed, we studied the same species in four landscapes with various habitat fragmentation levels, in which large amounts of field data were collected and analyzed using similar methodologies. We showed the existence of quantitative variations in dispersal behavior correlated with increased fragmentation. Dispersal propensity from habitat patches (for a given patch size), and mortality during dispersal (for a given patch connectivity) were lower in more fragmented landscapes. We suggest that these were the consequences of two different evolutionary responses of dispersal behavior at the individual level: (1) when fragmentation increased, the reluctance of individuals to cross habitat patch boundaries also increased; (2) when individuals dispersed, they flew straighter in the matrix, which is the best strategy to improve dispersal success. Such evolutionary responses could generate complex nonlinear patterns of dispersal changes at the metapopulation level according to habitat fragmentation. Due to the small size and increased isolation of habitat patches in fragmented landscapes, overall emigration rate and mortality during dispersal remained high. As a consequence, successful dispersal at the metapopulation scale remained limited. Therefore, to what extent the selection of individuals with a lower dispersal propensity and a higher survival during dispersal is able to limit detrimental effects of habitat fragmentation on dispersal success is unknown, and any conclusion that metapopulations would compensate for them is flawed.     

Effects of matrix characteristics and interpatch distance on functional connectivity in fragmented temperate rainforests

The connectivity of remnant patches of habitat may affect the persistence of species in fragmented landscapes. We evaluated the effects of the structural connectivity of forest patches (i.e., distance between patches) and matrix class (land-cover type) on the functional connectivity of 3 bird species (the White-crested Elaenia [Elaenia albiceps], the Green-backed Firecrown Hummingbird [Sephanoides sephaniodes], and the Austral Thrush [Turdus falklandii]). We measured functional connectivity as the rate at which each species crossed from one patch to another. We also evaluated whether greater functional connectivity translated into greater ecological connectivity (dispersal of fruit and pollen) by comparing among forest patches fruit set of a plant pollinated by hummingbirds and abundance of seedlings and adults of 2 plants with bird- and wind-dispersed seeds. Interpatch distance was strongly associated with functional connectivity, but its effect was not independent of matrix class. For one of the bird-dispersed plants, greater functional connectivity for White-crested Elaenias and Austral Thrushes (both frugivores) was associated with higher densities of this plant. The lack of a similar association for the wind-dispersed species suggests this effect is linked to the dispersal vector. The abundance of the hummingbird-pollinated species was not related to the presence of hummingbirds. Interpatch distance and matrix class affect animal movement in fragmented landscapes and may have a cascading effect on the distribution of some animal-dispersed species. On the basis of our results, we believe effort should be invested in optimizing patch configuration and modifying the matrix so as to mitigate the effects of patch isolation in fragmented landscapes.     

The control of emigration and its consequences for the survival of populations

Dispersal is the key process enhancing the long-term persistence of metapopulations in heterogeneous and dynamic landscapes. However, any individual emigrating from a occupied patch also increases the risk of local population extinction. The consequences of this increase for metapopulation persistence likely depend on the control of emigration. In this paper, we present results of individual-based simulations to compare the consequences of density-independent (DIE) and density-dependent (DDE) emigration on the extinction risk of local populations and a two-patch metapopulation. (1) For completely isolated patches extinction risk increases linearly with realised emigration rates in the DIE scenario. (2) For the DDE scenario extinction risk is nearly insensitive to emigration as longs as emigration probabilities remain below ≈0.2. Survival chances are up to half an order of magnitude larger than for populations with DIE. (3) For low dispersal mortality both modes of emigration increase survival of a metapopulation by ca. one order of magnitude. (4) For high dispersal mortality only DDE can improve the global survival chances of the metapopulation. (5) With DDE individuals are only removed from a population at high population density and the risk of extinction due to demographic stochasticity is thus much smaller compared to the DIE scenario.With density-dependent emigration prospects of metapopulations survival may thus be much higher compared to a system with density-independent emigration. Consequently, the knowledge about the factors driving emigration may significantly affect our conclusions concerning the conservation status of species.     

Metapopulation persistence in a dynamic landscape: more habitat or better stewardship?

Habitat loss and fragmentation has created metapopulations where there were once continuous populations. Ecologists and conservation biologists have become interested in the optimal way to manage and conserve such metapopulations. Several authors have considered the effect of patch disturbance and recovery on metapopulation persistence, but almost all such studies assume that every patch is equally susceptible to disturbance. We investigated the influence of protecting patches from disturbance on metapopulation persistence, and used a stochastic metapopulation model to answer the question: How can we optimally trade off returns from protection of patches vs. creation of patches? We considered the problem of finding, under budgetary constraints, the optimal combination of increasing the number of patches in the metapopulation network vs. increasing the number of protected patches in the network. We discovered that the optimal trade-off is dependent upon all of the properties of the system: the species dynamics, the dynamics of the landscape, and the relative costs of each action. A stochastic model and accompanying methodology are provided allowing a manager to determine the optimal policy for small metapopulations. We also provide two approximations, including a rule of thumb, for determining the optimal policy for larger metapopulations. The method is illustrated with an example inspired by information for the greater bilby, Macrotis lagotis, inhabiting southwestern Queensland, Australia. We found that given realistic costs for each action, protection of patches should be prioritized over patch creation for improving the persistence of the greater bilby during the next 20 years.     

Cougar Dispersal Patterns, Metapopulation Dynamics, and Conservation

Abstract: We examined cougar (   Puma concolor ) dispersal, emigration, and immigration in the San Andres Mountains, New Mexico, from 1985 to 1995 to quantify the effects of dispersal on the local population and surrounding subpopulations. We captured, tagged, and radio-collared animals to detect the arrival of new immigrants and dispersal characteristics of progeny. We found that cougars in southern New Mexico exhibited a metapopulation structure in which cougar subpopulations were separated by expanses of noncougar habitat and linked by dispersers. Of 43 progeny (n = 20 males , 23 females ) studied after independence, only 13 females exhibited philopatric behavior. Males dispersed significantly farther than females, were more likely to traverse large expanses of noncougar habitat, and were probably most responsible for nuclear gene flow between habitat patches. We estimated that an average of 8.5 progeny (i.e., cougars born in the study area) successfully emigrated from and 4.3 cougars successfully immigrated to the San Andres Mountains each year. Concurrently, an average of 4.1 progeny were recruited into the San Andres cougar population. Protected cougar subpopulations can contribute to metapopulation persistence by supplying immigrants to surrounding subpopulations that are affected by fragmentation or offtake by humans. Cougar population dynamics and dispersal behavior dictate that cougar management and conservation should be considered on a regional scale.     

The transition from isolated patches to a metapopulation in the eastern collared lizard in response to prescribed fires

Habitat fragmentation often arises from human-induced alterations to the matrix that reduce or eliminate dispersal between habitat patches. Elimination of dispersal increases local extinction and decreases recolonization. These phenomena were observed in the eastern collared lizard (Crotaphytus collaris collaris), which lives in the mid-continental highland region of the Ozarks (Missouri, USA) on glades: habitats of exposed bedrock that form desert-like habitats imbedded in a woodland matrix. With the onset of woodland fire suppression, glade habitats degenerated and the woodland matrix was altered to create a strong barrier to dispersal. By 1980, lizard populations in the Ozarks were rapidly going extinct. In response to this decline, some glades were restored by clearing and burning. Starting in 1984, collared lizard populations were translocated onto these restored habitats. The translocated populations persisted but did not colonize nearby glades or disperse among one another. In 1994 prescribed woodland fires were initiated, which unleashed much dispersal and colonizing behavior. Dispersal was highly nonrandom by both intrinsic variables (age, gender) and extrinsic variables (overall demography, glade population sizes, glade areas, landscape features), resulting in different classes of lizards being dominant in creating demographic cohesiveness among glades, colonizing new glades on a mountain, and colonizing new mountain systems. A dramatic transition was documented from isolated fragments, to a nonequilibrium colonizing metapopulation, and finally to a stable metapopulation. This transition is characterized by the convergence of rates of extinction and recolonization and a major alteration of dispersal probabilities and pattern in going from the nonequilibrium to stable metapopulation states.     

Conservation Corridors and Contagious Disease: A Cautionary Note

Recent conservation proposals frequently include the establishment of corridors to connect isolated patches of wildlife habitat. Much attention has been focused on the potential benefits of corridors with little note given to potentially adverse consequences. A simulation model is developed here to study the effect of corridors on the survival of a metapopulation in the presence of a fatal disease that is spread by direct contact between susceptible and infected individuals. For the disease modeled here, a landscape of patches connected by corridors generally suffers fewer metapopulation extinctions than a landscape of isolated patches. However, under a narrow range of conditions, results suggest that corridors may dramatically increase the probability of metapopulation extinction. This occurs when disease-induced mortality is low enough to allow infected individuals to spread the disease, but high enough to reduce population levels to the point that random demographic and environmental events cause frequent metapopulation extinctions. This has important implications for the design and management of conservation reserve networks. Although discussion focuses primarily on conservation corridors, the model results apply to any management techniques that increase the movement of individuals among populations.     

Modelling dispersal with diffusion and habitat selection: Analytical results for highly fragmented landscapes

Quantifying dispersal is crucial both for understanding ecological population dynamics, and for gaining insight into factors that affect the genetic structure of populations. The role of dispersal becomes pronounced in highly fragmented landscapes inhabited by spatially structured populations. We consider a landscape consisting of a set of habitat patches surrounded by unsuitable matrix, and model dispersal by assuming that the individuals follow a random walk with parameters that may be specific to the habitat type. We allow for spatial variation in patch quality, and account for edge-mediated behavior, the latter meaning that the individuals bias their movement towards the patches when close to an edge between a patch and the matrix. We employ a diffusion approximation of the random walk model to derive analytical expressions for various characteristics of the dispersal process. For example, we derive formulae for the time that an individual is expected to spend in its current patch i, and for the time that it will spend in the matrix, both conditional on the individual hitting next a given patch j before hitting any of the other patches or dying. The analytical formulae are based on the assumptions that the landscape is infinitely large, that the patches are circularly shaped, and that the patches are small compared to interpatch distances. We evaluate the effect of these assumptions by comparing the analytical results to numerical results in a real patch network that violates all of the three assumptions. We then consider a landscape that fulfills the assumptions, and show that in this case the analytical results are in a very good agreement with the numerical results. The results obtained here allow the construction of computationally efficient dispersal models that can be used as components of metapopulation models.