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.     

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.     

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.     

Simple discrete-time metapopulation models of patch occupancy

Simple models in theoretical ecology have a long-standing history of being used to understand how specific processes influence population dynamics as well as providing a foundation for future endeavors. The Levins model is the seminal example of this for continuous-time metapopulation dynamics. However, many natural populations have a distinct separation between processes and data is not collected continuously leading to the need for using a discrete-time model. Our goal is to develop a simple discrete-time metapopulation model of patch occupancy using difference equations. In our formulation, we consider the two fundamental processes of colonization and extinction that will be treated as sequential events and will only consider patch occupancy. To achieve this, we use a composition of two functions where one will reflect the extinction process and the other for the colonization process. Under some mild assumptions, we are able determine the dynamic behavior of the metapopulation. In addition, we provide numerous examples for the functions used to emulate the colonization and extinction processes. Our results illustrate that the dynamics of the model are tied to properties such as convexity and monotonicity of the colonization and extinction functions. In particular, if the model is non-monotone, then complex dynamics can arise such as cyclic and even chaotic behavior. Overall, our approach shows how certain properties of the colonization and extinction functions can influence metapopulation dynamics.     

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。     

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.     

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.     

用空间直观模型是否足以从斑块占据性资料中推断集合种群的动态过程?

在集合种群的研究中,经常要根据空间占据性数据应用斑块模型来推断种群的动态过程,在保护生物学应用中,斑块占据性模型的参数估测对于阐释集合种群动态和预测种群对生境破坏的反应极为重要。我们探讨了一种广泛应用的空间直观模型——率函数模型(Incidence function model)中参数估测的不确定性问题,通过构建由50个斑块组成的网络和两个假想的已知参数的集合种群,应用模拟模型产生集合种群随时间变化的斑块占据性数据系列：即快照(snapshot)。然后,根据这些快照,应用率函数模型和最大似然法估测种群动态参数。此外,我们还给出了传统的率函数模型的一个变形,这个变形包含了目标区效应(Target area effect)：即一个斑块的占据概率不但取决于空间隔离度,也取决于斑块本身面积的大小。结果表明：根据同一个集合种群不同的快照所估测的参数可以有很大差异,一个快照得出的参数提示的是占据性强但存活率低的集合种群,而另一个快照可能反映的是一个占据性弱但存活率高的集合种群。应用传统的率函数模型于一个包含了目标区效应的集合种群,导致斑块大小相关的灭绝率参数估测的正偏差。因此,仅根据一个快照的空间占据性数据来推测集合种群的过程有很大的不确定性[动物学报49(6)：787～794,2003]。     

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.     

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.     

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.     

Coherence,conservation and patch‐occupancy analysis

Spatial coherence (synchrony) among subpopulations poses a danger to the metacommunity, as it increases the risk of regional extinction. When this effect is significant, the use of inference techniques based on the stochastic patch occupancy model (SPOM) may be inadequate, since SPOMs assume that each habitat patch is either occupied or empty, thereby neglecting the intra‐patch dynamics. Here we suggest a general classification of the dynamics that allows the identification, in a model‐independent manner, of the regimes where coherence effects are strong. We also present a new technique, based on patch occupancy (presence/absence) data, for identifying the role of spatial coherence in the stabilization of a metapopulation. If the chance of a local extinction grows with the connectivity, this implies that spatial synchronization is too strong and that regional‐scale extinction becomes possible. When this scenario occurs, a decrease in the movement of individuals (habitat fragmentation, reduced dispersal rates) has a positive effect on the sustainability of the spatially distributed population. The results of individual based simulations of a spatially structured population are analyzed with SPOM and the regime where the two‐state approximation fails is identified.     

A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.     

The role of landscape-dependent disturbance and dispersal in metapopulation persistence

The fundamental processes that influence metapopulation dynamics (extinction and recolonization) will often depend on landscape structure. Disturbances that increase patch extinction rates will frequently be landscape dependent such that they are spatially aggregated and have an increased likelihood of occurring in some areas. Similarly, landscape structure can influence organism movement, producing asymmetric dispersal between patches. Using a stochastic, spatially explicit model, we examine how landscape-dependent correlations between dispersal and disturbance rates influence metapopulation dynamics. Habitat patches that are situated in areas where the likelihood of disturbance is low will experience lower extinction rates and will function as partial refuges. We discovered that the presence of partial refuges increases metapopulation viability and that the value of partial refuges was contingent on whether dispersal was also landscape dependent. Somewhat counterintuitively, metapopulation viability was reduced when individuals had a preponderance to disperse away from refuges and was highest when there was biased dispersal toward refuges. Our work demonstrates that landscape structure needs to be incorporated into metapopulation models when there is either empirical data or ecological rationale for extinction and/or dispersal rates being landscape dependent.     

Multi-patch deterministic and stochastic models for wildlife diseases

Spatial heterogeneity and host demography have a direct impact on the persistence or extinction of a disease. Natural or human-made landscape features such as forests, rivers, roads, and crops are important to the persistence of wildlife diseases. Rabies, hantaviruses, and plague are just a few examples of wildlife diseases where spatial patterns of infection have been observed. We formulate multi-patch deterministic and stochastic epidemic models and use these models to investigate problems related to disease persistence and extinction. We show in some special cases that a unique disease-free equilibrium exists. In these cases, a basic reproduction number ?₀ can be computed and shown to be bounded below and above by the minimum and maximum patch reproduction numbers ? _j , j=1, …, n. The basic reproduction number has a simple form when there is no movement or when all patches are identical or when the movement rate approaches infinity. Numerical examples of the deterministic and stochastic models illustrate the disease dynamics for different movement rates between three patches.     

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.     

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton–Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.