A Stochastic Metapopulation Model with Variability in Patch Size and Position

Analytically tractable metapopulation models usually assume that every patch is identical, which limits their application to real metapopulations. We describe a new single species model of metapopulation dynamics that allows variation in patch size and position. The state of the metapopulation is defined by the presence or absence of the species in each patch. For a system of n patches, this gives 2ⁿ possible states. We show how to construct and analyse a matrix describing transitions between all possible states by first constructing separate extinction and colonisation matrices. We illustrate the model′s application to metapopulations by considering an example of malleefowl, Leipoa ocellata, in southern Australia, and calculate extinction probabilities and quasi-stationary distributions. We investigate the relative importance of modelling the particular arrangement of patches and the variation in patch sizes for this metapopulation and we use the model to examine the effects of further habitat loss on extinction probabilities.     

Colonization and extinction dynamics of an annual plant metapopulation in an urban environment

The metapopulation framework considers that the spatiotemporal distribution of organisms results from a balance between the colonization and extinction of populations in a suitable and discrete habitat network. Recent spatially realistic metapopulation models have allowed patch dynamics to be investigated in natural populations but such models have rarely been applied to plants. Using a simple urban fragmented population system in which favourable habitat can be easily mapped, we studied patch dynamics in the annual plant Crepis sancta (Asteraceae). Using stochastic patch occupancy models (SPOMs) and multi‐year occupancy data we dissected extinction and colonization patterns in our system. Overall, our data were consistent with two distinct metapopulation scenarios. A metapopulation (sensu stricto) dynamic in which colonization occurs over a short distance and extinction is lowered by nearby occupied patches (rescue effect) was found in a set of patches close to the city centre, while a propagule rain model in which colonization occurs from a large external population was most consistent with data from other networks. Overall, the study highlights the importance of external seed sources in urban patch dynamics. Our analysis emphasizes the fact that plant distributions are governed not only by habitat properties but also by the intrinsic properties of colonization and dispersal of species. The metapopulation approach provides a valuable tool for understanding how colonization and extinction shape occupancy patterns in highly fragmented plant populations. Finally, this study points to the potential utility of more complex plant metapopulation models than traditionally used for analysing ecological and evolutionary processes in natural metapopulations.     

Structured models of metapopulation dynamics

I develop models of metapopulation dynamics that describe changes in the numbers of individuals within patches. These models are analogous to structured population models, with patches playing the role of individuals. Single species models which do not include the effect of immigration on local population dynamics of occupied patches typically lead to a unique equilibrium. The models can be used to study the distributions of numbers of individuals among patches, showing that both metapopulations with local outbreaks and metapopulations without outbreaks can occur in systems with no underlying environmental variability. Distributions of local population sizes (in occupied patches) can vary independently of the total population size, so both patterns of distributions of local population sizes are compatible with either rare or common species. Models which include the effect of immigration on local population dynamics can lead to two positive equilibria, one stable and one unstable, the latter representing a threshold between regional extinction and persistence.     

The relative contribution of local habitat and landscape context to metapopulation processes: a dynamic occupancy modeling approach

Changes in site occupancy across habitat patches have often been attributed to landscape features in fragmented systems, particularly when considering metapopulations. However, failure to include habitat quality of individual patches can mask the relative importance of local scale features in determining distributional changes. We employed dynamic occupancy modeling to compare the strength of local habitat variables and metrics of landscape patterns as drivers of metapopulation dynamics for a vulnerable, high‐elevation species in a naturally fragmented landscape. Repeat surveys of Bicknell's thrush Catharus bicknelli presence/non‐detection were conducted at 88 sites across Vermont, USA in 2006 and 2007. We used an organism‐based approach, such that at each site we measured important local‐scale habitat characteristics and quantified landscape‐scale features using a predictive habitat model for this species. We performed a principal component analysis on both the local and landscape features to reduce dimensionality. We estimated site occupancy, colonization, and extinction probabilities while accounting for imperfect detection. Univariate, additive, and interaction models of local habitat and landscape context were ranked using AICc scores. Both local and landscape scales were important in determining changes in occupancy patterns. An interaction between scales was detected for occupancy dynamics indicating that the relationship of the parameters to local‐scale habitat conditions can change depending on the landscape context and vice versa. An increase in both landscape‐ and local‐scale habitat quality increased occupancy and colonization probability while decreasing extinction risk. Colonization and extinction were both more strongly influenced by local habitat quality relative to landscape patterns. We also identified clear, qualitative thresholds for landscape‐scale features. Conservation of large habitat patches in high‐cover landscapes will help ensure persistence of Bicknell's thrushes, but only if local scale habitat quality is maintained. Our results highlight the importance of incorporating information beyond landscape characteristics when investigating patch occupancy patterns in metapopulations.     

Colonization–extinction dynamics of epixylic lichens along a decay gradient in a dynamic landscape

Metapopulation models are often used for understanding and predicting species dynamics in fragmented landscapes. Several models have been proposed depending on e.g. the relative importance of patch dynamics on the metapopulation dynamics. Dead wood is a dynamic substrate patch, and species that are confined to such patches have experienced a high degree of habitat loss in managed forests. Little is, however, known about how the population dynamics of epixylic species are affected by the fast dynamics of their substrate patches. We quantified the effect of local patch conditions and metapopulation processes on colonizations and extinctions of epixylic lichen species in a managed boreal forest landscape. This was done by twice surveying seven lichen metapopulations on 293 stumps in 30 stands of ages covering the duration of the dynamic patches (stumps). We also investigated the relative importance of local stochastic extinctions from stumps that remained available, and deterministic extinctions due to stump surface disappearance. We found importance of a decay gradient, surrounding metapopulation size, and local population sizes, in driving the colonization–extinction dynamics of epixylic lichens. The species were sorted along the stump decay gradient. Increasing surrounding metapopulation size was associated with increased colonization rates, and increasing local population size decreased lichen extinction rates. Finally, both local stochastic extinctions and deterministic extinctions due to patch disappearance occur, confirming that the long‐term persistence of epixylic lichens depends on colonization rates that compensate for stochastic population extinctions as well as deterministic extinctions.     

Spatial Heterogeneity in Habitat Quality and Cross-Scale Interactions in Metapopulations

Abstract Integration of habitat heterogeneity into spatially realistic metapopulation approaches reveals the potential for key cross-scale interactions. Broad-scale environmental gradients and land-use practices can create autocorrelation of habitat quality of suitable patches at intermediate spatial scales. Patch occupancy then depends not only on habitat quality at the patch scale but also on feedbacks from surrounding neighborhoods of autocorrelated patches. Metapopulation dynamics emerge from how demographic and dispersal processes interact with relevant habitat heterogeneity. We provide an empirical example from a metapopulation of round-tailed muskrats (Neofiber alleni) in which habitat quality of suitable patches was spatially autocorrelated most strongly within 1,000 m, which was within the expected dispersal range of the species. After controlling for factors typically considered in metapopulation studies—patch size, local patch quality, patch connectivity—we use a cross-variogram analysis to demonstrate that patch occupancy by muskrats was correlated with habitat quality across scales ≤1,171 m. We also discuss general consequences of spatial heterogeneity of habitat quality for metapopulations related to potential cross-scale interactions. We focus on spatially correlated extinctions and metapopulation persistence, hierarchical scaling of source–sink dynamics, and dispersal decisions by individuals in relation to information constraints.     

Spatial population structure in the banner-tailed kangaroo rat,Dipodomys spectabilis

I attempted to characterize spatial units of local dynamics and dispersal in banner-tailed kangaroo rats (Dipodomys spectabilis), to determine if spatial structure influenced population dynamics in the way predicted by current metapopulation models. D. spectabilis exhibited a hierarchical spatial structure. Local populations that appeared as discrete entities on a scale of kilometers were subdivided into clusters of mounds on a scale of meters. This structure, however, cannot be characyerized in terms of the discrete habitat patches envisioned by the metapopulation models. Occupied areas were statistically distinguishable from the surrounding matrix, but this difference was only quantitative. There were no discrete boundaries between occupied areas and the matrix. Habitat within occupied areas was heterogeneous, and occupied areas in different locations were statistically distinguishable from each other. High heterogeneity within occupied areas, and high contrast among them, make it difficult to define what is a suitable habitat patch for D. spectabilis. On a smaller spatial scale, there was significant aggregation of resident mounds within occupied areas. These aggregations, however, do not correspond to discrete habitat patches. Rather, they appear to result from an interaction between fine-scale habitat heterogeneity and limited dispersal due to natal philopatry and low adult vagility. These complications make it difficult to identify habitat patches independent of the species' distribution. For species like D. spectabilis that are patchily distributed but do not occupy discrete habitat patches, a patch occupancy approach does not seem appropriate for describing spatial structure. Hierarchical spatial structure underscores the need for a framework that incorporates multiple scales of spatial structure, rather than one that pre-imposes a single spatial scale as being important for population dynamics. A framework that (i) considers patchiness as a combination of both habitat heterogeneity, and life-history and behavioral characteristics, and (ii) incorporates hierarchical spatial structure, appears to be the most suitable for conceptualizing spatial dynamics of behaviorally complex vertebrates such as D. spectabilis.     

Toward ecologically scaled landscape indices

Nature conservation is increasingly based on a landscape approach rather than a species approach. Landscape planning that includes nature conservation goals requires integrated ecological tools. However, species differ widely in their response to landscape change. We propose a framework of ecologically scaled landscape indices that takes into account this variation. Our approach is based on a combination of field studies of spatially structured populations (metapopulations) and model simulations in artificial landscapes. From these, we seek generalities in the relationship among species features, landscape indices, and metapopulation viability. The concept of ecological species profiles is used to group species according to characteristics that are important in metapopulations' response to landscape change: individual area requirements as the dominant characteristic of extinction risk in landscape patches and dispersal distance as the main determinant of the ability to colonize patches. The ecological profiles and landscape indices are then integrated into two ecologically scaled landscape indices (ESLI): average patch carrying capacity and average patch connectivity. The field data show that the fraction of occupied habitat patches is correlated with the two ESLI. To put the ESLI into a perspective of metapopulation persistence, we determine the viability for six ecological profiles at different degrees of habitat fragmentation using a metapopulation model and computer-generated landscapes. The model results show that the fraction of occupied patches is a good indicator for metapopulation viability. We discuss how ecological profiles, ESLI, and the viability threshold can be applied for landscape planning and design in nature conservation.     

The influence of patch demographics on metapopulations, with particular reference to successional landscapes

The classical view of metapopulations relates the regional abundance of a species to the balance between the extinction and colonization dynamics of identical local populations. Species in successional landscapes may represent the most appropriate examples of classical metapopulations. However, Levins‐type metapopulation models do not explicitly separate population loss due to successional habitat change from other causes of extinction. A further complication is that the chance of population loss due to successional habitat change may be related to the age of a patch. I developed simple patch occupancy models to include succession and included consideration of patch age structure to address two related questions: what are the implications of changes in patch demographic rates and when is a move to a structured patch occupancy model justified? Age‐related variation in patch demography could increase or decrease the equilibrium fraction of the available habitat occupied by a species when compared to the predictions of an unstructured model. Metapopulation persistence was enhanced when the age class of patches with the highest species occupancy suffered relatively low losses to habitat succession. Conversely, when the age class of patches with the highest species occupancy also had relatively high successional loss rates, extinction thresholds were higher that would be predicted by a simple unstructured model. Hence age‐related variation in patch successional rate introduces biases into the predictions of simple unstructured models. Such biases can be detected from field surveys of the fraction of occupied and unoccupied patches in each age class. Where a bias is demonstrated, unstructured models will not be adequate for making predictions about the effects of changing parameters on metapopulation size. Thinking in successional terms emphasizes how landscapes might be managed to enhance or reduce the patch occupancy by any particular metapopulation     

The use of spatial concepts as a basis for designing a viable-habitat network: Conserving redstart (Phoenicurus phoenicurus) populations in Sherwood Forest, England

This paper presents a method for the design of a habitat network in a fragmented landscape. It employs metapopulation dynamics and connectivity to develop spatial rules which, together with the requirements of a target species, can be used to guide the creation of a network to improve population persistence. Using these guidelines, two conservation strategies of patch enlargement and corridor creation are developed. The spatial models used to produce the habitat network are described. The resulting maps show the network consisting of the source patches that could be occupied by stable populations and a number of patches, which could support viable local populations, situated in the proximity of the sources or linked to them. The approach could be used to inform decision-making in nature conservation and landscape planning.     

Habitat quality of source patches and connectivity in fragmented landscapes

Because spatial connectivity is critical to dispersal success and persistence of species in highly fragmented landscapes, the way that we envision and measure connectivity is consequential for biodiversity conservation. Connectivity metrics used for predictive modeling of spatial turnover and patch occupancy for metapopulations, such as with Incidence Function Models (IFM), incorporate distances to and sizes of possible source populations. Here, our focus is on whether habitat quality of source patches also is considered in these connectivity metrics. We propose that effective areas (weighted by habitat quality) of source patches should be better surrogates for population size and dispersal potential compared to unadjusted patch areas. Our review of a representative sample of the literature revealed that only 12.5% of studies incorporated habitat quality of source patches into IFM-type connectivity metrics. Quality of source patches generally was not taken into account in studies even if habitat quality of focal patches was included in analyses. We provide an empirical example for a metapopulation of a rare wetland species, the round-tailed muskrat (Neofiber alleni), demonstrating that a connectivity metric based on effective areas of source patches better predicts patch colonization and occupancy than a metric that used simple patch areas. The ongoing integration of landscape ecology and metapopulation dynamics could be hastened by incorporating habitat quality of source patches into spatial connectivity metrics applied to species conservation in fragmented landscapes.     

Experimental beetle metapopulations respond positively to dynamic landscapes and reduced connectivity

Interactive effects of multiple environmental factors on metapopulation dynamics have received scant attention. We designed a laboratory study to test hypotheses regarding interactive effects of factors affecting the metapopulation dynamics of red flour beetle, Tribolium castaneum. Within a four-patch landscape we modified resource level (constant and diminishing), patch connectivity (high and low) and patch configuration (static and dynamic) to conduct a 2(3) factorial experiment, consisting of 8 metapopulations, each with 3 replicates. For comparison, two control populations consisting of isolated and static subpopulations were provided with resources at constant or diminishing levels. Longitudinal data from 22 tri-weekly counts of beetle abundance were analyzed using bayesian Poisson generalized linear mixed models to estimate additive and interactive effects of factors affecting abundance. Constant resource levels, low connectivity and dynamic patches yielded greater levels of adult beetle abundance. For a given resource level, frequency of colonization exceeded extinction in landscapes with dynamic patches when connectivity was low, thereby promoting greater patch occupancy. Negative density dependence of pupae on adults occurred and was stronger in landscapes with low connectivity and constant resources; these metapopulations also demonstrated greatest stability. Metapopulations in control landscapes went extinct quickly, denoting lower persistence than comparable landscapes with low connectivity. When landscape carrying capacity was constant, habitat destruction coupled with low connectivity created asynchronous local dynamics and refugia within which cannibalism of pupae was reduced. Increasing connectivity may be counter-productive and habitat destruction/recreation may be beneficial to species in some contexts.     

Single-species metapopulation dynamics: concepts, models and observations

This paper outlines a conceptual and theoretical framework for single-species metapopulation dynamics based on the Levins model and its variants. The significance of the following factors to metapopulation dynamics are explored: evolutionary changes in colonization ability; habitat patch size and isolation; compensatory effects between colonization and extinction rates; the effect of immigration on local dynamics (the rescue effect); and heterogeneity among habitat patches. The rescue effect may lead to alternative stable equilibria in metapopulation dynamics. Heterogeneity among habitat patches may give rise to a bimodal equilibrium distribution of the fraction of patches occupied in an assemblage of species (the core-satellite distribution). A new model of incidence functions is described, which allows one to estimate species' colonization and extinction rates on islands colonized from mainland. Four distinct kinds of stochasticity affecting metapopulation dynamics are discussed with examples. The concluding section describes four possible scenarios of metapopulation extinction.     

Extinction dynamics in mainland-island metapopulations: an N-patch stochastic model

A generalization of the well-known Levins’ model of metapopulations is studied. The generalization consists of (i) the introduction of immigration from a mainland, and (ii) assuming the dynamics is stochastic, rather than deterministic. A master equation, for the probability that n of the patches are occupied, is derived and the stationary probability P _s (n), together with the mean and higher moments in the stationary state, determined. The time-dependence of the probability distribution is also studied: through a Gaussian approximation for general n when the boundary at n = 0 has little effect, and by calculating P(0, t), the probability that no patches are occupied at time t, by using a linearization procedure. These analytic calculations are supplemented by carrying out numerical solutions of the master equation and simulations of the stochastic process. The various approaches are in very good agreement with each other. This allows us to use the forms for P _s 0) and P(0, t) in the linearization approximation as a basis for calculating the mean time for a metapopulation to become extinct. We give an analytical expression for the mean time to extinction derived within a mean field approach. We devise a simple method to apply our mean field approach even to complex patch networks in realistic model metapopulations. After studying two spatially extended versions of this nonspatial metapopulation model—a lattice metapopulation model and a spatially realistic model—we conclude that our analytical formula for the mean extinction time is generally applicable to those metapopulations which are really endangered, where extinction dynamics dominates over local colonization processes. The time evolution and, in particular, the scope of our analytical results, are studied by comparing these different models with the analytical approach for various values of the parameters: the rates of immigration from the mainland, the rates of colonization and extinction, and the number of patches making up the metapopulation.     

Population turnover and habitat dynamics in Notonecta (Hemiptera: Notonectidae) metapopulations

Simple metapopulation models assume that local populations occur in patches of uniform quality habitat separated by non-habitat. However field metapopulations tend to show considerable spatial and temporal variation in patch quality, and hence probability of occupancy. This may have implications for the adequacy of simple metapopulation models in describing and predicting regional population dynamics of natural systems. This study investigated the effects of habitat characteristics on landscape-scale occupancy dynamics of two species of backswimmer (Notonecta, Hemiptera: Notonectidae) in small freshwater ponds. The results demonstrated clear links between habitat, pond occupancy and population turnover, particularly local extinction. There were considerable changes in the habitat of individual ponds between years, but local changes were not spatially correlated and the frequency distribution of habitat conditions at the landscape level remained similar in different years. Stable occupancy levels of Notonecta species appears to result from a balance of the rates of creation and loss of suitable habitat due to spatially uncorrelated habitat change. Systems such as this, where turnover is driven by habitat dynamics, demonstrate the potential value of incorporating the dynamics of habitat change into metapopulation models. Such developments are likely to improve predictions of landscape-scale occupancy dynamics, whilst also allowing patch-level predictions of occupancy, based on local habitat conditions. Received: 18 August 1999 / Accepted: 3 December 1999     

Merging spatial and temporal structure within a metapopulation model

Current research recognizes that both the spatial and temporal structure of the landscape influence species persistence. Patch models that incorporate the spatial structure of the landscape have been used to investigate static habitat destruction by comparing persistence results within nested landscapes. Other researchers have incorporated temporal structure into their models by making habitat suitability a dynamic feature of the landscape. In this article, we present a spatially realistic patch model that allows patches to be in one of three states: uninhabitable, habitable, or occupied. The model is analytically tractable and allows us to explore the interactions between the spatial and temporal structure of the landscape as perceived by the target species. Extinction thresholds are derived that depend on habitat suitability, mean lifetime of a patch, and metapopulation capacity. We find that a species is able to tolerate more ephemeral destruction, provided that the rate of the destruction does not exceed the scale of its own metapopulation dynamics, which is dictated by natural history characteristics and the spatial structure of the landscape. This model allows for an expansion of the classic definition of a patch and should prove useful when considering species inhabiting complex dynamic landscapes, for example, agricultural landscapes.     

Area-dependent migration by ringlet butterflies generates a mixture of patchy population and metapopulation attributes

Interpretation of spatially structured population systems is critically dependent on levels of migration between habitat patches. If there is considerable movement, with each individual visiting several patches, there is one ”patchy population”; if there is intermediate movement, with most individuals staying within their natal patch, there is a metapopulation; and if (virtually) no movement occurs, then the populations are separate (Harrison 1991, 1994). These population types actually represent points along a continuum of much to no mobility in relation to patch structure. Therefore, interpretation of the effects of spatial structure on the dynamics of a population system must be accompanied by information on mobility. We use empirical data on movements by ringlet butterflies, Aphantopus hyperantus, to investigate two key issues that need to be resolved in spatially-structured population systems. First, do local habitat patches contain largely independent local populations (the unit of a metapopulation), or merely aggregations of adult butterflies (as in patchy populations)? Second, what are the effects of patch area on migration in and out of the patches, since patch area varies considerably within most real population systems, and because human landscape modification usually results in changes in habitat patch sizes? Mark-release-recapture (MRR) data from two spatially structured study systems showed that 63% and 79% of recaptures remained in the same patch, and thus it seems reasonable to call both systems metapopulations, with some capacity for separate local dynamics to take place in different local patches. Per capita immigration and emigration rates declined with increasing patch area, while the resident fraction increased. Actual numbers of emigrants either stayed the same or increased with area. The effect of patch area on movement of individuals in the system are exactly what we would have expected if A. hyperantus were responding to habitat geometry. Large patches acted as local populations (metapopulation units) and small patches simply as locations with aggregations (units of patchy populations), all within 0.5 km². Perhaps not unusually, our study system appears to contain a mixture of metapopulation and patchy-population attributes.     

异质种群动态模型:破碎化景观动态模拟的新途径

景观破碎化导致物种以异质种群方式存活,使得基于异质种群动态模拟破碎化景观动态成为可能。异质种群动态模型的发展为景观动态模拟奠定了良好基础。根据空间处理方式的不同,异质种群模型可分为三大类,可不同程度地用于描述破碎化景观动态。(1)空间不确定异质种群模型,假定所有局域种群间均等互联,模型中不包含空间信息,仅能用于景观斑块动态描述;(2)空间确定异质种群模型,假设局域种群在二维空间上以规则格子形式排列,是一种准现实的空间处理方式,可用于景观动态的简单描述;(3)空间现实异质种群模型,包含了破碎化景观中局域种群的几何特征,可直接用于真实景观动态的模拟研究。空间现实的和基于个体的异质种群模型不但是未来异质种群模型发展的主流,也将成为未来破碎化景观动态研究的重要工具。为了更加准确完整地描述破碎化景观动态,不但应该综合运用已有的各种异质种群模型方法,更要引进新模型来刎画多物种、多变量、高维度、复杂连接的破碎化景观格局与过程。     

Metapopulation theory for fragmented landscapes

We review recent developments in spatially realistic metapopulation theory, which leads to quantitative models of the dynamics of species inhabiting highly fragmented landscapes. Our emphasis is in stochastic patch occupancy models, which describe the presence or absence of the focal species in habitat patches. We discuss a number of ecologically important quantities that can be derived from the full stochastic models and their deterministic approximations, with a particular aim of characterizing the respective roles of the structure of the landscape and the properties of the species. These quantities include the threshold condition for persistence, the contributions that individual habitat patches make to metapopulation dynamics and persistence, the time to metapopulation extinction, and the effective size of a metapopulation living in a heterogeneous patch network.     

Relative importance of host plant patch geometry and habitat quality on the patterns of occupancy,extinction and density of the monophagous butterfly <Emphasis Type="Italic">Iolana iolas</Emphasis>

Habitat fragmentation is a major cause of species rarity and decline because it increases local population extinctions and reduces recolonisation rates of remnant patches. Although two major patch characteristics (area and connectivity) have been used to predict distribution patterns in fragmented landscapes, other factors can affect the occurrence of a species as well as the probability of it becoming extinct. In this paper, we study the spatial structure and dynamics of the butterfly Iolana iolas in a 75-patch network of its host plant (Colutea hispanica) to determine the relative importance of patch area, connectivity and habitat quality characteristics on occupancy, extinction and density over the period 2003–2006. Occupancy in 2003, incidence (proportion of years occupied) and probability of extinction were mostly affected by patch area. Smaller patches were less likely to be occupied because they had a higher probability of extinction, partly due to environmental stochasticity. The density of I. iolas was negatively related to patch area in all study years. Only in 2004 was the density of I. iolas positively influenced by fruit production per plant. Our results suggest that for I. iolas, and probably for other specialist butterflies with clearly delimited resource requirements, metapopulation dynamics can be satisfactorily predicted using only geometric variables because most habitat characteristics are subsumed in patch area. However, this hypothesis should be subject to further testing under diverse environmental conditions to evaluate the extent of its generalisation.