How noise and coupling influence leading indicators of population extinction in a spatially extended ecological system

Anticipating critical transitions in spatially extended systems is a key topic of interest to ecologists. Gradually declining metapopulations are an important example of a spatially extended biological system that may exhibit a critical transition. Theory for spatially extended systems approaching extinction that accounts for environmental stochasticity and coupling is currently lacking. Here, we develop spatially implicit two-patch models with additive and multiplicative forms of environmental stochasticity that are slowly forced through population collapse, through changing environmental conditions. We derive patch-specific expressions for candidate indicators of extinction and test their performance via a simulation study. Coupling and spatial heterogeneities decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations, and the noise regime and the degree of coupling together determine trends in summary statistics. This theory may be readily applied to other spatially extended ecological systems, such as coupled infectious disease systems on the verge of elimination.     

Heteroskedasticity as a leading indicator of desertification in spatially explicit data

Regime shifts are abrupt transitions between alternate ecosystem states including desertification in arid regions due to drought or overgrazing. Regime shifts may be preceded by statistical anomalies such as increased autocorrelation, indicating declining resilience and warning of an impending shift. Tests for conditional heteroskedasticity, a type of clustered variance, have proven powerful leading indicators for regime shifts in time series data, but an analogous indicator for spatial data has not been evaluated. A spatial analog for conditional heteroskedasticity might be especially useful in arid environments where spatial interactions are critical in structuring ecosystem pattern and process. We tested the efficacy of a test for spatial heteroskedasticity as a leading indicator of regime shifts with simulated data from spatially extended vegetation models with regular and scale‐free patterning. These models simulate shifts from extensive vegetative cover to bare, desert‐like conditions. The magnitude of spatial heteroskedasticity increased consistently as the modeled systems approached a regime shift from vegetated to desert state. Relative spatial autocorrelation, spatial heteroskedasticity increased earlier and more consistently. We conclude that tests for spatial heteroskedasticity can contribute to the growing toolbox of early warning indicators for regime shifts analyzed with spatially explicit data.     

Dynamics of populations on the verge of extinction

Theoretical considerations suggest that extinction in dispersal-limited populations is necessarily a threshold-like process that is analogous to a critical phase transition in physics. We use this analogy to find robust, common features in the dynamics of extinctions, and suggest early warning signals which may indicate that a population is endangered. As the critical threshold of extinction is approached, the population spontaneously fragments into discrete subpopulations and, consequently, density regulation fails. The population size declines and its spatial variance diverges according to scaling laws. Therefore, we can make robust predictions exactly in the range where prognosis is vital, on the verge of extinction.     

A first principles derivation of animal group size distributions

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.     

Scale-free extinction dynamics in spatially structured host-parasitoid systems

Much of the work on extinction events has focused on external perturbations of ecosystems, such as climatic change, or anthropogenic factors. Extinction, however, can also be driven by endogenous factors, such as the ecological interactions between species in an ecosystem. Here we show that endogenously driven extinction events can have a scale-free distribution in simple spatially structured host-parasitoid systems. Due to the properties of this distribution there may be many such simple ecosystems that, although not strictly permanent, persist for arbitrarily long periods of time. We identify a critical phase transition in the parameter space of the host-parasitoid systems, and explain how this is related to the scale-free nature of the extinction process. Based on these results, we conjecture that scale-free extinction processes and critical phase transitions of the type we have found may be a characteristic feature of many spatially structured, multi-species ecosystems in nature. The necessary ingredient appears to be competition between species where the locally inferior type disperses faster in space. If this condition is satisfied then the eventual outcome depends subtly on the strength of local superiority of one species versus the dispersal rate of the other.     

Effects of climate‐driven primary production change on marine food webs: implications for fisheries and conservation

Climate change is altering the rate and distribution of primary production in the world's oceans. Primary production is critical to maintaining biodiversity and supporting fishery catches, but predicting the response of populations to primary production change is complicated by predation and competition interactions. We simulated the effects of change in primary production on diverse marine ecosystems across a wide latitudinal range in Australia using the marine food web model Ecosim. We link models of primary production of lower trophic levels (phytoplankton and benthic producers) under climate change with Ecosim to predict changes in fishery catch, fishery value, biomass of animals of conservation interest, and indicators of community composition. Under a plausible climate change scenario, primary production will increase around Australia and generally this benefits fisheries catch and value and leads to increased biomass of threatened marine animals such as turtles and sharks. However, community composition is not strongly affected. Sensitivity analyses indicate overall positive linear responses of functional groups to primary production change. Responses are robust to the ecosystem type and the complexity of the model used. However, model formulations with more complex predation and competition interactions can reverse the expected responses for some species, resulting in catch declines for some fished species and localized declines of turtle and marine mammal populations under primary productivity increases. We conclude that climate‐driven primary production change needs to be considered by marine ecosystem managers and more specifically, that production increases can simultaneously benefit fisheries and conservation. Greater focus on incorporating predation and competition interactions into models will significantly improve the ability to identify species and industries most at risk from climate change.     

Population fluctuations,power laws and mixtures of lognormal distributions

A number of investigators have invoked a cascading local interaction model to account for power‐law‐distributed fluctuations in ecological variables. Invoking such a model requires that species be tightly coupled, and that local interactions among species influence ecosystem dynamics over a broad range of scales. Here we reanalyse bird population data used by Keitt & Stanley (1998, Dynamics of North American breeding bird populations. Nature, 393, 257–260) to support a cascading local interaction model. We find that the power law they report can be attributed to mixing of lognormal distributions. More tentatively, we propose that mixing of distributions accounts for other empirical power laws reported in the ecological literature.     

Timescales of population rarity and commonness in random environments

This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate (lambda) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If lambda < 1, the population process is generally transient with escape towards extinction. When lambda > or = 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When lambda >1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when lambda is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When lambda = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent -3/2. We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the -3/2 power law generally approximates the law of rarity of 'weakly invading' species with lambda values close to one. Some preliminary results for the dynamics of multispecific systems are presented.     

Extinction and invasion do not add up in noisy dynamic ecological networks

Species extinction and invasion concurrently affect the composition and properties of ecological communities, yet their effects have largely been studied separately, and with more focus on species and ecological functional groups than the whole-community level. We adopted a dynamic ecological network approach to compare the effects of simultaneous single-species primary extinction and invasion on a set of ecosystem metrics to the effects of extinction and invasion in isolation. We also investigated the relationship between the impact and reversibility of extinction or invasion through reintroduction or eradication, respectively. We used Monte Carlo simulations of bioenergetic ecological network models that combined trophic and mutualistic interactions, contained either prey-dependent or ratio-dependent trophic functional responses, and incorporated either white or pink environmental stochasticity. As the separate extinction or invasion impact increased, the simultaneous extinction–invasion impact increased but was decreasingly additive of the two separate impacts, across all ecosystem metrics. Greater extinction or invasion impact was associated with lower reversibility for most model types and ecosystem metrics. There were also systematic differences between models with prey- and ratio-dependent functional responses. These results highlight the importance of considering the combined effects of extinction and invasion in ecological studies, management and restoration.     

Robustness of early warning signals of regime shifts in time-delayed ecological models

Various ecological and other complex dynamical systems may exhibit abrupt regime shifts or critical transitions, wherein they reorganize from one stable state to another over relatively short time scales. Because of potential losses to ecosystem services, forecasting such unexpected shifts would be valuable. Using mathematical models of regime shifts, ecologists have proposed various early warning signals of imminent shifts. However, their generality and applicability to real ecosystems remain unclear because these mathematical models are considered too simplistic. Here, we investigate the robustness of recently proposed early warning signals of regime shifts in two well-studied ecological models, but with the inclusion of time-delayed processes. We find that the average variance may either increase or decrease prior to a regime shift and, thus, may not be a robust leading indicator in time-delayed ecological systems. In contrast, changing average skewness, increasing autocorrelation at short time lags, and reddening power spectra of time series of the ecological state variable all show trends consistent with those of models with no time delays. Our results provide insights into the robustness of early warning signals of regime shifts in a broader class of ecological systems.     

Biotic replacement and mass extinction of the Ediacara biota

The latest Neoproterozoic extinction of the Ediacara biota has been variously attributed to catastrophic removal by perturbations to global geochemical cycles, ‘biotic replacement’ by Cambrian-type ecosystem engineers, and a taphonomic artefact. We perform the first critical test of the ‘biotic replacement’ hypothesis using combined palaeoecological and geochemical data collected from the youngest Ediacaran strata in southern Namibia. We find that, even after accounting for a variety of potential sampling and taphonomic biases, the Ediacaran assemblage preserved at Farm Swartpunt has significantly lower genus richness than older assemblages. Geochemical and sedimentological analyses confirm an oxygenated and non-restricted palaeoenvironment for fossil-bearing sediments, thus suggesting that oxygen stress and/or hypersalinity are unlikely to be responsible for the low diversity of communities preserved at Swartpunt. These combined analyses suggest depauperate communities characterized the latest Ediacaran and provide the first quantitative support for the biotic replacement model for the end of the Ediacara biota. Although more sites (especially those recording different palaeoenvironments) are undoubtedly needed, this study provides the first quantitative palaeoecological evidence to suggest that evolutionary innovation, ecosystem engineering and biological interactions may have ultimately caused the first mass extinction of complex life.     

Do biotic interactions modulate ecosystem functioning along stress gradients? Insights from semi-arid plant and biological soil crust communities

Climate change will exacerbate the degree of abiotic stress experienced by semi-arid ecosystems. While abiotic stress profoundly affects biotic interactions, their potential role as modulators of ecosystem responses to climate change is largely unknown. Using plants and biological soil crusts, we tested the relative importance of facilitative–competitive interactions and other community attributes (cover, species richness and species evenness) as drivers of ecosystem functioning along stress gradients in semi-arid Mediterranean ecosystems. Biotic interactions shifted from facilitation to competition along stress gradients driven by water availability and temperature. These changes were, however, dependent on the spatial scale and the community considered. We found little evidence to suggest that biotic interactions are a major direct influence upon indicators of ecosystem functioning (soil respiration, organic carbon, water-holding capacity, compaction and the activity of enzymes related to the carbon, nitrogen and phosphorus cycles) along stress gradients. However, attributes such as cover and species richness showed a direct effect on ecosystem functioning. Our results do not agree with predictions emphasizing that the importance of plant–plant interactions will be increased under climate change in dry environments, and indicate that reductions in the cover of plant and biological soil crust communities will negatively impact ecosystems under future climatic conditions.     

Scaling environmental change through the community-level: a trait-based response-and-effect framework for plants

Predicting ecosystem responses to global change is a major challenge in ecology. A critical step in that challenge is to understand how changing environmental conditions influence processes across levels of ecological organization. While direct scaling from individual to ecosystem dynamics can lead to robust and mechanistic predictions, new approaches are needed to appropriately translate questions through the community level. Species invasion, loss, and turnover all necessitate this scaling through community processes, but predicting how such changes may influence ecosystem function is notoriously difficult. We suggest that community‐level dynamics can be incorporated into scaling predictions using a trait‐based response–effect framework that differentiates the community response to environmental change (predicted by response traits) and the effect of that change on ecosystem processes (predicted by effect traits). We develop a response‐and‐effect functional framework, concentrating on how the relationships among species' response, effect, and abundance can lead to general predictions concerning the magnitude and direction of the influence of environmental change on function. We then detail several key research directions needed to better scale the effects of environmental change through the community level. These include (1) effect and response trait characterization, (2) linkages between response‐and‐effect traits, (3) the importance of species interactions on trait expression, and (4) incorporation of feedbacks across multiple temporal scales. Increasing rates of extinction and invasion that are modifying communities worldwide make such a research agenda imperative.     

Spatial correlation as leading indicator of catastrophic shifts

Generic early-warning signals such as increased autocorrelation and variance have been demonstrated in time-series of systems with alternative stable states approaching a critical transition. However, lag times for the detection of such leading indicators are typically long. Here, we show that increased spatial correlation may serve as a more powerful early-warning signal in systems consisting of many coupled units. We first show why from the universal phenomenon of critical slowing down, spatial correlation should be expected to increase in the vicinity of bifurcations. Subsequently, we explore the applicability of this idea in spatially explicit ecosystem models that can have alternative attractors. The analysis reveals that as a control parameter slowly pushes the system towards the threshold, spatial correlation between neighboring cells tends to increase well before the transition. We show that such increase in spatial correlation represents a better early-warning signal than indicators derived from time-series provided that there is sufficient spatial heterogeneity and connectivity in the system.     

Multitrophic diversity effects of network degradation

Predicting the functional consequences of biodiversity loss in realistic, multitrophic communities remains a challenge. No existing biodiversity–ecosystem function study to date has simultaneously incorporated information on species traits, network topology, and extinction across multiple trophic levels, while all three factors are independently understood as critical drivers of post‐extinction network structure and function. We fill this gap by comparing the functional consequences of simulated species loss both within (monotrophic) and across (bitrophic) trophic levels, in an ecological interaction network estimated from spatially explicit field data on tropical fecal detritus producer and consumers (mammals and dung beetles). We simulated trait‐ordered beetle and mammal extinction separately (monotrophic extinction) and the coextinction of beetles following mammal loss (bitrophic extinction), according to network structure. We also compared the diversity effects of bitrophic extinction models using a standard monotrophic function (the daily production or consumption of fecal detritus) and a unique bitrophic functional metric (the proportion of daily detritus production that is consumed). We found similar mono‐ and bitrophic diversity effects, regardless of which species traits were used to drive extinctions, yet divergent predictions when different measures of function were used. The inclusion of information on network structure had little apparent effect on the qualitative relationship between diversity and function. These results contribute to our growing understanding of the functional consequences of biodiversity from real systems and underscore the importance of species traits and realistic functional metrics to assessments of the ecosystem impacts of network degradation through species loss.     

Spatial flows and the regulation of ecosystems

Spatial flows of materials and organisms across ecosystem boundaries are ubiquitous. Understanding the consequences of these flows should be a basic goal of ecosystem science, and yet it has received scant theoretical treatment to date. Here, using a simple, open, nutrient-limited ecosystem model with trophic interactions, we explore theoretically how spatial flows affect the functioning of local ecosystems, how physical mass-balance constraints interact with biological demographic constraints in the regulation of this functioning, and how failure to consider these constraints explicitly can lead to models that are ecologically inconsistent. In particular, we show that standard prey-dependent models for trophic interactions may lead to implausible outcomes when embedded in an ecosystem context with appropriate mass flows and mass-balance constraints. Our analysis emphasizes the need for integration of population, community, and ecosystem perspectives in ecology and the critical consequences of assuming closed versus open systems.     

Testing the effects of plant species loss on multiple ecosystem functions based on extinction scenarios

Ecosystem functions are threatened by continuing global loss of biodiversity. We simultaneously investigated three ecosystem functions and forage nutrient values following potential species extinction scenarios (dominant species removal, rare species removal, end-member species removal and random species removal) in a Mongolian grassland. ANPP, forage nutrient values, litter decomposition, and soil respiration were measured one and/or two years after plant removal. DNA samples of microorganisms extracted from the soil were subjected to metagenomics analysis. Finally, we calculated the multifunctionality, and examined the relationship of multifunctionality with plant and microorganism diversity. Among ecosystem functions, ANPP and litter decomposition rate decreased under random and rare species extinction scenarios, respectively, and forage quality increased when only dominant species had been removed. Diversity and species composition of soil microorganism were not affected by plant species richness or removal scenario. Only genus-level diversity of bacteria and ANPP were significantly and positively correlated with microbial diversity. Taken together, decreasing species richness of plants and soil organisms rarely impaired multifunctionality. Ecosystem functions were relatively robust to realistic disturbances and species extinction in natural grasslands. However, as each function responded differently to the different sets of species removed, the consequences of a realistic non-random extinction scenario for multiple ecosystem functions should be critical to the management of biodiversity loss caused by different disturbances.     

Ecosystem services and management options as blanket indicators of ecosystem health

A pragmatic and integrative approach to evaluation of the environment combines ecosystem sciences, health sciences, and social sciences. Each has a crucial role to play: the ecosystem sciences provide information on the complex dynamics of ecosystems as they are influenced by stress and disturbance; the health sciences provide a methodology for systematic diagnosis of pathology, taxonomy of ills, and models for preventive as well as rehabilitative modes; the social sciences bring to the fore the importance of human values which are part and parcel of any health evaluation. The complexity of stress-response systems precludes anything approximating a complete understanding of mechanisms underpinning ecosystem transformations. However, the loss of ecosystem services and management options appears to be a general phenomenon that permits an overall evaluation of ecosystem health in both aquatic and terrestrial systems. Such blanket indicators take into account both the impairment of ecosystem function and societal values. This is illustrated by the history of ecosystem transformation in the Laurentian Lower Great Lakes and in the overharvested forest ecosystems of Eastern Canada. In both cases, cultural stress resulted in losses in highly valued ecosystem services and management options. These losses have been partially compensated for by new technologies that have permitted commercial use of the remaining lower quality resources. This process itself, however, may be pathological, reinforcing a degradation sequence rather than serving to restore ecosystem health.     

Indicators quantifying small pelagic fish interactions: application using a trophic model of the southern Benguela ecosystem

Three indicators quantifying interactions between species are developed for an upwelling system to provide useful measures for the comparison of marine ecosystem structure and function. Small pelagic fish are dominant in upwelling systems, and by definition, they are pivotal in a wasp-waist upwelling system. The indicator of interaction strength (IS) quantifies the effect that a change in biomass of one group has on abundance of other groups. The functional impact (FI) indicator quantifies the trophic impacts of species on their own and other functional groups or feeding guilds. The trophic replacement (TR) indicator quantifies the trophic similarity between a species that is removed from an ecosystem and other species in that ecosystem, i.e. it quantifies the ability of one group to trophically replace another. A trophic model of the southern Benguela ecosystem is used as an example for the application of the indicators. The strong similarities in trophic functioning of the southern Benguela ecosystem in the anchovy-dominated system of the 1980s, and the 1990s when there was a shift towards greater sardine abundance, are explained by the mutual trophic replacement abilities of anchovy and sardine. Differences between the proposed indicators and mixed trophic impact assessment are highlighted, mainly resulting from the static versus dynamic nature of the models upon which they are based. Trophic indicators such as those presented here, together with other kinds of ecosystem indicators, may assist in defining operational frameworks for ecosystem-based fisheries management.