Critical slowing down as an indicator of transitions in two-species models

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.     

Measuring complexity to infer changes in the dynamics of ecological systems under stress

Despite advances in our mechanistic understanding of ecological processes, the inherent complexity of real-world ecosystems still limits our ability in predicting ecological dynamics especially in the face of on-going environmental stress. Developing a model is frequently challenged by structure uncertainty, unknown parameters, and limited data for exploring out-of-sample predictions. One way to address this challenge is to look for patterns in the data themselves in order to infer the underlying processes of an ecological system rather than to build system-specific models. For example, it has been recently suggested that statistical changes in ecological dynamics can be used to infer changes in the stability of ecosystems as they approach tipping points. For computer scientists such inference is similar to the notion of a Turing machine: a computational device that could execute a program (the process) to produce the observed data (the pattern). Here, we make use of such basic computational ideas introduced by Alan Turing to recognize changing patterns in ecological dynamics in ecosystems under stress. To do this, we use the concept of Kolmogorov algorithmic complexity that is a measure of randomness. In particular, we estimate an approximation to Kolmogorov complexity based on the Block Decomposition Method (BDM). We apply BDM to identify changes in complexity in simulated time-series and spatial datasets from ecosystems that experience different types of ecological transitions. We find that in all cases, K_BDM complexity decreased before all ecological transitions both in time-series and spatial datasets. These trends indicate that loss of stability in the ecological models we explored is characterized by loss of complexity and the emergence of a regular and computable underlying structure. Our results suggest that Kolmogorov complexity may serve as tool for revealing changes in the dynamics of ecosystems close to ecological transitions.     

Land Transitions in the Tropics: Going Beyond the Case Studies

Estimates of the percent of Earth's land surface that has either been transformed or degraded by human activity range between 39 and 50 percent, with agriculture accounting for the vast majority of these changes. Although much of the focus of research on land use and cover change in the tropics has been on deforestation, ongoing socioeconomic changes both locally and globally have made land transitions in the tropics extremely fluid. In addition, feedbacks between land cover change and human behavior constrain the extent and trajectories of land transitions. The sustainability of land use systems in the tropics depends on an understanding of coupled human–natural systems that can lead to general frameworks for management and prediction. The unprecedented availability of land use/cover data together with ecological data collected at large spatial scales offer exciting opportunities for advancing our understanding of socioecological systems. We rely on six studies of land transitions in the tropics to illustrate some promising approaches and pose critical questions to guide this body of research.     

Flickering as an early warning signal

Most work on generic early warning signals for critical transitions focuses on indicators of the phenomenon of critical slowing down that precedes a range of catastrophic bifurcation points. However, in highly stochastic environments, systems will tend to shift to alternative basins of attraction already far from such bifurcation points. In fact, strong perturbations (noise) may cause the system to “flicker” between the basins of attraction of the system’s alternative states. As a result, under such noisy conditions, critical slowing down is not relevant, and one would expect its related generic leading indicators to fail, signaling an impending transition. Here, we systematically explore how flickering may be detected and interpreted as a signal of an emerging alternative attractor. We show that—although the two mechanisms differ—flickering may often be reflected in rising variance, lag-1 autocorrelation and skewness in ways that resemble the effects of critical slowing down. In particular, we demonstrate how the probability distribution of a flickering system can be used to map potential alternative attractors and their resilience. Thus, while flickering systems differ in many ways from the classical image of critical transitions, changes in their dynamics may carry valuable information about upcoming major changes.     

Ecosystems off track: rate‐induced critical transitions in ecological models

Theory suggests that gradual environmental change may erode the resilience of ecosystems and increase their susceptibility to critical transitions. This notion has received a lot of attention in ecology in recent decades. An important question receiving far less attention is whether ecosystems can cope with the rapid environmental changes currently imposed. The importance of this question was recently highlighted by model studies showing that elevated rates of change may trigger critical transitions, whereas slow environmental change would not. This paper aims to provide a mechanistic understanding of these rate‐induced critical transitions to facilitate identification of rate sensitive ecosystems. Analysis of rate sensitive ecological models is challenging, but we demonstrate how rate‐induced transitions in an elementary model can still be understood. Our analyses reveal that rate‐induced transitions 1) occur if the rate of environmental change is high compared to the response rate of ecosystems, 2) are driven by rates, rather than magnitudes, of change and 3) occur once a critical rate of change is exceeded. Disentangling rate‐induced transitions from classical transitions in observations would be challenging. However, common features of rate‐sensitive models suggest that ecosystems with coupled fast–slow dynamics, exhibiting repetitive catastrophic shifts or displaying periodic spatial patterns are more likely to be rate sensitive. Our findings are supported by experimental studies showing rate‐dependent outcomes. Rate sensitivity of models suggests that the common definition of ecological resilience is not suitable for a subset of real ecosystems and that formulating limits to magnitudes of change may not always safeguard against ecosystem degradation. Synthesis Understanding and predicting ecosystem response to environmental change is one of the key challenges in ecology. Model studies have suggested that slow, gradual environmental change beyond some critical threshold can trigger so‐called critical transitions and abrupt ecosystem degradation. An important question remains however whether ecosystems can cope with the ongoing rapid anthropogenic environmental changes to which they are currently imposed. In this study we demonstrate that in some ecological models elevated rates of change can trigger critical transitions even if slow environmental change of the same magnitude would not. Such rateinduced critical transitions in models suggest that concepts like resilience and planetary boundaries may not always be sufficient to explain and prevent ecosystem degradation.     

Self-reproducing systems: structure, niche relations and evolution.

A formal definition of a self-reproducing system is proposed using Petri nets. A potential self-reproducing system is a set of places in the Petri net such that the number of tokens in each place increases due to some sequence of internal transitions (a transition is called internal to the marked subset of places if at least one of its starting places and one of its terminating places belongs to that subset). An actual self-reproducing system is a system that compensates the outflow of its components by reproduction. In a suitable environment every potential self-reproducing system becomes an actual one. Each Petri net can be considered as an ecosystem with the web of ecological niches bound together with trophic and other relations. The stationary dynamics of the ecosystem is characterized by the set of filled niches. The process of evolution is described in terms of niche composition change. Perspectives of the theory of self-reproducing systems in biology are discussed.     

Recovery after mass extinction: evolutionary assembly in large-scale biosphere dynamics

Biotic recoveries following mass extinctions are characterized by a process in which whole ecologies are reconstructed from low-diversity systems, often characterized by opportunistic groups. The recovery process provides an unexpected window to ecosystem dynamics. In many aspects, recovery is very similar to ecological succession, but important differences are also apparently linked to the innovative patterns of niche construction observed in the fossil record. In this paper, we analyse the similarities and differences between ecological succession and evolutionary recovery to provide a preliminary ecological theory of recoveries. A simple evolutionary model with three trophic levels is presented, and its properties (closely resembling those observed in the fossil record) are compared with characteristic patterns of ecological response to disturbances in continuous models of three-level ecosystems.     

Localized states qualitatively change the response of ecosystems to varying conditions and local disturbances

The response of dynamical systems to varying conditions and disturbances is a fundamental aspect of their analysis. In spatially extended systems, particularly in pattern-forming systems, there are many possible responses, including critical transitions, gradual transitions and locally confined responses. Here, we use the context of vegetation dynamics in drylands in order to study the response of pattern-forming ecosystems to oscillating precipitation and local disturbances. We focus on two precipitation ranges, a bistability range of bare soil with a patterned vegetation state, and a bistability range of uniform vegetation with a patterned vegetation state. In these ranges, there are many different stable states, which allow for both abrupt and gradual transitions between the system states to occur. We find that large amplitude oscillations of the precipitation rate can lead to a collapse of the vegetation in one range, while in the other range, they result in the convergence to a patterned state with a preferred wavelength. In addition, we show that a series of local disturbances results in the collapse of the vegetation in one range, while it drives the system toward fluctuations around a finite average biomass in the other range. Moreover, it is shown that under certain conditions, local disturbances can actually increase the overall vegetation density. These significant differences in the system response are attributed to the existence of localized states in one of the bistability ranges.     

On signals of phase transitions in salmon population dynamics

Critical slowing down (CSD) reflects the decline in resilience of equilibria near a bifurcation and may reveal early warning signals (EWS) of ecological phase transitions. We studied CSD in the recruitment dynamics of 120 stocks of three Pacific salmon (Oncorhynchus spp.) species in relation to critical transitions in fishery models. Pink salmon (Oncorhynchus gorbuscha) exhibited increased variability and autocorrelation in populations that had a growth parameter, r, close to zero, consistent with EWS of extinction. However, models and data for sockeye salmon (Oncorhynchus nerka) indicate that portfolio effects from heterogeneity in age-at-maturity may obscure EWS. Chum salmon (Oncorhynchus keta) show intermediate results. The data do not reveal EWS of Ricker-type bifurcations that cause oscillations and chaos at high r. These results not only provide empirical support for CSD in some ecological systems, but also indicate that portfolio effects of age structure may conceal EWS of some critical transitions.     

Rapid evolution and the convergence of ecological and evolutionary time

Recent studies have documented rates of evolution of ecologically important phenotypes sufficiently fast that they have the potential to impact the outcome of ecological interactions while they are underway. Observations of this type go against accepted wisdom that ecological and evolutionary dynamics occur at very different time scales. While some authors have evaluated the rapidity of a measured evolutionary rate by comparing it to the overall distribution of measured evolutionary rates, we believe that ecologists are mainly interested in rapid evolution because of its potential to impinge on ecological processes. We therefore propose that rapid evolution be defined as a genetic change occurring rapidly enough to have a measurable impact on simultaneous ecological change. Using this definition we propose a framework for decomposing rates of ecological change into components driven by simultaneous evolutionary change and by change in a non‐evolutionary factor (e.g. density dependent population dynamics, abiotic environmental change). Evolution is judged to be rapid in this ecological context if its contribution to ecological change is large relative to the contribution of other factors. We provide a worked example of this approach based on a theoretical predator–prey interaction [ Abrams, P. & Matsuda, H. (1997) . Evolution, 51, 1740], and find that in this system the impact of prey evolution on predator per capita growth rate is 63% that of internal ecological dynamics. We then propose analytical methods for measuring these contributions in field situations, and apply them to two long‐term data sets for which suitable ecological and evolutionary data exist. For both data sets relatively high rates of evolutionary change have been found when measured as character change in standard deviations per generation (haldanes). For Darwin's finches evolving in response to fluctuating rainfall [ Grant, P.R. & Grant, B.R. (2002) . Science, 296, 707], we estimate that evolutionary change has been more rapid than ecological change by a factor of 2.2. For a population of freshwater copepods whose life history evolves in response to fluctuating fish predation [ Hairston, N.G. Jr & Dillon, T.A. (1990) . Evolution, 44, 1796], we find that evolutionary change has been about one quarter the rate of ecological change – less than in the finch example, but nevertheless substantial. These analyses support the view that in order to understand temporal dynamics in ecological processes it is critical to consider the extent to which the attributes of the system under investigation are simultaneously changing as a result of rapid evolution.     

Social structure and life-history patterns in western gorillas (Gorilla gorilla gorilla)

Life-history traits and ecological conditions have an important influence on primate social systems. Most of what we know about the life-history patterns and social structure of gorillas comes from studies of eastern gorillas (Gorilla beringei sp.), which live under dramatically different ecological conditions compared to western gorillas (Gorilla gorilla sp.). In this paper we present new data on western gorilla social structure and life histories from four study sites, and make comparisons with eastern gorilla populations. Data were obtained from two study sites with gorilla groups undergoing the habituation process (Lossi, Democratic Republic of Congo and Bai Hokou, Central African Republic) and two "bai" studies (Maya Nord and Mbeli Bai, Republic of Congo). The size and structure of these groups were similar to those seen in eastern gorillas. However, differences in the occurrence of various group transitions (group formations, changes between one-male and multimale composition, and group disintegrations) exist, and western gorillas notably exhibit much higher rates of male emigration and correspondingly fewer multimale groups compared to mountain gorillas. Certain phenomena have been observed only rarely, including predation by leopards. The preliminary data show no significant differences in birth rates between western gorillas and mountain gorillas. The ecological variability across gorilla habitats likely explains the flexibility in the social system of gorillas, but we need more information on the social relationships and ecology of western gorillas to elucidate the causes for the similarities and differences between western and eastern gorillas on the levels of individuals, social groups, and population dynamics.     

When can herbivores slow or reverse the spread of an invading plant? A test case from Mount St. Helens

Here we study the spatial dynamics of a coinvading consumer-resource pair. We present a theoretical treatment with extensive empirical data from a long-studied field system in which native herbivorous insects attack a population of lupine plants recolonizing a primary successional landscape created by the 1980 volcanic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its herbivores, we develop a system of integrodifference equations to study plant-herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low-density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of "invasion momentum," beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents to limit the spread of pest species.     

Coevolution of slow-fast populations: evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics

We study the interplay of ecological and evolutionary dynamics in communities composed of populations with contrasting time-scales. In such communities, genetic variation of individual traits can cause population transitions between stationary and cyclic ecological regimes, hence abrupt variations in fitness. Such abrupt variations raise ridges in the adaptive landscape, where the populations are poised between equilibrium and cyclic coexistence and along which evolutionary trajectories can remain sliding for long times or halt at special points called evolutionary pseudo-equilibria. These novel phenomena should be generic to all systems in which ecological interactions cause fitness to vary discontinuously. They are demonstrated by the analysis of a predator-prey community, with one adaptive trait for each population. The eco-evolutionary dynamics of the system show a number of other distinctive features, including evolutionary extinction and two forms of Red Queen dynamics. One of them is characterized by intermittent bouts of cyclic oscillations of the two populations.     

Zero sum, the niche, and metacommunities: long-term dynamics of community assembly

Recent models of community assembly, structure, and dynamics have incorporated, to varying degrees, three mechanistic processes: resource limitation and interspecific competition, niche requirements of species, and exchanges between a local community and a regional species pool. Synthesizing 30 years of data from an intensively studied desert rodent community, we show that all of these processes, separately and in combination, have influenced the structural organization of this community and affected its dynamical response to both natural environmental changes and experimental perturbations. In addition, our analyses suggest that zero-sum constraints, niche differences, and metacommunity processes are inextricably linked in the ways that they affect the structure and dynamics of this system. Explicit consideration of the interaction of these processes should yield a deeper understanding of the assembly and dynamics of other ecological communities. This synthesis highlights the role that long-term data, especially when coupled with experimental manipulations, can play in assessing the fundamental processes that govern the structure and function of ecological communities.     

Shifts in the dynamics of productivity signal ecosystem state transitions at the biome‐scale

Understanding ecosystem dynamics and predicting directional changes in ecosystem in response to global changes are ongoing challenges in ecology. Here we present a framework that links productivity dynamics and ecosystem state transitions based on a spatially continuous dataset of aboveground net primary productivity (ANPP) from the temperate grassland of China. Across a regional precipitation gradient, we quantified spatial patterns in ANPP dynamics (variability, asymmetry and sensitivity to rainfall) and related these to transitions from desert to semi‐arid to mesic steppe. We show that these three indices of ANPP dynamics displayed distinct spatial patterns, with peaks signalling transitions between grassland types. Thus, monitoring shifts in ANPP dynamics has the potential for predicting ecosystem state transitions in the future. Current ecosystem models fail to capture these dynamics, highlighting the need to incorporate more nuanced ecological controls of productivity in models to forecast future ecosystem shifts.     

熵权模糊综合评价法在城市生态安全评价中的应用

为了拓新城市复合生态系统生态安全评价方法的研究,根据压力-状态-响应模型,构建了一个有3个要素和33项具体指标的城市生态安全水平度量的指标体系,并运用客观的熵权法赋权、模糊综合评价法以及划分的等级标准对我国5个经济发达城市的生态安全水平进行量化分析.结果表明,苏州市和北京市对“较安全”级别的隶属度最大,分别为0.376和0.286;深圳市、上海市和广州市则处于“临界安全”状态.另外,深圳市具有较大的系统压力,苏州市的系统状态和系统响应表现最优.与其它评价方法相比,该方法评价过程简易,结果定量和相对客观可信.     

Back-and-forth hermaphroditism: phylogenetic context of reproductive system evolution in subdioecious Daphne laureola

Recent phylogenetic analyses of sexual reproductive systems supported the evolutionary pathway from hermaphroditism to dioecy via gynodioecy in different groups of angiosperms. In this study, we explore the evolution of sexual reproductive systems in Daphne laureola L. (Thymelaeaceae), a species with variation in reproductive system among population. Sequences from the ITS region of the nuclear ribosomal cistron and two plastid markers (psbA-trnH and ndhF) were analyzed and used to map the population reproductive system along the molecular phylogeny. Our results support D. laureola as a monophyletic lineage with three different clades within the Iberian Peninsula. The hermaphroditic populations belong to two different clades, whereas gynodioecy is ubiquitous but characteristic of the third clade, which grouped together all the North-Western Iberian populations sampled, including the apparently oldest haplotype sampled. Gynodioecy appears as the most likely basal condition of the 13 analyzed populations, but different evolutionary transitions in reproductive sexual system were traced within each D. laureola clade. Both ecological conditions and (meta)population dynamics may help explain plant reproductive system evolution at the microevolutionary scale. Phylogenetic studies in which the historical relationships between populations differing in reproductive system can be ascertained will help to clarify the process.     

Forecasting species ranges by statistical estimation of ecological niches and spatial population dynamics

Aim The study and prediction of species–environment relationships is currently mainly based on species distribution models. These purely correlative models neglect spatial population dynamics and assume that species distributions are in equilibrium with their environment. This causes biased estimates of species niches and handicaps forecasts of range dynamics under environmental change. Here we aim to develop an approach that statistically estimates process‐based models of range dynamics from data on species distributions and permits a more comprehensive quantification of forecast uncertainties. Innovation We present an approach for the statistical estimation of process‐based dynamic range models (DRMs) that integrate Hutchinson's niche concept with spatial population dynamics. In a hierarchical Bayesian framework the environmental response of demographic rates, local population dynamics and dispersal are estimated conditional upon each other while accounting for various sources of uncertainty. The method thus: (1) jointly infers species niches and spatiotemporal population dynamics from occurrence and abundance data, and (2) provides fully probabilistic forecasts of future range dynamics under environmental change. In a simulation study, we investigate the performance of DRMs for a variety of scenarios that differ in both ecological dynamics and the data used for model estimation. Main conclusions Our results demonstrate the importance of considering dynamic aspects in the collection and analysis of biodiversity data. In combination with informative data, the presented framework has the potential to markedly improve the quantification of ecological niches, the process‐based understanding of range dynamics and the forecasting of species responses to environmental change. It thereby strengthens links between biogeography, population biology and theoretical and applied ecology.     

Signal extraction from long-term ecological data using Bayesian and non-Bayesian state-space models

Emerging ecological time series from long-term ecological studies and remote sensing provide excellent opportunities for ecologists to study the dynamic patterns and governing processes of ecological systems. However, signal extraction from long-term time series often requires system learning (e.g., estimation of true system state) to process the large amount of information, to reconstruct system state, to account for measurement error, and to handle missing data. State-space models (SSMs) are a natural choice for these tasks and thus have received increasing attention in ecological and environmental studies. Data-based learning using SSMs that connect ecological processes to the measurement of system state becomes a useful technique in the ecological informatics toolkit. The present study illustrates the use of the Kalman filter (KF), an estimator of SSMs, with case studies of population dynamics. The examples of the SSM applications include the reconstruction of system state using the KF method and Markov chain Monte Carlo methods, estimation of measurement-error variances in the estimates of animal population abundance using basic structural models (BSMs), and estimation of missing values using the KF and Kalman smoother. Estimation of measurement-error variances by BSMs does not require knowledge of the functional form that generates the time series data. Instead, BSMs approximate the trajectory or deterministic skeleton of a system dynamics in a semi-parametric fashion, and provide a robust estimator of measurement-error variances. The present study also compares Bayesian SSMs with non-Bayesian SSMs. The joint use of the KF method or its extensions and Markov chain Monte Carlo (MCMC) methods is a promising approach to the parameter estimation of SSMs.