Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes

We compute the mean patch occupancy for a stochastic, spatially explicit patch-occupancy metapopulation model on a dynamic, correlated landscape, using a mathematically exact perturbation expansion about a mean-field limit that applies when dispersal range is large. Stochasticity in the metapopulation and landscape dynamics gives negative contributions to patch occupancy, the former being more important at high occupancy and the latter at low occupancy. Positive landscape correlations always benefit the metapopulation, but are only significant when the correlation length is comparable to, or smaller than, the dispersal range. Our analytical results allow us to consider the importance of spatial kernels in all generality. We find that the shape of the landscape correlation function is typically unimportant, and that the variance is overwhelmingly the most important property of the colonisation kernel. However, short-range singularities in either the colonisation kernel or landscape correlations can give rise to qualitatively different behaviour.     

Generalizing Levins metapopulation model in explicit space: models of intermediate complexity

A recent study [Harding and McNamara, 2002. A unifying framework for metapopulation dynamics. Am. Nat. 160, 173-185] presented a unifying framework for the classic Levins metapopulation model by incorporating several realistic biological processes, such as the Allee effect, the Rescue effect and the Anti-rescue effect, via appropriate modifications of the two basic functions of colonization and extinction rates. Here we embed these model extensions on a spatially explicit framework. We consider population dynamics on a regular grid, each site of which represents a patch that is either occupied or empty, and with spatial coupling by neighborhood dispersal. While broad qualitative similarities exist between the spatially explicit models and their spatially implicit (mean-field) counterparts, there are also important differences that result from the details of local processes. Because of localized dispersal, spatial correlation develops among the dynamics of neighboring populations that decays with distance between patches. The extent of this correlation at equilibrium differs among the metapopulation types, depending on which processes prevail in the colonization and extinction dynamics. These differences among dynamical processes become manifest in the spatial pattern and distribution of “clusters” of occupied patches. Moreover, metapopulation dynamics along a smooth gradient of habitat availability show significant differences in the spatial pattern at the range limit. The relevance of these results to the dynamics of disease spread in metapopulations is discussed.     

Metapopulation Persistence in Random Fragmented Landscapes

Habitat destruction and land use change are making the world in which natural populations live increasingly fragmented, often leading to local extinctions. Although local populations might undergo extinction, a metapopulation may still be viable as long as patches of suitable habitat are connected by dispersal, so that empty patches can be recolonized. Thus far, metapopulations models have either taken a mean-field approach, or have modeled empirically-based, realistic landscapes. Here we show that an intermediate level of complexity between these two extremes is to consider random landscapes, in which the patches of suitable habitat are randomly arranged in an area (or volume). Using methods borrowed from the mathematics of Random Geometric Graphs and Euclidean Random Matrices, we derive a simple, analytic criterion for the persistence of the metapopulation in random fragmented landscapes. Our results show how the density of patches, the variability in their value, the shape of the dispersal kernel, and the dimensionality of the landscape all contribute to determining the fate of the metapopulation. Using this framework, we derive sufficient conditions for the population to be spatially localized, such that spatially confined clusters of patches act as a source of dispersal for the whole landscape. Finally, we show that a regular arrangement of the patches is always detrimental for persistence, compared to the random arrangement of the patches. Given the strong parallel between metapopulation models and contact processes, our results are also applicable to models of disease spread on spatial networks.     

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.     

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.     

Effect of habitat fragmentation on dispersal in the butterfly Proclossiana eunomia

Comparison of dispersal rates of the bog fritillary butterfly between continuous and fragmented landscapes indicates that between patch dispersal is significantly lower in the fragmented landscape, while population densities are of the same order of magnitude. Analyses of the dynamics of the suitable habitat for the butterfly in the fragmented landscape reveal a severe, non linear increase in spatial isolation of patches over a time period of 30 years (i.e. 30 butterfly generations), but simulations of the butterfly metapopulation dynamics using a structured population model show that the lower dispersal rates in the fragmented landscape are far above the critical threshold leading to metapopulation extinction. These results indicate that changes in individual behaviour leading to the decrease of dispersal rates in the fragmented landscape were rapidly selected for when patch spatial isolation increased. The evidence of such an adaptive answer to habitat fragmentation suggests that dispersal mortality is a key factor for metapopulation persistence in fragmented landscapes. We emphasise that landscape spatial configuration and patch isolation have to be taken into account in the debate about large-scale conservation strategies.     

Effects of demographic parameters on metapopulation size and persistence: an analytical stochastic model

An analytical stochastic metapopulation model is developed. It describes how individuals will be distributed among patches as a function of density-dependent birth, death and emigration rates, and the probability of successful dispersal. The model includes demographic stochasticity, but not catastrophes, environmental stochasticity or variation in patch size and suitability. All patches are equally likely to be colonized by migrants. The model predicts: (a) mean and variance of the number of individuals per patch; (b) probability distribution of individuals per patch; (c) mean number of individuals in transit; and (d) turn-over rate and expected persistence time of a single patch. The model shows that (a) dispersal rates must be intermediate in order to ensure metapopulation persistence; (b) the mean number of individuals per patch is often well below the carrying capacity; (c) long transit times and/or high mortality during dispersal reduce the mean number of individuals per patch; (d) density-dependent emigration responses will usually increase metapopulation size and persistence compared with density-independent dispersal; (e) an increase in the per capita net growth rate can both increase and decrease metapopulation size and persistence depending on whether dispersal rates are high or low; (f) density-independent birth, death, and emigration rates lead to a spatial pattern described by the negative binomial distribution.     

Transient dynamics in metapopulation response to perturbation

Transient time in population dynamics refers to the time it takes for a population to return to population-dynamic equilibrium (or close to it) following a perturbation in the environment or in population size. Depending on the direction of the perturbation, transient time may either denote the time until extinction (or until the population has decreased to a lower equilibrium level), or the recovery time needed to reach a higher equilibrium level. In the metapopulation context, the length of the transient time is set by the interplay between population dynamics and landscape structure. Assuming a spatially realistic metapopulation model, we show that transient time is a product of four factors: the strength of the perturbation, the ratio between the metapopulation capacity of the landscape and a threshold value determined by the properties of the species, and the characteristic turnover rate of the species, adjusted by a factor depending on the structure of the habitat patch network. Transient time is longest following a large perturbation, for a species which is close to the threshold for persistence, for a species with slow turnover, and in a habitat patch network consisting of only a few dynamically important patches. We demonstrate that the essential behaviour of the n-dimensional spatially realistic Levins model is captured by the one-dimensional Levins model with appropriate parameter transformations.     

The effect of conspecific attraction on metapopulation dynamics

Random dispersal direction is assumed in all current metapopulation models. This assumption is called into question by recent experiments demonstrating that some species disperse preferentially to sites occupied by conspecifies. We incorporate conspecific attraction into two metapopulation models which differ in type of dispersal, the Levins model and a two-dimensional stepping-stone model. In both models, conspecific attraction lowers the proportion of occupied habitat patches within a metapopulation at equilibrium.     

Two‐species metacommunity dynamics mediated by habitat preference

The Levins model is a simple and widely used metapopulation model that describes temporal changes in the regional abundance of a single species and has increasingly been applied to metacommunity contexts including multiple species. Although a fundamental assumption commonly made when using the model is that species randomly move between habitat patches, most organisms exhibit habitat preference in reality. A method of incorporating habitat preference (directed dispersal) into the Levins metapopulation model was developed in a previous study. In the current study, we extended the approach to explore two‐species metacommunity dynamics (i.e. competition and predation) mediated by habitat preference. Our results theoretically revealed that coexistence of competing metapopulations requires conspecific aggregation and heterospecific segregation whereas the conspecific segregation of prey and effective avoidance of unsuitable prey‐free patches are crucial for persistence of predator metapopulations. In addition, we qualitatively and quantitatively demonstrated the effect of habitat preference on the outcomes of interspecific interactions. The present study opens a new research avenue in metacommunity ecology in complex nature and contributes to improved landscape management for the conservation of species (e.g. territorial and group‐living animals) and biodiversity.     

Metapopulations in multifractal landscapes: on the role of spatial aggregation

The use of fractals in ecology is currently pervasive over many areas. However, very few studies have linked fractal properties of landscapes to generating ecological mechanisms and dynamics. In this study I show that lacunarity (a measure of the landscape texture) is a well suited ecologically scaled landscape index that can be explicitly incorporated in metapopulation models such as the classical Levins equation. I show that the average lacunarity of an aggregated landscape is linearly correlated to the habitat that a species with local spatial processed information may perceive. Lacunarity is a computationally feasible index to measure, and is related to the metapopulation capacity of landscapes. A general approach to multifractal landscapes has been conceived, and some analytical results for self-similar landscapes are outlined, including the specific effect of landscape heterogeneity, decoupled from that of contagion by dispersal. Spatially explicit simulations show agreement with the semi-implicit method presented.     

Modelling epiphyte metapopulation dynamics in a dynamic forest landscape

We combine simulations with spatial statistics to estimate the parameters of a metapopulation model for the epiphytic lichen Lobaria pulmonaria specializing on aspen ( Populus tremula ) and goat willow ( Salix caprea ) in Fennoscandian boreal old-growth forests. We estimated the parameters of a forest landscape model (FIN-LANDIS) by repeatedly running simulations and selecting the set of parameters for tree ecology and fire regime that reproduced empirical host tree density and spatial patterns. Second, we tested which variables were important in epiphyte colonization and estimated the dispersal kernel. Third, we run a metapopulation model for the lichen across the estimated landscape scenarios and selected values for the remaining parameters that reproduced the empirical patterns of epiphyte occurrence. There was little variation in predicted dynamics, occupancy and spatial patterns between replicate metapopulation simulations. However, more data would be required for accurately estimating the parameters of FIN-LANDIS primarily because of the inherent stochasticity in large scale forest fires. Following the beginning of fire suppression in the study area 150 years ago, the model predicts that lichen occupancy first increases but subsequently declines. The lower occupancy in the past than at present is explained by high rate of tree destruction by fires, which increases local extinction rate in patch-tracking metapopulations. In the absence of fires, the occupancy increases because of lower extinction rate, but without forest fires or alternative means of host tree regeneration, the lichen is predicted to go ultimately extinct because of severely reduced density of aspen and goat willow.     

Size and genetic composition of the colonizing propagules in a butterfly metapopulation

The amount and spatial distribution of genetic variation that is maintained in a metapopulation depends critically on the colonization process. Here, we use molecular markers to determine the number and genetic relatedness of individuals establishing new local populations in a large metapopulation of the Glanville fritillary butterfly Melitaea cinxia. The empirical results are compared with the predictions of a dispersal model based on a diffusion approximation of correlated random walk, which serves as a base‐line hypothesis about the rate and pattern of colonization. The results show that half of the new local populations consisted of a single larval group of full sibs and hence necessarily of the offspring of a single female. If the colonization involved two or more larval groups, these were usually oviposited by two different females that were unrelated to each other. The pattern of colonizations is thus intermediate between the propagule pool and the migrant pool models. These results elucidate the generation of genetic stochasticity, which may influence the dynamics of small populations. The dispersal model predicted well the pattern of habitat occupancy and the pattern of colonizations in relation to landscape structure, though which particular habitat patches became colonized was influenced also by measures of habitat quality not included in the model.     

Long distance dispersal and landscape occupancy in a metapopulation of the cranberry fritillary butterfly

Movements between habitat patches in a patchy population of the butterfly Boloria aquilonaris were monitored using capture-mark-recapture methods during three successive generations. For each data set, the inverse cumulative proportion of individuals moving 100 m distance classes was fitted to the negative exponential function and the inverse power function. In each case, the negative exponential function provided a better fit than the inverse power function. Two dispersal kernels were generated using both negative exponential and inverse power functions. These dispersal kernels were used to predict movements between 14 suitable sites in a landscape of 220 km². The negative exponential function generated a dispersal kernel predicting extremely low probabilities for movements exceeding 1 km. The inverse power function generated probabilities predicting that between site movements were possible, according to metapopulation size. CMR studies in the landscape revealed that long distance movements occurred at each generation, corresponding to predictions of the inverse power function dispersal kernel. A total of 26 movements between sites (up to 13 km) were detected, together with recolonisation of empty sites. The spatial scale of the metapopulation dynamics is larger than ever reported on butterflies and long distance movements clearly matter to the persistence of this species in a highly fragmented landscape.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

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.     

Host–parasitoid spatial models: the interplay of demographic stochasticity and dynamics

Host–parasitoid metapopulation models have typically been deterministic models formulated with population numbers as a continuous variable. Spatial heterogeneity in local population abundance is a typical (and often essential) feature of these models and means that, even when average population density is high, some patches have small population sizes. In addition, large temporal population fluctuations are characteristic of many of these models, and this also results in periodically small local population sizes. Whenever population abundances are small, demographic stochasticity can become important in several ways. To investigate this problem, we have reformulated a deterministic, host–parasitoid metapopulation as an integer-based model in which encounters between hosts and parasitoids, and the fecundity of individuals are modelled as stochastic processes. This has a number of important consequences: (1) stochastic fluctuations at small population sizes tend to be amplified by the dynamics to cause massive population variability, i.e. the demographic stochasticity has a destabilizing effect; (2) the spatial patterns of local abundance observed in the deterministic counterpart are largely maintained (although the area of ''spatial chaos'' is extended); (3) at small population sizes, dispersal by discrete individuals leads to a smaller fraction of new patches being colonized, so that parasitoids with small dispersal rates have a greater tendency for extinction and higher dispersal rates have a larger competitive advantage; and (4) competing parasitoids that could coexist in the deterministic model due to spatial segregation cannot now coexist for any combination of parameters.     

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.     

Metapopulation persistence in dynamic landscapes: the role of dispersal distance

Species associated with transient habitats need efficient dispersal strategies to ensure their regional survival. Using a spatially explicit metapopulation model, we studied the effect of the dispersal range on the persistence of a metapopulation as a function of the local population and landscape dynamics (including habitat patch destruction and subsequent regeneration). Our results show that the impact of the dispersal range depends on both the local population and patch growth. This is due to interactions between dispersal and the dynamics of patches and populations via the number of potential dispersers. In general, long-range dispersal had a positive effect on persistence in a dynamic landscape compared to short-range dispersal. Long-range dispersal increases the number of couplings between the patches and thus the colonisation of regenerated patches. However, long-range dispersal lost its advantage for long-term persistence when the number of potential dispersers was low due to small population growth rates and/or small patch growth rates. Its advantage also disappeared with complex local population dynamics and in a landscape with clumped patch distribution.