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.     

Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure

We describe a mathematically exact method for the analysis of spatially structured Markov processes. The method is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity. As an example, we consider a spatial version of the Levins metapopulation model, in which the habitat patches are distributed in the d-dimensional landscape Rd in a random (but possibly correlated) manner. Assuming that the dispersal kernel is characterized by a length scale L, we examine how the behavior of the metapopulation deviates from the mean-field model for a finite but large L. For example, we show that the equilibrium fraction of occupied patches is given by p(0)+c/L(d)+O(L(-3d/2)), where p(0) is the equilibrium state of the Levins model and the constant c depends on p(0), the dispersal kernel, and the structure of the landscape. We show that patch occupancy can be increased or decreased by spatial structure, but is always decreased by stochasticity. Comparison with simulations show that the analytical results are not only asymptotically exact (as L-->infinity), but a good approximation also when L is relatively small.     

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.     

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.     

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.     

Non-equilibria in small metapopulations: comparing the deterministic Levins model with its stochastic counterpart

In this paper, we examine, for small metapopulations, the stochastic analog of the classical Levins metapopulation model. We study its basic model output, the expected time to metapopulation extinction, for systems which are brought out of equilibrium by imposing sudden changes in patch number and the colonization and extinction parameters. We find that the expected metapopulation extinction time shows different behavior from the relaxation time of the original, deterministic, Levins model. This relaxation time is therefore limited in value for predicting the behavior of the stochastic model. However, predictions about the extinction time for deterministically unviable cases remain qualitatively the same. Our results further suggest that, if we want to counteract the effects of habitat loss or increased dispersal resistance, the optimal conservation strategy is not to restore the original situation, that is, to create habitat or decrease resistance against dispersal. As long as the costs for different management options are not too dissimilar, it is better to improve the quality of the remaining habitat in order to decrease the local extinction rate.     

Is metapopulation patch occupancy in nature well predicted by the Levins model?

Although the Levins model has made important theoretical contributions to ecology, its empirical support has not been conclusively established yet. We used published colonization and extinction data from 55 metapopulations to calculate their Levins equilibrium patch occupancy. Over all species, there were not significant differences between the observed patch occupancies and the Levins model's estimates. However, invertebrates and vertebrate species with some degree of threat had patch occupancies larger than the model's expectancies. A temporal sampling effect was found for invertebrate species, with departure from the Levins model decreasing as the length of the study period increased. There was a negative relationship between patch occupancy and extinction probability, as expected under the “rescue effect”. The high rates at which invertebrates produce propagules could lead the Levins model to underestimate patch occupancy, whereas the observed patch occupancy of threatened species may be a transient phenomenon that results from extinction probabilities that increase over time. Therefore, the Levins model captures the metapopulation dynamics of a wide range of species in a simple formula whereas its equilibrium point can be used as evidence of metapopulation stability. Although mechanistic models provide more precise and accurate metapopulation predictions, they also can sacrifice the generality and simplicity of the Levins model.     

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.     

Convergence of a structured metapopulation model to Levins’s model

We consider a structured metapopulation model describing the dynamics of a single species, whose members are located in separate patches that are linked through migration according to a mean field rule. Our main aim is to find conditions under which its equilibrium distribution is reasonably approximated by that of the unstructured model of Levins (1969). We do this by showing that the (positive) equilibrium distribution converges, as the carrying capacity of each population goes to infinity together with appropriate scalings on the other parameters, to a bimodal distribution, consisting of a point mass at 0, together with a positive part which is closely approximated by a shifted Poisson centred near the carrying capacity. Under this limiting régime, we also give simpler approximate formulae for the equilibrium distribution. We conclude by showing how to compute persistence regions in parameter space for the exact model, and then illustrate all our results with numerical examples. Our proofs are based on Steins method.Supported in part by Schweizer Nationalfonds Projekt Nrs 20–61753.00 and 20–67909.02Supported in part by CNR of Italy under Grant n. 00.0142.ST74     

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     

How Levins’ dynamics emerges from a Ricker metapopulation model

Understanding the dynamics of metapopulations close to extinction is of vital importance for management. Levins-like models, in which local patches are treated as either occupied or empty, have been used extensively to explore the extinction dynamics of metapopulations, but they ignore the important role of local population dynamics. In this paper, we consider a stochastic metapopulation model where local populations follow a stochastic, density-dependent dynamics (the Ricker model), and use this framework to investigate the behaviour of the metapopulation on the brink of extinction. We determine under which circumstances the metapopulation follows a time evolution consistent with Levins’ dynamics. We derive analytical expressions for the colonisation and extinction rates (c and e) in Levins-type models in terms of reproduction, survival and dispersal parameters of the local populations, providing an avenue to parameterising Levins-like models from the type of information on local demography that is available for a number of species. To facilitate applying our results, we provide a numerical algorithm for computing c and e.     

How populations persist when asexuality requires sex: the spatial dynamics of coping with sperm parasites

The twofold cost of sex implies that sexual and asexual reproduction do not coexist easily. Asexual forms tend to outcompete sexuals but may eventually suffer higher extinction rates, creating tension between short- and long-term advantages of different reproductive modes. The 'short-sightedness' of asexual reproduction takes a particularly intriguing form in gynogenetic species complexes, in which an asexual species requires sperm from a related sexual host species to trigger embryogenesis. Asexuals are then predicted to outcompete their host, after which neither species can persist. We examine whether spatial structure can explain continued coexistence of the species complex, and assess the evidence based on data on the Amazon molly (Poecilia formosa). A modification of the Levins metapopulation model creates two regions of good prospects for coexistence, connected by a region of poorer patch occupancy levels. In the first case, mate discrimination and/or niche differentiation keep local extinction rates low, and most patches contain both species; the other possibility resembles host-parasite dynamics where parasites frequently drive the host locally extinct. Several dynamical features are counterintuitive and relate to the parasitic nature of interactions in the species complex: for example, high local extinction rates of the asexual species can be beneficial for its own persistence. This creates a link from the evolution of sexual reproduction to that of prudent predation.     

Adaptive walks on changing landscapes: Levins' approach extended

The assumption that trade-offs exist is fundamental in evolutionary theory. Levins (Am. Nat. 96 (1962) 361-372) introduced a widely adopted graphical method for analyzing evolution towards an optimal combination of two quantitative traits, which are traded off. His approach explicitly excluded the possibility of density- and frequency-dependent selection. Here we extend Levins method towards models, which include these selection regimes and where therefore fitness landscapes change with population state. We employ the same kind of curves Levins used: trade-off curves and fitness contours. However, fitness contours are not fixed but a function of the resident traits and we only consider those that divide the trait space into potentially successful mutants and mutants which are not able to invade ('invasion boundaries'). The developed approach allows to make a priori predictions about evolutionary endpoints and about their bifurcations. This is illustrated by applying the approach to several examples from the recent literature.     

Alternative causes for range limits: a metapopulation perspective

All species have limited distributions at broad geographical scales. At local scales, the distribution of many species is influenced by the interplay of the three factors of habitat availability, local extinctions and colonization dynamics. We use the standard Levins metapopulation model to illustrate how gradients in these three factors can generate species' range limits. We suggest that the three routes to range limits have radically different evolutionary implications. Because the Levins model makes simplifying assumptions about the spatial coupling of local populations, we present numerical studies of spatially explicit metapopulation models that complement the analytical model. The three routes to range limits give rise to distinct spatiotemporal patterns. Range limits in one species can also arise because of environmental gradients impinging upon other species. We briefly discuss a predator–prey example, which illustrates indirect routes to range limits in a metacommunity context.     

The effect of landscape pattern on Mediterranean vegetation dynamics: A modelling approach using functional types

Abstract. In the framework of land use changes in the Mediterranean area, I asked to what extent different landscape structures might determine long‐term dynamics in Mediterranean ecosystems. To answer this question, a spatially explicit model was developed (the Melca model), incorporating two functional types of woody species dominant in Mediterranean ecosystems: a resprouter (R) and a non‐resprouter fire‐recruiter (seeders, S). The model was used as a tool for generating hypotheses on the possible consequences of different landscape scenarios. Thus, five different hierarchically structured random landscapes were generated, all having the same cover for the two functional types but different landscape structure (ranging from highly heterogeneous to homogeneous landscapes). After a 100‐yr simulation, plant cover and spatial pattern had changed and the changes were different for the different initial spatial configurations, suggesting that long‐term vegetation dynamics is spatially dependent (the resultant dynamics are sensitive to the initial spatial structure). In the landscapes where R‐type species had a low number of large patches and S‐species had a large number of small patches, the number of R‐patches increased and their size decreased, while the number of S‐patches decreased. In these cases, the final cover of the two types changed little from the initial cover. Landscapes with a large number of small R‐patches interspersed with S‐patches had a decrease in the number of R‐patches, an increase in the number of S‐patches and a decrease in the size of S‐patches. In these landscapes, final cover was significantly changed, increasing in R‐type and decreasing in S‐type species. These results suggest that low spatial autocorrelation (low aggregation) favours R‐type species. Implications for land management are also discussed.     

THE EFFECTS OF THE RELATIVE GEOGRAPHIC SCALES OF GENE FLOW AND SELECTION ON MORPH FREQUENCIES IN THE WALKING-STICK TIMEMA CRISTINAE

Gene frequencies in large populations are determined by a balance between selection and gene flow between neighborhoods of different selection regimes. This balance is affected by the area of the patches of a given selection regime relative to the gene-flow distance. If patches are small relative to gene-flow distance, similarity in the total area occupied by different patch types is a crucial condition for the stability of polymorphisms. However, if patches are larger than the gene-flow distance, then the relative area of different patch types is less important because of reduced gene flow resulting from isolation by distance. Two morphs (striped and unstriped) of the walking-stick Timema cristinae were each strongly associated with patches of distinct species of food plants on which they are most cryptic. The frequency of a morph was high on the plant on which it is most cryptic when either: (1) the area occupied by the food plant (patch) was very large; (2) the patch was completely isolated from other patches; or (3) the patch was larger than adjacent patches. Results (1) and (2) are consistent with isolation-by-distance models, and result (3) is consistent with Levene's multiple-niche polymorphism model.     

The potential for community level evaluations based on loop analysis

In this paper we present results obtained with a computer simulation in which a community, described by Levins in his presentation of loop analysis (Levins, R., 1975, Evolution in communities near equilibrium, in: Ecology and Evolution of Communities, M.L. Cody and J.M. Diamond (eds) (Harvard University Press, Cambridge, Mass.) pp. 16-50), is analysed. We show how our simulation accurately reproduces Levins' calculations and further, (i) how our simulation can be used to answer questions raised by Levins but never answered, (ii) how the simulation can be used to dissect a community in order to analyse the roles played by the various entities, and (iii) how predictions relating to the evolution of this community, proposed by Levins, can be analysed with some interesting and unexpected results. In particular, it becomes clear that discussion of the type of selection that may take place needs to be done in the framework of the community in which it occurs.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.