Stochastic models for mainland-island metapopulations in static and dynamic landscapes

This paper has three primary aims: to establish an effective means for modelling mainland-island metapopulations inhabiting a dynamic landscape; to investigate the effect of immigration and dynamic changes in habitat on metapopulation patch occupancy dynamics; and to illustrate the implications of our results for decision-making and population management. We first extend the mainland-island metapopulation model of Alonso and McKane [Bull. Math. Biol. 64:913–958, 2002] to incorporate a dynamic landscape. It is shown, for both the static and the dynamic landscape models, that a suitably scaled version of the process converges to a unique deterministic model as the size of the system becomes large. We also establish that, under quite general conditions, the density of occupied patches, and the densities of suitable and occupied patches, for the respective models, have approximate normal distributions. Our results not only provide us with estimates for the means and variances that are valid at all stages in the evolution of the population, but also provide a tool for fitting the models to real metapopulations. We discuss the effect of immigration and habitat dynamics on metapopulations, showing thatmainland-like patches heavily influence metapopulation persistence, and we argue for adopting measures to increase connectivity between this large patch and the other island-like patches. We illustrate our results with specific reference to examples of populations of butterfly and the grasshopperBryodema tuberculata.     

Genetic diversity and genetic differentiation in Daphnia metapopulations with subpopulations of known age

If colonization of empty habitat patches causes genetic bottlenecks, freshly founded, young populations should be genetically less diverse than older ones that may have experienced successive rounds of immigration. This can be studied in metapopulations with subpopulations of known age. We studied allozyme variation in metapopulations of two species of water fleas (Daphnia) in the skerry archipelago of southern Finland. These populations have been monitored since 1982. Screening 49 populations of D. longispina and 77 populations of D. magna, separated by distances of 1.5-2180 m, we found that local genetic diversity increased with population age whereas pairwise differentiation among pools decreased with population age. These patterns persisted even after controlling for several potentially confounding ecological variables, indicating that extinction and recolonization dynamics decrease local genetic diversity and increase genetic differentiation in these metapopulations by causing genetic bottlenecks during colonization. We suggest that the effect of these bottlenecks may be twofold, namely decreasing genetic diversity by random sampling and leading to population-wide inbreeding. Subsequent immigration then may not only introduce new genetic material, but also lead to the production of noninbred hybrids, selection for which may cause immigrant alleles to increase in frequency, thus leading to increased genetic diversity in older populations.     

Evolutionarily Stable Dispersal Rates Do Not Always Increase with Local Extinction Rates

Earlier models on the evolution of dispersal have suggested that evolutionarily stable dispersal rates should increase with the frequency of local extinctions. Most metapopulation models assume site saturation (i.e., no local population dynamics), yet the majority of species distributed as metapopulations rarely attain carrying capacity in all occupied patches. In this article, we relax this assumption and examine the evolutionarily stable dispersal rate under nonsaturated but still competitive demographic conditions. Contrary to previous predictions, we show that increasing local extinction rates may allow decreasing dispersal rates to evolve.     

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

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

Metapopulation-level adaptation of insect host plant preference and extinction-colonization dynamics in heterogeneous landscapes

Species living in highly fragmented landscapes typically occur as metapopulations with frequent turnover of local populations. The turnover rate depends on population sizes and connectivities, but it may also depend on the phenotypic and genotypic composition of populations. The Glanville fritillary butterfly (Melitaea cinxia) in Finland uses two host plant species, which show variation in their relative abundances at two spatial scales: locally among individual habitat patches and regionally among networks of patches. Female butterflies in turn exhibit spatial variation in genetically determined host plant preference within and among patch networks. Emigration, immigration and establishment of new populations have all been shown to be strongly influenced by the match between the host plant composition of otherwise suitable habitat patches and the host plant preference of migrating butterflies. The evolutionary consequences of such biased migration and colonization with respect to butterfly phenotypes might differ depending on spatial configuration and plant species composition of the patches in heterogeneous patch networks. Using a spatially realistic individual-based model we show that the model-predicted evolution of host plant preference due to biased migration explains a significant amount of the observed variation in host plant use among metapopulations living in dissimilar networks. This example illustrates how the ecological extinction-colonization dynamics may be linked with the evolutionary dynamics of life history traits in metapopulations.     

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     

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.     

Evolution of Complex Density-Dependent Dispersal Strategies

The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, “triple-threshold” strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.     

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.     

Delayed population explosion of an introduced butterfly

1. The causes of lagged population and geographical range expansions after species introductions are poorly understood, and there are relatively few detailed case studies. 2. We document the 29-year history of population dynamics and structure for a population of Euphydryas gillettii Barnes that was introduced to the Colorado Rocky Mountains, USA in 1977. 3. The population size remained low (< 200 individuals) and confined to a single habitat patch (approximately 2.25 ha) to 1998. These values are similar to those of many other populations within the natural geographical range of the species. 4. However, by 2002 the population increased dramatically to > 3000 individuals and covered approximately 70 ha, nearly all to the south of the original site. The direction of population expansion was the same as that of predominant winds. 5. By 2004, the butterfly's local distribution had retracted mainly to three habitat patches. It thus exhibited a 'surge/contraction' form of population growth. Searches within 15 km of the original site yielded no other new populations. 6. In 2005, butterfly numbers crashed, but all three habitat patches remained occupied. The populations within each patch did not decrease in the same proportions, suggesting independent dynamics that are characteristic of metapopulations. 7. We postulate that this behaviour results, in this species, in establishment of satellite populations and, given appropriate habitat structure, may result in lagged or punctuated expansions of introduced populations.     

Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species

Patch occupancy theory predicts that a trade-off between competition and dispersal should lead to regional coexistence of competing species. Empirical investigations, however, find local coexistence of superior and inferior competitors, an outcome that cannot be explained within the patch occupancy framework because of the decoupling of local and spatial dynamics. We develop two-patch metapopulation models that explicitly consider the interaction between competition and dispersal. We show that a dispersal-competition trade-off can lead to local coexistence provided the inferior competitor is superior at colonizing empty patches as well as immigrating among occupied patches. Immigration from patches that the superior competitor cannot colonize rescues the inferior competitor from extinction in patches that both species colonize. Too much immigration, however, can be detrimental to coexistence. When competitive asymmetry between species is high, local coexistence is possible only if the dispersal rate of the inferior competitor occurs below a critical threshold. If competing species have comparable colonization abilities and the environment is otherwise spatially homogeneous, a superior ability to immigrate among occupied patches cannot prevent exclusion of the inferior competitor. If, however, biotic or abiotic factors create spatial heterogeneity in competitive rankings across the landscape, local coexistence can occur even in the absence of a dispersal-competition trade-off. In fact, coexistence requires that the dispersal rate of the overall inferior competitor not exceed a critical threshold. Explicit consideration of how dispersal modifies local competitive interactions shifts the focus from the patch occupancy approach with its emphasis on extinction-colonization dynamics to the realm of source-sink dynamics. The key to coexistence in this framework is spatial variance in fitness. Unlike in the patch occupancy framework, high rates of dispersal can undermine coexistence, and hence diversity, by reducing spatial variance in fitness.     

Effects of connectivity and recurrent local disturbances on community structure and population density in experimental metacommunities

Metacommunity theory poses that the occurrence and abundance of species is a product of local factors, including disturbance, and regional factors, like dispersal among patches. While metacommunity ideas have been broadly tested there is relatively little work on metacommunities subject to disturbance. We focused on how localized disturbance and dispersal interact to determine species composition in metacommunities. Experiments conducted in simple two-patch habitats containing eight protozoa and rotifer species tested how dispersal altered community composition in both communities that were disturbed and communities that connected to refuge communities not subject to disturbance. While disturbance lowered population densities, in disturbed patches connected to undisturbed patches this was ameliorated by immigration. Furthermore, species with high dispersal abilities or growth rates showed the fastest post-disturbance recovery in presence of immigration. Connectivity helped to counteract the negative effect of disturbances on local populations, allowing mass-effect-driven dispersal of individuals from undisturbed to disturbed patches. In undisturbed patches, however, local population sizes were not significantly reduced by emigration. The absence of a cost of dispersal for undisturbed source populations is consistent with a lack of complex demography in our system, such as age- or sex-specific emigration. Our approach provides an improved way to separate components of population growth from organisms' movement in post-disturbance recovery of (meta)communities. Further studies are required in a variety of ecosystems to investigate the transient dynamics resulting from disturbance and dispersal.     

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.     

Short-term studies underestimate 30-generation changes in a butterfly metapopulation

Most studies of rare and endangered species are based on work carried out within one generation, or over one to a few generations of the study organism. We report the results of a study that spans 30 generations (years) of the entire natural range of a butterfly race that is endemic to 35 km(2) of north Wales, UK. Short-term studies (surveys in single years and dynamics over 4 years) of this system led to the prediction that the regional distribution would be quite stable, and that colonization and extinction dynamics would be relatively unimportant. However, a longer-term study revealed unexpectedly high levels of population turnover (local extinction and colonization), affecting 18 out of the 20 patches that were occupied at any time during the period. Modelling the system (using the 'incidence function model' (IFM) for metapopulations) also showed higher levels of colonization and extinction with increasing duration of the study. The longer-term dynamics observed in this system can be compared, at a metapopulation level, with the increased levels of variation observed with increasing time that have been observed in single populations. Long-term changes may arise from local changes in the environment that make individual patches more or less suitable for the butterfly, or from unusual colonization or extinction events that take metapopulations into alternative states. One implication is that metapopulation and population viability analyses based on studies that cover only a few animal or plant generations may underestimate extinction threats.     

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.     

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.     

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.     

Predators Reduce Extinction Risk in Noisy Metapopulations

Background

Spatial structure across fragmented landscapes can enhance regional population persistence by promoting local “rescue effects.” In small, vulnerable populations, where chance or random events between individuals may have disproportionately large effects on species interactions, such local processes are particularly important. However, existing theory often only describes the dynamics of metapopulations at regional scales, neglecting the role of multispecies population dynamics within habitat patches.

Findings

By coupling analysis across spatial scales we quantified the interaction between local scale population regulation, regional dispersal and noise processes in the dynamics of experimental host-parasitoid metapopulations. We find that increasing community complexity increases negative correlation between local population dynamics. A potential mechanism underpinning this finding was explored using a simple population dynamic model.

Conclusions

Our results suggest a paradox: parasitism, whilst clearly damaging to hosts at the individual level, reduces extinction risk at the population level.     

Evolution of dispersal in metapopulations with local density dependence and demographic stochasticity

In this paper, we predict the outcome of dispersal evolution in metapopulations based on the following assumptions: (i) population dynamics within patches are density-regulated by realistic growth functions; (ii) demographic stochasticity resulting from finite population sizes within patches is accounted for; and (iii) the transition of individuals between patches is explicitly modelled by a disperser pool. We show, first, that evolutionarily stable dispersal rates do not necessarily increase with rates for the local extinction of populations due to external disturbances in habitable patches. Second, we describe how demographic stochasticity affects the evolution of dispersal rates: evolutionarily stable dispersal rates remain high even when disturbance-related rates of local extinction are low, and a variety of qualitatively different responses of adapted dispersal rates to varied levels of disturbance become possible. This paper shows, for the first time, that evolution of dispersal rates may give rise to monotonically increasing or decreasing responses, as well as to intermediate maxima or minima.