Generalizations of the Moran effect explaining spatial synchrony in population fluctuations

The Moran effect for populations separated in space states that the autocorrelations in the population fluctuations equal the autocorrelation in environmental noise, assuming the same linear density regulation in all populations. Here we generalize the Moran effect to include also nonlinear density regulation with spatial heterogeneity in local population dynamics as well as in the effects of environmental covariates by deriving a simple expression for the correlation between the sizes of two populations, using diffusion approximation to the theta-logistic model. In general, spatial variation in parameters describing the dynamics reduces population synchrony. We also show that the contribution of a covariate to spatial synchrony depends strongly on spatial heterogeneity in the covariate or in its effect on local dynamics. These analyses show exactly how spatial environmental covariation can synchronize fluctuations of spatially segregated populations with no interchange of individuals even if the dynamics are nonlinear.     

局域种群的Allee效应和集合种群的同步性

从包含Allee效应的局域种群出发,建立了耦合映像格子模型,即集合种群模型.通过分析和计算机模拟表明:(1)当局域种群受到Allee效应强度较大时,集合种群同步灭绝;(2)而当Allee效应强度相对较弱时,通过稳定局域种群动态(减少混沌)使得集合种群发生同步波动,而这种同步波动能够增加集合种群的灭绝风险;(3)斑块间的连接程度对集合种群同步波动的发生有很大的影响,适当的破碎化有利于集合种群的续存.全局迁移和Allee效应结合起来增加了集合种群同步波动的可能,从而增加集合种群的灭绝风险.这些结果对理解同步性的机理、利用同步机理来制定物种保护策略和害虫防治都有重要的意义.     

Does red noise increase or decrease extinction risk? Single extreme events versus series of unfavorable conditions

Recent theoretical studies have shown contrasting effects of temporal correlation of environmental fluctuations (red noise) on the risk of population extinction. It is still debated whether and under which conditions red noise increases or decreases extinction risk compared with uncorrelated (white) noise. Here, we explain the opposing effects by introducing two features of red noise time series. On the one hand, positive autocorrelation increases the probability of series of poor environmental conditions, implying increasing extinction risk. On the other hand, for a given time period, the probability of at least one extremely bad year ("catastrophe") is reduced compared with white noise, implying decreasing extinction risk. Which of these two features determines extinction risk depends on the strength of environmental fluctuations and the sensitivity of population dynamics to these fluctuations. If extreme (catastrophic) events can occur (strong noise) or sensitivity is high (overcompensatory density dependence), then temporal correlation decreases extinction risk; otherwise, it increases it. Thus, our results provide a simple explanation for the contrasting previous findings and are a crucial step toward a general understanding of the effect of noise color on extinction risk.     

Viability in a pink environment: why "white noise" models can be dangerous

Analysis of long time series suggests that environmental fluctuations may be accurately represented by 1/ f   noise (pink noise), where temporal correlation is found at several scales, and the range of fluctuations increases over time. Previous studies on the effects of coloured noise on population dynamics used first or second order autoregressive noise. I examined the importance of coloured noise for extinction risk using true 1/ f   noise. I also considered the problem of estimating extinction risk with a limited sample of environmental variation. Pink noise environments increased extinction risk in random walk models where environmental variation affected the growth rate. However, pink noise environments decreased extinction risk in the Ricker model where environmental variation modified the carrying capacity. Underestimation of environmental variance almost always yielded underestimation of extinction risk. For either population viability analysis or management, we should carefully consider the long-term behaviour of the environment as well as how we include environmental noise in population models.     

The extended Moran effect and large-scale synchronous fluctuations in the size of great tit and blue tit populations

1. Synchronous fluctuations of geographically separated populations are in general explained by the Moran effect, i.e. a common influence on the local population dynamics of environmental variables that are correlated in space. Empirical support for such a Moran effect has been difficult to provide, mainly due to problems separating out effects of local population dynamics, demographic stochasticity and dispersal that also influence the spatial scaling of population processes. Here we generalize the Moran effect by decomposing the spatial autocorrelation function for fluctuations in the size of great tit Parus major and blue tit Cyanistes caeruleus populations into components due to spatial correlations in the environmental noise, local differences in the strength of density regulation and the effects of demographic stochasticity. 2. Differences between localities in the strength of density dependence and nonlinearity in the density regulation had a small effect on population synchrony, whereas demographic stochasticity reduced the effects of the spatial correlation in environmental noise on the spatial correlations in population size by 21.7% and 23.3% in the great tit and blue tit, respectively. 3. Different environmental variables, such as beech mast and climate, induce a common environmental forcing on the dynamics of central European great and blue tit populations. This generates synchronous fluctuations in the size of populations located several hundred kilometres apart. 4. Although these environmental variables were autocorrelated over large areas, their contribution to the spatial synchrony in the population fluctuations differed, dependent on the spatial scaling of their effects on the local population dynamics. We also demonstrate that this effect can lead to the paradoxical result that a common environmental variable can induce spatial desynchronization of the population fluctuations. 5. This demonstrates that a proper understanding of the ecological consequences of environmental changes, especially those that occur simultaneously over large areas, will require information about the spatial scaling of their effects on local population dynamics.     

Environment,but not migration rate,influences extinction risk in experimental metapopulations

Ecological theory suggests that several demographic factors influence metapopulation extinction risk, including synchrony in population size between subpopulations, metapopulation size and the magnitude of fluctuations in population size. Theoretically, each of these is influenced by the rate of migration between subpopulations. Here we report on an experiment where we manipulated migration rate within metapopulations of the freshwater zooplankton Daphnia magna to examine how migration influenced each of these demographic variables, and subsequent effects on metapopulation extinction. In addition, our experimental procedures introduced unplanned but controlled differences between metapopulations in light intensity, enabling us to examine the relative influences of environmental and demographic factors. We found that increasing migration rate increased subpopulation synchrony. We failed to detect effects of migration on population size and fluctuations in population size at the metapopulation or subpopulation level, however. In contrast, light intensity did not influence synchrony, but was positively correlated with population size and negatively correlated with population fluctuation. Finally, synchrony did not influence time to extinction, while population size and the magnitude of fluctuations did. We conclude that environmental factors had a greater influence on extinction risk than demographic factors, and that metapopulation size and fluctuation were more important to extinction risk than metapopulation synchrony.     

Spatial scales of population synchrony of two competing species: effects of harvesting and strength of competition

Theoretical analyses of single‐species models have revealed that the degree of synchrony in fluctuations of geographically separated populations increases with increasing spatial covariation in environmental fluctuations and increased interchange of individuals, but decreases with local strength of density dependence. Here we extend these results to include interspecific competition between two species as well as harvesting. We show that the effects of interspecific competition on the geographical scale of population synchrony are dependent on the pattern of spatial covariation of environmental variables. If the environmental noise is uncorrelated between the competing species, competition generally increases the spatial scale of population synchrony of both species. Otherwise, if the environmental noises are strongly correlated between species, competition generally increases the spatial scale of population synchrony of at least one, but also often of both species. The magnitude of these spatial scaling effects is, however, strongly influenced by the dispersal capacity of the two competing species. If the species are subject to proportional harvesting, this may synchronise population dynamics over large geographical areas, affecting the vulnerability of harvested species to environmental changes. However, the strength of interspecific competition may strongly modify this effect of harvesting on the spatial scale of population synchrony. For example, harvesting of one species may affect the spatial distribution of competing species that are not subject to harvesting. These analytical results provide an important illustration of the importance of applying an ecosystem rather than a single‐species perspective when developing harvest strategies for a sustainable management of exploited species.     

Modelling the role of social behavior in the persistence of the alpine marmot Marmota marmota

A general rule of thumb for biological conservation obtained from simple models of hypothetical species is that for populations with strong environmental noise moderate increases in habitat size or quality do not substantially reduce extinction risk. However, whether this rule also holds for real species with complex behavior, such as social species with breeding units and reproductive suppression, is uncertain. Here we present a population viability analysis of the alpine marmot Marmota marmota, which displays marked social behavior, i.e. it lives in social groups of up to twenty individuals. Our analysis is based on a long‐term field study carried out in the Bavarian Alps since 1982. During the first fifteen years of this study, 687 marmots were individually marked and the movements and fate of 98 dispersing marmots were recorded with radio‐telemetry. Thus, in contrast to most other viability analyses of spatially structured populations, good data about dispersal exist. A model was constructed which is individual‐based, spatially explicit at the scale of clusters of neighbouring territories, and spatially implicit at larger scales. The decisive aspect of marmot life history, winter mortality, is described by logistic regression where mortality is increased by age and the severity of winter, and decreased by the number of subdominant individuals present in a group. Model predictions of group size distribution are in good agreement with the results of the field study. The model shows that the effect of sociality on winter mortality is very effective in buffering environmental harshness and fluctuations. This underpins theoretical results stating that the appropriate measure of the strength of environmental noise is the ratio between the variance of population growth rate and the intrinsic rate of increase. The lessons from our study for biological conservation are that simple, unstructured models may not be sufficient to assess the viability of species with complex behavioral traits, and that even moderate increases in habitat capacity may substantially reduce extinction risk even if environmental fluctuations seem high.     

Larval dispersal dampens population fluctuation and shapes the interspecific spatial distribution patterns of rocky intertidal gastropods

Many marine benthic invertebrates pass through a planktonic larval stage whereas others spend their entire lifetimes in benthic habitats. Recent studies indicate that non‐planktonic species show relatively greater fine‐scale patchiness than do planktonic species, but the underlying mechanisms remain unknown. One hypothesis for such a difference is that larval dispersal enhances the connectivity of populations and buffers population fluctuations and reduces local extinction risk, consequently increasing patch occupancy rate and decreasing spatial patchiness. If this mechanism does indeed play a significant role, then the distribution of non‐planktonic species should be more aggregated – both temporally and spatially – than the distribution of species with a planktonic larval stage. To test this prediction, we compared 1) both the spatial and the temporal abundance–occupancy relationships and 2) both the spatial and the temporal mean–variance relationships of population size across species of rocky intertidal gastropods with differing dispersive traits from the Pacific coast of Japan. We found that, compared to planktonic species, non‐planktonic species exhibited 1) a smaller occupancy rate for any given level of mean population size and 2) greater variations in population size, both spatially and temporally. This suggests that the macroecological patterns observed in this study (i.e. the abundance–occupancy relationships and mean–variance relationships of population size across species) were shaped by the effect of larval dispersal dampening population fluctuation, which works over both space and time. While it has been widely assumed that larval dispersal enhances population fluctuations, larval dispersal may in fact enhance the connectively of populations and buffer population fluctuations and reduce local extinction risks.     

Optimizing Metapopulation Sustainability through a Checkerboard Strategy

The persistence of a spatially structured population is determined by the rate of dispersal among habitat patches. If the local dynamic at the subpopulation level is extinction-prone, the system viability is maximal at intermediate connectivity where recolonization is allowed, but full synchronization that enables correlated extinction is forbidden. Here we developed and used an algorithm for agent-based simulations in order to study the persistence of a stochastic metapopulation. The effect of noise is shown to be dramatic, and the dynamics of the spatial population differs substantially from the predictions of deterministic models. This has been validated for the stochastic versions of the logistic map, the Ricker map and the Nicholson-Bailey host-parasitoid system. To analyze the possibility of extinction, previous studies were focused on the attractiveness (Lyapunov exponent) of stable solutions and the structure of their basin of attraction (dependence on initial population size). Our results suggest that these features are of secondary importance in the presence of stochasticity. Instead, optimal sustainability is achieved when decoherence is maximal. Individual-based simulations of metapopulations of different sizes, dimensions and noise types, show that the system''s lifetime peaks when it displays checkerboard spatial patterns. This conclusion is supported by the results of a recently published Drosophila experiment. The checkerboard strategy provides a technique for the manipulation of migration rates (e.g., by constructing corridors) in order to affect the persistence of a metapopulation. It may be used in order to minimize the risk of extinction of an endangered species, or to maximize the efficiency of an eradication campaign.     

Effects on population persistence: the interaction between environmental noise colour,intraspecific competition and space

It is accepted that accurate estimation of risk of population extinction, or persistence time, requires prediction of the effect of fluctuations in the environment on population dynamics. Generally, the greater the magnitude, or variance, of environmental stochasticity, the greater the risk of population extinction. Another characteristic of environmental stochasticity, its colour, has been found to affect population persistence. This is important because real environmental variables, such as temperature, are reddened or positively temporally autocorrelated. However, recent work has disagreed about the effect of reddening environmental stochasticity. Ripa and Lundberg (1996) found increasing temporal autocorrelation (reddening) decreased the risk of extinction, whereas a simple and powerful intuitive argument (Lawton 1988) predicts increased risk of extinction with reddening. This study resolves the apparent contradiction, in two ways, first, by altering the dynamic behaviour of the population models. Overcompensatory dynamics result in persistence times increasing with increased temporal autocorrelation; undercompensatory dynamics result in persistence times decreasing with increased temporal autocorrelation. Secondly, in a spatially subdivided population, with a reasonable degree of spatial heterogeneity in patch quality, increasing temporal autocorrelation in the environment results in decreasing persistence time for both types of competition. Thus, the inclusion of coloured noise into ecological models can have subtle interactions with population dynamics.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.     

Migration, coherence and persistence in a fragmented landscape

The chance of local extinction is high during periods of small population size. Accordingly, a metapopulation made of local communities that support internal population cycling may face the threat of regional extinction if the local dynamics is coherent (synchronized). These systems achieve maximum sustainability at an intermediate level of migration that allows recolonization but prevents synchronization. Here we implement an individual-based simulation technique to examine the maximum persistence condition for a system of patch habitats connected by passive migration. The models discussed in this paper take into consideration realistic elements of metapopulations, such as migration cost, disordered spatial structure, frustration and environmental noise. It turns out that the state with maximum anti-correlation between neighboring patches is the most sustainable one, even in the presence of these complications. The results suggest, at least for small systems, a model independent conservation strategy: coherence between neighboring local communities has, in general, a negative impact, and population will benefit from intervention that increases anti-correlations.     

The effect of static and dynamic spatially structured disturbances on a locally dispersing population

Previous models of locally dispersing populations have shown that in the presence of spatially structured fixed habitat heterogeneity, increasing local spatial autocorrelation in habitat generally has a beneficial effect on such populations, increasing equilibrium population density. It has also been shown that with large-scale disturbance events which simultaneously affect contiguous blocks of sites, increasing spatial autocorrelation in the disturbances has a harmful effect, decreasing equilibrium population density. Here, spatial population models are developed which include both of these spatially structured exogenous influences, to determine how they interact with each other and with the endogenously generated spatial structure produced by the population dynamics. The models show that when habitat is fragmented and disturbance occurs at large spatial scales, the population cannot persist no matter how large its birth rate, an effect not seen in previous simpler models of this type. The behavior of the model is also explored when the local autocorrelation of habitat heterogeneity and disturbance events are equal, i.e. the two effects occur at the same spatial scale. When this scale parameter is very small, habitat fragmentation prevents the population from persisting because sites attempting to reproduce will drop most of their offspring on unsuitable sites; when the parameter is very large, large-scale disturbance events drive the population to extinction. Population levels reach their maximum at intermediate values of the scale parameter, and the critical values in the model show that the population will persist most easily at these intermediate scales of spatial influences. The models are investigated via spatially explicit stochastic simulations, traditional (infinite-dispersal) and improved (local-dispersal) mean-field approximations, and pair approximations.     

A perfect storm: the combined effects on population fluctuations of autocorrelated environmental noise, age structure, and density dependence

While it is widely appreciated that climate can affect the population dynamics of various species, a mechanistic understanding of how climate interacts with life-history traits to influence population fluctuations requires development. Here we build a general density-dependent age-structured model that accounts for differential responses in life-history traits to increasing population density. We show that as the temporal frequency of favorable environmental conditions increases, population fluctuations also increase provided that unfavorable environmental conditions still occur. As good years accumulate and the number of individuals in a population increases, successive life-history traits become vulnerable to density dependence once a return to unfavorable conditions prevails. The stronger this ratcheting of density dependence in life-history traits by autocorrelated climatic conditions, the larger the population fluctuations become. Highly fecund species, and those in which density dependence occurs in juvenile and adult vital rates at similar densities, are most sensitive to increases in the frequency of favorable conditions. Understanding the influence of global warming on temporal correlation in regional environmental conditions will be important in identifying those species liable to exhibit increased population fluctuations that could lead to their extinction.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

Extinctions in competitive communities forced by coloured environmental variation

Understanding the relationships between environmental fluctuations, population dynamics and species interactions in natural communities is of vital theoretical and practical importance. This knowledge is essential in assessing extinction risks in communities that are, for example, pressed by changing environmental conditions and increasing exploitation. We developed a model of density dependent population renewal, in a Lotka–Volterra competitive community context, to explore the significance of interspecific interactions, demographic stochasticity, population growth rate and species abundance on extinction risk in populations under various autocorrelation (colour) regimes of environmental forcing. These factors were evaluated in two cases, where either a single species or the whole community was affected by the external forcing. Species' susceptibility to environmental noise with different autocorrelation structure depended markedly on population dynamics, species' position in the abundance hierarchy and how similarly community members responded to external forcing. We also found interactions between demographic stochasticity and environmental noise leading to a reversal in extinction probabilities from under- to overcompensatory dynamics. We compare our results with studies of single species populations and contrast possible mechanisms leading to extinctions. Our findings indicate that abundance rank, the form of population dynamics, and the colour of environmental variation interact in affecting species extinction risk. These interactions are further modified by interspecific interactions within competitive communities as the interactions filter and modulate the environmental noise.     

Extinction risk under coloured environmental noise

Positively autocorrelated red environmental noise is characterized by a strong dependence of expected sample variance on sample length. This dependence has to be taken into account when assessing extinction risk under red and white uncorrelated environmental noise. To facilitate a comparison between red and white noise, their expected variances can be scaled to be equal, but only at a chosen time scale. We show with a simple one-dimensional population dynamics model that the different but equally reasonable choices of the time scale yield qualitatively different results on the dependence of extinction risk on the colour of environmental noise: extinction risk might increase as well as decrease when the temporal correlation of noise increases.     

Climate and spatio-temporal variation in the population dynamics of a long distance migrant, the white stork

1. A central question in ecology is to separate the relative contribution of density dependence and stochastic influences to annual fluctuations in population size. Here we estimate the deterministic and stochastic components of the dynamics of different European populations of white stork Ciconia ciconia. We then examined whether annual changes in population size was related to the climate during the breeding period (the 'tap hypothesis' sensu Saether, Sutherland & Engen (2004, Advances in Ecological Research, 35, 185 209) or during the nonbreeding period, especially in the winter areas in Africa (the 'tube hypothesis'). 2. A general characteristic of the population dynamics of this long-distance migrant is small environmental stochasticity and strong density regulation around the carrying capacity with short return times to equilibrium. 3. Annual changes in the size of the eastern European populations were correlated by rainfall in the wintering areas in Africa as well as local weather in the breeding areas just before arrival and in the later part of the breeding season and regional climate variation (North Atlantic Oscillation). This indicates that weather influences the population fluctuations of white storks through losses of sexually mature individuals as well as through an effect on the number of individuals that manages to establish themselves in the breeding population. Thus, both the tap and tube hypothesis explains climate influences on white stork population dynamics. 4. The spatial scale of environmental noise after accounting for the local dynamics was 67 km, suggesting that the strong density dependence reduces the synchronizing effects of climate variation on the population dynamics of white stork. 5. Several climate variables reduced the synchrony of the residual variation in population size after accounting for density dependence and demographic stochasticity, indicating that these climate variables had a synchronizing effect on the population fluctuations. In contrast, other climatic variables acted as desynchronizing agents. 6. Our results illustrate that evaluating the effects of common environmental variables on the spatio-temporal variation in population dynamics require estimates and modelling of their influence on the local dynamics.