Spatially structured metapopulation models: global and local assessment of metapopulation capacity

We model metapopulation dynamics in finite networks of discrete habitat patches with given areas and spatial locations. We define and analyze two simple and ecologically intuitive measures of the capacity of the habitat patch network to support a viable metapopulation. Metapopulation persistence capacity lambda(M) defines the threshold condition for long-term metapopulation persistence as lambda(M)>delta, where delta is defined by the extinction and colonization rate parameters of the focal species. Metapopulation invasion capacity lambda(I) sets the condition for successful invasion of an empty network from one small local population as lambda(I)>delta. The metapopulation capacities lambda(M) and lambda(I) are defined as the leading eigenvalue or a comparable quantity of an appropriate "landscape" matrix. Based on these definitions, we present a classification of a very general class of deterministic, continuous-time and discrete-time metapopulation models. Two specific models are analyzed in greater detail: a spatially realistic version of the continuous-time Levins model and the discrete-time incidence function model with propagule size-dependent colonization rate and a rescue effect. In both models we assume that the extinction rate increases with decreasing patch area and that the colonization rate increases with patch connectivity. In the spatially realistic Levins model, the two types of metapopulation capacities coincide, whereas the incidence function model possesses a strong Allee effect characterized by lambda(I)=0. For these two models, we show that the metapopulation capacities can be considered as simple sums of contributions from individual habitat patches, given by the elements of the leading eigenvector or comparable quantities. We may therefore assess the significance of particular habitat patches, including new patches that might be added to the network, for the metapopulation capacities of the network as a whole. We derive useful approximations for both the threshold conditions and the equilibrium states in the two models. The metapopulation capacities and the measures of the dynamic significance of particular patches can be calculated for real patch networks for applications in metapopulation ecology, landscape ecology, and conservation biology.     

Long-term persistence of species and the SLOSS problem

The single large or several small (SLOSS) problem has been addressed in a large number of empirical and theoretical studies, but no coherent conclusion has yet been reached. Here I study the SLOSS problem in the context of metapopulation dynamics. I assume that there is a fixed total amount A(0) of habitat available, and I derive formulas for the optimal number n and area A of habitat patches, where n=A(0)/A. I consider optimality in two ways. First, I attempt to maximize the time to metapopulation extinction, which is a relevant measure for metapopulation viability for rare and threatened species. Second, I attempt to maximize the metapopulation capacity of the habitat patch network, which corresponds both with maximizing the distance to the deterministic extinction threshold and with maximizing the fraction of occupied patches. I show that in the typical case, a small number of large patches maximizes the metapopulation capacity, while an intermediate number of habitat patches maximizes the time to extinction. The main conclusion stemming from the analysis is that the optimal number of patches is largely affected by the relationship between habitat patch area and rates of immigration, emigration and local extinction. Here this relationship is summarized by a single factor zeta, termed the patch area scaling factor.     

Metapopulation dynamics of the bog fritillary butterfly: modelling the effect of habitat fragmentation

Population viability analysis (PVA) and metapopulation theory are valuable tools to model the dynamics of spatially structured populations. In this article we used a spatially realistic population dynamic model to simulate the trajectory of a Proclossiana eunomia metapopulation in a network of habitat patches located in the Belgian Ardenne. Sensitivity analysis was used to evaluate the relative influence of the different parameters on the model output. We simulated habitat loss by removing a percentage of the original habitat, proportionally in each habitat patch. Additionally, we evaluated isolation and fragmentation effects by removing and dividing habitat patches from the network, respectively. The model predicted a slow decline of the metapopulation size and occupancy. Extinction risks predicted by the model were highly sensitive to environmental stochasticity and carrying capacity. For a determined level of habitat destruction, the expected lifetime of the metapopulation was highly dependent on the spatial configuration of the landscape. Moreover, when the proportion of removed habitat is above 40% of the original habitat, the loss of whole patches invariably leads to the strongest reduction in metapopulation viability.     

Rapid assessment of metapopulation viability under climate and land-use change

In order to predict species response to climate and land-use change, numerically fast and easily applicable assessment tools for species survival are required. We present a set of formulae to calculate the mean lifetime of a metapopulation in a spatially heterogeneous and dynamic landscape subject to habitat patch diminution, loss and/or spatial shift of the habitat network. The formulae require as inputs (i) information about the number, location and size of the habitat patches for several time steps to quantify landscape dynamics in terms of patch destruction, diminution or shifting rates and (ii) data on species traits such as their vulnerability to environmental variation and their dispersal ability to quantify local colonisation and extinction rates. We validate the formulae with a spatially explicit simulation. The analysis is complemented by a protocol for the easy use of the approach and practical application examples. A software implementation is available on request from the authors.     

Matrix Matters:?Differences of Grand Skink Metapopulation Parameters in Native Tussock Grasslands and Exotic Pasture Grasslands

Modelling metapopulation dynamics is a potentially very powerful tool for conservation biologists. In recent years, scientists have broadened the range of variables incorporated into metapopulation modelling from using almost exclusively habitat patch size and isolation, to the inclusion of attributes of the matrix and habitat patch quality. We investigated the influence of habitat patch and matrix characteristics on the metapopulation parameters of a highly endangered lizard species, the New Zealand endemic grand skink (Oligosoma grande) taking into account incomplete detectability. The predictive ability of the developed zxmetapopulation model was assessed through cross-validation of the data and with an independent data-set. Grand skinks occur on scattered rock-outcrops surrounded by indigenous tussock (bunch) and pasture grasslands therefore implying a metapopulation structure. We found that the type of matrix surrounding the habitat patch was equally as important as the size of habitat patch for estimating occupancy, colonisation and extinction probabilities. Additionally, the type of matrix was more important than the physical distance between habitat patches for colonisation probabilities. Detection probability differed between habitat patches in the two matrix types and between habitat patches with different attributes such as habitat patch composition and abundance of vegetation on the outcrop. The developed metapopulation models can now be used for management decisions on area protection, monitoring, and the selection of translocation sites for the grand skink. Our study showed that it is important to incorporate not only habitat patch size and distance between habitat patches, but also those matrix type and habitat patch attributes which are vital in the ecology of the target species.     

The effective size of a metapopulation living in a heterogeneous patch network

I analyze stochastic patch occupancy models (SPOMs), which record habitat patches as empty or occupied. A problem with SPOMs has been that if the spatial structure of a heterogeneous habitat patch network is taken into account, the computational effort needed to analyze a SPOM grows as a power of 2n, where n is the number of habitat patches. I propose a computationally feasible approximation method, which approximates the behavior of a heterogeneous SPOM by an "ideal" metapopulation inhabiting a network of identical and equally connected habitat patches. The transformation to the ideal metapopulation is based on weighting the individual patch occupancies by the dynamic values of the habitat patches, which may be calculated from the deterministic mean-field approximation of the original SPOM. Conceptually, the method resembles the calculation of the effective size of a population in the context of population genetics. I demonstrate how the method may be applied to SPOMs with flexible structural assumptions and with spatially correlated and temporally varying parameter values. I apply the method to a real habitat patch network inhabited by the Glanville fritillary butterfly, illustrating that the metapopulation dynamics of this species are essentially driven by temporal variability in the environmental conditions.     

Are spatially correlated or uncorrelated disturbance regimes better for the survival of species?

The survival of species in dynamic landscapes (characterised by patch destruction and subsequent regeneration) depends on both the species' attributes and the disturbance pattern. Using a spatially explicit model we explored how the mean time to extinction of a metapopulation depends on the spatial correlation of patch destruction in relation to the population growth and dispersal abilities of species. Two contrasting answers are possible. On the one hand, increasing spatial correlation of patch destruction increases the spatial correlation of population growth and this is known to decrease metapopulation persistence. On the other hand, spatially correlated patch destruction and regeneration can lead to clustered habitat patches and this is known to increase metapopulation persistence. Therefore, we hypothesised that some species are better off under spatially correlated and alternatively uncorrelated disturbance regimes. However, contrary to this hypothesis, in all kinds of cases spatial correlation reduced metapopulation persistence. We found this to be due to the fact that the spatial correlation of patch destruction causes increasing temporal fluctuations in the regional carrying capacity of the metapopulation and is hence generally disadvantageous for long-term persistence. The main consequence for conservation biology is that reducing spatial correlation in disturbances is likely to be a reliable strategy in a dynamic landscape that will benefit practically all species with a low risk of adverse side effects .     

Predicting metapopulation lifetime from macroscopic network properties

This paper presents a comparatively simple approximation formula for the mean life time of a metapopulation in a habitat network where habitat patch arrangement may be irregular and patch sizes differ. It is based on previous work on the development of an analytical approximation formula by Frank and Wissel [K. Frank, C. Wissel, A formula for the mean lifetime of metapopulations in heterogeneous landscapes, Am. Nat. 159 (2002) 530] and extends it by abstracting from individual patch locations. The mean metapopulation lifetime is expressed as a function of four macroscopic network properties: the ratio of dispersal range and network size, the ratio of range of environmental correlation and network size, and the total number and (geometric mean) size of the patches. The analysis takes into account that (ceteris paribus) patches close to the boundary of the habitat network contribute less to metapopulation survival than patches close to the centre of the network. Ignoring this fact can lead to a substantial overestimation of the mean metapopulation lifetime. Due to its numerical simplicity, the formula can be used as a conservation objective function even in complex network design problems where the number of patches to be allocated is very large. Numerical tests of the formula show that it performs very well within a wide range of network structures.     

Using connectivity metrics in conservation planning – when does habitat quality matter?

Aim The objective of conservation planning is often to prioritize patches based on their estimated contribution to metapopulation or metacommunity viability. The contribution that an individual patch makes will depend on its intrinsic characteristics, such as habitat quality, as well as its location relative to other patches, its connectivity. Here we systematically evaluate five patch value metrics to determine the importance of including an estimate of habitat quality into the metrics. Location We tested the metrics in landscapes designed to represent different degrees of variability in patch quality and different levels of patch aggregation. Methods In each landscape, we simulated population dynamics using a spatially explicit, continuous time metapopulation model linked to within patch logistic growth models. We tested five metrics that are used to estimate the contribution that a patch makes to metapopulation viability: two versions of the probability of connectivity index, two versions of patch centrality (a graph theory metric) and the metapopulation capacity metric. Results All metrics performed best in environments where patch quality was very variable and high quality patches were aggregated. Metrics that incorporated some measure of patch quality did better in all environments, but did particularly well in environments with high variance of patch quality and spatial aggregation of good quality patches. Main conclusions Including an estimate of patch quality significantly increased the ability of a given connectivity metric to rank correctly habitat patches according to their contribution to metapopulation viability. Incorporating patch quality is particularly important in landscapes where habitat quality is highly variable and good quality patches are spatially aggregated. However, caution should be used when applying patch metrics to homogeneous landscapes, even if good estimates of patch quality are available. Our results demonstrate that landscape structure and the degree of variability in patch quality need to be assessed prior to selecting a suitable method for estimating patch value.     

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.     

Metapopulation models for extinction threshold in spatially correlated landscapes

Simple analytical models assuming homogeneous space have been used to examine the effects of habitat loss and fragmentation on metapopulation size. The models predict an extinction threshold, a critical amount of suitable habitat below which the metapopulation goes deterministically extinct. The consequences of non-random loss of habitat for species with localized dispersal have been studied mainly numerically. In this paper, we present two analytical approaches to the study of habitat loss and its metapopulation dynamic consequences incorporating spatial correlation in both metapopulation dynamics as well as in the pattern of habitat destruction. One approach is based on a measure called metapopulation capacity, given by the dominant eigenvalue of a "landscape" matrix, which encapsulates the effects of landscape structure on population extinctions and colonizations. The other approach is based on pair approximation. These models allow us to examine analytically the effects of spatial structure in habitat loss on the equilibrium metapopulation size and the threshold condition for persistence. In contrast to the pair approximation based approaches, the metapopulation capacity based approach allows us to consider species with long as well as short dispersal range and landscapes with spatial correlation at different scales. The two methods make dissimilar assumptions, but the broad conclusions concerning the consequences of spatial correlation in the landscape structure are the same. Our results show that increasing correlation in the spatial arrangement of the remaining habitat increases patch occupancy, that this increase is more evident for species with short-range than long-range dispersal, and that to be most beneficial for metapopulation size, the range of spatial correlation in landscape structure should be at least a few times greater than the dispersal range of the species.     

Effects of matrix heterogeneity on animal dispersal: from individual behavior to metapopulation-level parameters

Mounting theoretical and empirical evidence shows that matrix heterogeneity may have contrasting effects on metapopulation dynamics by contributing to patch isolation in nontrivial ways. We analyze the movement properties during interpatch dispersal in a metapopulation of Iberian lynx (Lynx pardinus). On a daily temporal scale, lynx habitat selection defines two types of matrix habitats where individuals may move: open and dispersal habitats (avoided and used as available, respectively). There was a strong and complex impact of matrix heterogeneity on movement properties at several temporal scales (hourly and daily radiolocations and the entire dispersal event). We use the movement properties on the hourly temporal scale to build a simulation model to reconstruct individual dispersal events. The two most important parameters affecting model predictions at both the individual (daily) and metapopulation scales were related to the movement capacity (number of movement steps per day and autocorrelation in dispersal habitat) followed by the parameters representing the habitat selection in the matrix. The model adequately reproduced field estimates of population-level parameters (e.g., interpatch connectivity, maximum and final dispersal distances), and its performance was clearly improved when including the effect of matrix heterogeneity on movement properties. To assume there is a homogeneous matrix results in large errors in the estimate of interpatch connectivity, especially for close patches separated by open habitat or corridors of dispersal habitat, showing how important it is to consider matrix heterogeneity when it is present. Movement properties affect the interaction of dispersing individuals with the landscape and can be used as a mechanistic representation of dispersal at the metapopulation level. This is so when the effect of matrix heterogeneity on movement properties is evaluated under biologically meaningful spatial and temporal scales.     

Metapopulation persistence in fragmented landscapes: significant interactions between genetic and demographic processes

We formulated a mathematical model in order to study the joint influence of demographic and genetic processes on metapopulation viability. Moreover, we explored the influence of habitat structure, matrix quality and disturbance on the interplay of these processes. We showed that the conditions that allow metapopulation persistence under the synergistic action of genetic and demographic processes depart significantly from predictions based on a mere superposition of the effects of each process separately. Moreover, an optimal dispersal rate exists that maximizes the range of survival rates of dispersers under which metapopulation persists and at the same time allows the largest sustainable patch removal and patch‐size reduction. The relative impact of patch removal and patch‐size reduction depends both on matrix quality and the dispersal strategy of the species: metapopulation persistence is more affected by patch‐size reduction (patch removal) for low (high)‐dispersing species, in presence of a low (high) quality matrix. Avoidance of inbreeding, through increased dispersal when the rate of inbreeding in a population is large, has positive effects on low‐dispersing species, but impairs the persistence of high‐dispersing species. Finally, size heterogeneity between patches largely influences metapopulation dynamics; the presence of large patches, even at the expense of other patches being smaller, can have positive effects on persistence in particular for species of low dispersing ability.     

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.     

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.     

Dispersal-induced desynchronization: from metapopulations to metacommunities

The conceptualization of fragmented populations in terms of metapopulation theory has become standard over the last three decades. It is well known that increases in between‐patch migration rates cause more synchronous population fluctuations and that this coherence increases the risk of global metapopulation extinction. Because species’ migration rates and the probability of individuals surviving migration events depend on the effective distance between patches, the benefit of improving conservation corridors or the matrix between habitat patches has been questioned. As populations occur in the context of larger communities, moving from a metapopulation to a metacommunity model framework is a natural extension to address the generality of these conclusions. We show how considering a metacommunity can modify the conclusion that decreasing the effective distance between habitat patches (via improving matrix quality or other measures) necessarily increases the degree of metapopulation synchrony. We show that decreases in effective between‐patch distance may deter population synchrony because of the simultaneous effect this change has on the migration patterns of other species. These results indicate that species interactions need to be considered when the effect of conservation measures on population synchrony, and ultimately persistence, is addressed.     

Potential metapopulation structure and the effects of habitat quality on population size of the endangered False Ringlet butterfly

The False Ringlet (Coenonympha oedippus) is a European butterfly species, endangered due to the severe loss and fragmentation of its habitat. In Hungary, two remaining populations of the butterfly occur in lowland Purple Moorgrass meadows. We studied a metapopulation occupying twelve habitat patches in Central Hungary. Our aim was to reveal what measures of habitat quality affect population size and density of this metapopulation, estimate dispersal parameters and describe phenology of subpopulations. Local population sizes and dispersal parameters were estimated from an extensive mark–release–recapture dataset, while habitat quality was characterized by groundwater level, cover of grass tussocks, bush cover, height of vegetation and grass litter at each habitat patch. The estimated size of the metapopulation was more than 3,000 individuals. We estimated a low dispersal capacity, especially for females, indicating a very low probability of (re)colonization. Butterfly abundance and density in local populations increased with higher grass litter, lower groundwater level and larger area covered by tussocks. We suppose that these environmental factors affect butterfly abundance by determining the microclimatic conditions for both larvae and adult butterflies. Our results suggest that the long-term preservation of the studied metapopulation needs the maintenance of high quality habitat patches by appropriate mowing regime and water regulation. Management also should facilitate dispersal to strengthen metapopulation structure with creating stepping-stones or gradually increase habitat quality in present matrix.     

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.     

Heterogeneity in patch quality buffers metapopulations from pathogen impacts

Many wildlife species persist on a network of ephemerally occupied habitat patches connected by dispersal. Provisioning of food and other resources for conservation management or recreation is frequently used to improve local habitat quality and attract wildlife. Resource improvement can also facilitate local pathogen transmission, but the landscape-level consequences of provisioning for pathogen spread and habitat occupancy are poorly understood. Here, we develop a simple metapopulation model to investigate how heterogeneity in patch quality resulting from resource improvement influences long-term metapopulation occupancy in the presence of a virulent pathogen. We derive expressions for equilibrium host–pathogen outcomes in terms of provisioning effects on individual patches (through decreased patch extinction rates) and at the landscape level (the fraction of high-quality, provisioned patches), and highlight two cases of practical concern. First, if occupancy in the unprovisioned metapopulation is sufficiently low, a local maximum in occupancy occurs for mixtures of high- and low-quality patches, such that further increasing the number of high-quality patches both lowers occupancy and allows pathogen invasion. Second, if the pathogen persists in the unprovisioned metapopulation, further provisioning can result in all patches becoming infected and in a global minimum in occupancy. This work highlights the need for more empirical research on landscape-level impacts of local resource provisioning on pathogen dynamics.     

Contrasting movement patterns in two species of chequerspot butterflies, Euphydryas aurinia and Melitaea phoebe, in the same patch network

Abstract.  1. Mark–release–recapture studies were conducted on two species of chequerspot butterfly, Euphydryas aurinia and Melitaea phoebe , in the same habitat patch network in Yanjiaping, a small basin in the Taihang Mountains, north-west of Beijing, China, in 2000.
2. Euphydryas aurinia tended to stay in the habitat patches and to move to neighbouring patches, whereas M. phoebe moved widely among patches in the entire network.
3. The parameters of the virtual migration model showed higher daily emigration propensity in M. phoebe and in E. aurinia males than in E. aurinia females, and significantly greater average daily movement distance in M. phoebe than in E. aurinia .
4. The results are consistent with the previous findings showing genetic structuring among local populations of E. aurinia but not among local populations of M. phoebe .
5. Based on the genetic and ecological results, it was concluded that E. aurinia has a classic metapopulation in the study area, whereas M. phoebe appears to have a source–sink metapopulation.
6. In 2000, when there was an overall increase in the abundance of the two species, the limited mobility of E. aurinia resulted in an increase in the average local population size, whereas the increase in the number of local populations in M. phoebe was due to its high mobility .