Resilience in a two-population system: interactions between Allee effects and connectivity

Resilience in ecosystems and resistance to regime shifts has been a major focus in ecological research. How migration and general network dynamics affect the resilience of populations or induce regime shift cascades is a particularly challenging open question in theoretical ecology. We focus on regime shifts in populations with variable-strength Allee effects to demonstrate the effect of migration on resilience in two-population systems with critical transitions. The result is a mathematical model that justifies the assumption that resilience can be averaged across connected populations and suggests several management strategies to either avoid or induce regime shift cascades.     

Allee Effects, Propagule Pressure and the Probability of Establishment: Risk Analysis for Biological Invasions

Colonization is of longstanding interest in theoretical ecology and biogeography, and in the management of weeds and other invasive species, including insect pests and emerging infectious diseases. Due to accelerating invasion rates and widespread economic costs and environmental damages caused by invasive species, colonization theory has lately become a matter of considerable interest. Here we review the concept of propagule pressure to inquire if colonization theory might provide quantitative tools for risk assessment of biological invasions. By formalizing the concept of propagule pressure in terms of stochastic differential equation models of population growth, we seek a synthesis of invasion biology and theoretical population biology. We focus on two components of propagule pressure that affect the chance of invasion: (1) the number of individuals initially introduced, and (2) the rate of subsequent immigration. We also examine how Allee effects, which are expected to be common in newly introduced populations, may inhibit establishment of introduced propagules. We find that the establishment curve (i.e., the chance of invasion as a function of initial population size), can take a variety of shapes depending on immigration rate, carrying capacity, and the severity of Allee effects. Additionally, Allee effects can cause the stationary distribution of population sizes to be bimodal, which we suggest is a possible explanation for time lags commonly observed between the detection of an introduced population and widespread invasion of the landscape.     

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.     

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.     

Allee effects and population regulation: a test for biotic resistance against an invasive leafroller by resident parasitoids

Resident natural enemies can impact invasive species by causing Allee effects, leading to a reduction in establishment success of small founder populations, or by regulating or merely suppressing the abundance of established populations. Epiphyas postvittana, the Light Brown Apple Moth, an invasive leafroller in California, has been found to be attacked by a large assemblage of resident parasitoids that cause relatively high rates of parasitism. Over a 4-year period, we measured the abundance and per capita growth rates of four E. postvittana populations in California and determined parasitism rates. We found that at two of the sites, parasitism caused a component Allee effect, a reduction in individual survivorship at lower E. postvittana population densities, although it did not translate into a demographic Allee effect, an impact on per capita population growth rates at low densities. Instead, E. postvittana populations at all four sites exhibited strong compensatory density feedback throughout the entire range of densities observed at each site. As we found no evidence for a negative relationship between per capita population growth rates and parasitism rates, we concluded that resident parasitoids were unable to regulate E. postvittana populations in California. Despite a lack of evidence for regulation or a demographic Allee effect, the impact of resident parasitoids on E. postvittana populations is substantial and demonstrates significant biotic resistance against this new invader.     

Allee effects may slow the spread of parasites in a coastal marine ecosystem

Allee effects are thought to mediate the dynamics of population colonization, particularly for invasive species. However, Allee effects acting on parasites have rarely been considered in the analogous process of infectious disease establishment and spread. We studied the colonization of uninfected wild juvenile Pacific salmon populations by ectoparasitic salmon lice (Lepeophtheirus salmonis) over a 4-year period. In a data set of 68,376 fish, we observed 85 occurrences of precopular pair formation among 1,259 preadult female and 613 adult male lice. The probability of pair formation was dependent on the local abundance of lice, but this mate limitation is likely offset somewhat by mate-searching dispersal of males among host fish. A mathematical model of macroparasite population dynamics that incorporates the empirical results suggests a high likelihood of a demographic Allee effect, which can cause the colonizing parasite populations to die out. These results may provide the first empirical evidence for Allee effects in a macroparasite. Furthermore, the data give a rare detailed view of Allee effects in colonization dynamics and suggest that Allee effects may dampen the spread of parasites in a coastal marine ecosystem.     

Divergent density feedback control of migratory predator recovery following sex‐biased perturbations

Uncertainty in risks posed by emerging stressors such as synthetic hormones impedes conservation efforts for threatened vertebrate populations. Synthetic hormones often induce sex‐biased perturbations in exposed animals by disrupting gonad development and early life‐history stage transitions, potentially diminishing per capita reproductive output of depleted populations and, in turn, being manifest as Allee effects. We use a spatially explicit biophysical model to evaluate how sex‐biased perturbation in life‐history traits of individuals (maternal investment in egg production and male‐skewed sex allocation in offspring) modulates density feedback control of year‐class strength and recovery trajectories of a long‐lived, migratory fish—shovelnose sturgeon (Scaphirhynchus platorynchus)—under spatially and temporally dynamic synthetic androgen exposure and habitat conditions. Simulations show that reduced efficiency of maternal investment in gonad development prolonged maturation time, increased the probability of skipped spawning, and, in turn, shrunk spawner abundance, weakening year‐class strength. However, positive density feedback disappeared (no Allee effect) once the exposure ceased. By contrast, responses to the demographic perturbation manifested as strong positive density feedback; an abrupt shift in year‐class strength and spawner abundance followed after more than two decades owing to persistent negative population growth (a strong Allee effect), reaching an alternative state without any sign of recovery. When combined with the energetic perturbation, positive density feedback of the demographic perturbation was dampened as extended maturation time reduced the frequency of producing male‐biased offspring, allowing the population to maintain positive growth rate (a weak Allee effect) and gradually recover. The emergent patterns in long‐term population projections illustrate that sex‐biased perturbation in life‐history traits can interactively regulate the strength of density feedback in depleted populations such as Scaphirhynchus sturgeon to further diminish reproductive capacity and abundance, posing increasingly greater conservation challenges in chemically altered riverscapes.     

Spatial variance and spatial skewness: leading indicators of regime shifts in spatial ecological systems

Ecosystems can undergo large-scale changes in their states, known as catastrophic regime shifts, leading to substantial losses to services they provide to humans. These shifts occur rapidly and are difficult to predict. Several early warning signals of such transitions have recently been developed using simple models. These studies typically ignore spatial interactions, and the signal provided by these indicators may be ambiguous. We employ a simple model of collapse of vegetation in one and two spatial dimensions and show, using analytic and numerical studies, that increases in spatial variance and changes in spatial skewness occur as one approaches the threshold of vegetation collapse. We identify a novel feature, an increasing spatial variance in conjunction with a peaking of spatial skewness, as an unambiguous indicator of an impending regime shift. Once a signal has been detected, we show that a quick management action reducing the grazing activity is needed to prevent the collapse of vegetated state. Our results show that the difficulties in obtaining the accurate estimates of indicators arising due to lack of long temporal data can be alleviated when high-resolution spatially extended data are available. These results are shown to hold true independent of various details of model or different spatial dispersal kernels such as Gaussian or heavily fat tailed. This study suggests that spatial data and monitoring multiple indicators of regime shifts can play a key role in making reliable predictions on ecosystem stability and resilience. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Population Genetic Consequences of the Allee Effect and the Role of Offspring-Number Variation

A strong demographic Allee effect in which the expected population growth rate is negative below a certain critical population size can cause high extinction probabilities in small introduced populations. But many species are repeatedly introduced to the same location and eventually one population may overcome the Allee effect by chance. With the help of stochastic models, we investigate how much genetic diversity such successful populations harbor on average and how this depends on offspring-number variation, an important source of stochastic variability in population size. We find that with increasing variability, the Allee effect increasingly promotes genetic diversity in successful populations. Successful Allee-effect populations with highly variable population dynamics escape rapidly from the region of small population sizes and do not linger around the critical population size. Therefore, they are exposed to relatively little genetic drift. It is also conceivable, however, that an Allee effect itself leads to an increase in offspring-number variation. In this case, successful populations with an Allee effect can exhibit less genetic diversity despite growing faster at small population sizes. Unlike in many classical population genetics models, the role of offspring-number variation for the population genetic consequences of the Allee effect cannot be accounted for by an effective-population-size correction. Thus, our results highlight the importance of detailed biological knowledge, in this case on the probability distribution of family sizes, when predicting the evolutionary potential of newly founded populations or when using genetic data to reconstruct their demographic history.     

Disturbance, invasion and re-invasion: managing the weed-shaped hole in disturbed ecosystems

We aim to develop a simple model to explore how disturbance and propagule pressure determine conditions for successful invasion in systems where recruitment occurs only in disturbed sites. Disturbance is often thought to favour invaders as it allows recruitment; however, the effects of disturbance are more complicated when it results in mortality of the invader. When disturbance rates in both invader occupied and unoccupied sites are the same, recruitment and mortality effects are exactly balanced, and successful invasion is independent of the disturbance regime. Differences in the disturbance rates between invader occupied and unoccupied sites can occur through invader modification or management of disturbance. Under these conditions, we found a novel mechanism for the generation of an Allee effect, which occurs when the invader promotes disturbance in sites it already occupies. When Allee effects occur one-off, large-scale disturbances can result in permanent, dramatic shifts in invader abundance; and conversely, reducing the population below a critical threshold can cause extinction.     

Leading indicators of trophic cascades

Regime shifts are large, long-lasting changes in ecosystems. They are often hard to predict but may have leading indicators which are detectable in advance. Potential leading indicators include wider swings in dynamics of key ecosystem variables, slower return rates after perturbation and shift of variance towards lower frequencies. We evaluated these indicators using a food web model calibrated to long-term whole-lake experiments. We investigated whether impending regime shifts driven by gradual increase in exploitation of the top predator can create signals that cascade through food webs and be discerned in phytoplankton. Substantial changes in standard deviations, return rates and spectra occurred near the switch point, even two trophic levels removed from the regime shift in fishes. Signals of regime shift can be detected well in advance, if the driver of the regime shift changes much more slowly than the dynamics of key ecosystem variables which can be sampled frequently enough to measure the indicators. However, the regime shift may occur long after the driver has passed the critical point, because of very slow transient dynamics near the critical point. Thus, the ecosystem can be poised for regime shift by the time the signal is discernible. Field tests are needed to evaluate these indicators.     

Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion

We formulated a spatially explicit stochastic population model with an Allee effect in order to explore how invasive species may become established. In our model, we varied the degree of migration between local populations and used an Allee effect with variable birth and death rates. Because of the stochastic component, population sizes below the Allee effect threshold may still have a positive probability for successful invasion. The larger the network of populations, the greater the probability of an invasion occurring when initial population sizes are close to or above the Allee threshold. Furthermore, if migration rates are low, one or more than one patch may be successfully invaded, while if migration rates are high all patches are invaded.     

Allee effects, aggregation, and invasion success

Understanding the factors that influence successful colonization can help inform ecological theory and aid in the management of invasive species. When founder populations are small, individual fitness may be negatively impacted by component Allee effects through positive density dependence (e.g., mate limitation). Reproductive and survival mechanisms that suffer due to a shortage of conspecifics may scale up to be manifest in a decreased per-capita population growth rate (i.e., a demographic Allee effect). Mean-field population level models are limited in representing how component Allee effects scale up to demographic Allee effects when heterogeneous spatial structure influences conspecific availability. Thus, such models may not adequately characterize the probability of establishment. In order to better assess how individual level processes influence population establishment and spread, we developed a spatially explicit individual-based stochastic simulation of a small founder population. We found that increased aggregation can affect individual fitness and subsequently impact population growth; however, relatively slow dispersal—in addition to initial spatial structure—is required for establishment, ultimately creating a tradeoff between probability of initial establishment and rate of subsequent spread. Since this result is sensitive to the scaling up of component Allee effects, details of individual dispersal and interaction kernels are key factors influencing population level processes. Overall, we demonstrate the importance of considering both spatial structure and individual level traits in assessing the consequences of Allee effects in biological invasions.     

Modeling Responses of Leafy Spurge Dispersal to Control Strategies

Leafy spurge （Euphorbia esula L.） has substantial negative effects on grassland biodiversity, productivity, and economic benefit in North America. To predict these negative impacts, we need an appropriate plant-spread model which can simulate the response of an invading population to different control strategies. In this study, using a stochastic map lattice approach we generated a spatially explicitly stochastic process-based model to simulate dispersal trajectories of leafy spurge under various control scenarios. The model integrated dispersal curve, propagule pressure, and population growth of leafy spurge at local and short-temporal scales to capture spread features of leafy spurge at large spatial and long-temporal scales. Our results suggested that narrow-, medium-, and fat-tailed kernels did not differ in their ability to predict spread, in contrast to previous works. For all kernels, Allee effects were significantly present and could explain the lag phase （three decades） before leafy spurge spread accelerated. When simulating from the initial stage of introduction, Allee effects were critical in predicting spread rate of leafy spurge, because the prediction could be seriously affected by the low density period of leafy spurge community. No Allee effects models were not able to simulate spread rate well in this circumstance. When applying control strategies to the current distribution, Allee effects could stop the spread of leafy spurge; no Allee effects models, however, were able to slow but not stop the spread. The presence of Allee effects had significant ramifications on the efficiencies of control strategies. For both Allee and no Allee effects models, the later that control strategies were implemented, the more effort had to be input to achieve similar control results.     

Alternate attractors in the population dynamics of a tree-killing bark beetle

Among the most striking changes in ecosystems are those that happen abruptly and resist return to the original condition (i.e., regime shifts). This frequently involves conspicuous changes in the abundance of one species (e.g., an oubreaking pest or keystone species). Alternate attractors in population dynamics could explain switches between low and high levels of abundance, and could underlie some cases of regime shifts in ecosystems; this longstanding theoretical possibility has been difficult to test in nature. We compared the ability of an alternate attractors model versus two competing models to explain population fluctuations in the tree-killing bark beetle, Dendroctonus frontalis. Frequency distributions of abundance were distinctly bimodal, a prediction of the alternate attractors model, strongly indicating the lack of a single, noisy equilibrium. Time series abundance data refuted the existence of strong delayed density-dependence or nonlinearities, as required by the endogenous cycles model. The model of alternate attractors was further supported by the existence of positive density-dependence at intermediate beetle abundances. Experimental manipulations show that interactions with competitors and shared enemies could create a locally stable equilibrium in small populations of D. frontalis. High variation among regions and years in the abundance of predators and competitors could permit switches between alternate states. Dendroctonus frontalis now provides the strongest case that we know of for alternate attractors in natural population dynamics. The accompanying demographic instability appears to underlie spatially extensive outbreaks that have lasting impacts on forest ecosystems. Understanding feedbacks in populations with alternate attractors can help to identify thresholds underlying regime shifts, and potentially manage them to avoid undesirable impacts.     

干旱区生态系统稳态转换及其预警信号——基于景观格局特征的识别方法

在日益加剧的气候变化和土地开垦、放牧等人类活动干扰下,具有多稳态特征的干旱区生态系统可能会经历从相对健康状态到退化状态的稳态转换,导致生态系统的功能下降。早期预警信号的识别是生态系统稳态转换研究的热点,也是管理实践中防止生态系统退化的关键环节。以往预警信号研究聚焦于通用信号如自相关性、方差等统计学指标,然而这些指标对于具有特定机制的干旱区生态系统可能并不适用。基于干旱区景观格局特征所发展起来的空间指标为生态系统稳态转换提供了独特的空间视角,对于理解干旱区生态系统退化过程和机理具有科学意义和实践价值。介绍了干旱区生态系统稳态转换现象及其转换机制;聚焦景观生态学的指标和方法,从空间视角总结基于干旱区景观格局特征的关键预警指标(植被覆盖度、植被斑块形态、植被斑块大小频率分布和水文连通性等),重点剖析这些关键指标的概念、量化方法、识别特征及其实践应用;最后针对指标的优势和局限性对未来的研究方向进行展望,包括发掘潜在景观指标,加强干旱区生态系统变化的多种驱动要素的相互作用机制研究,开展多时空尺度的实证研究,构建生态系统稳态转换预警信号的整体分析框架,以及加强指标阈值的量化研究等方面。     

Dispersion in time and space affect mating success and Allee effects in invading gypsy moth populations

1. Understanding why invading populations sometimes fail to establish is of considerable relevance to the development of strategies for managing biological invasions. 2. Newly arriving populations tend to be sparse and are often influenced by Allee effects. Mating failure is a typical cause of Allee effects in low-density insect populations, and dispersion of individuals in space and time can exacerbate mate-location failure in invading populations. 3. Here we evaluate the relative importance of dispersal and sexual asynchrony as contributors to Allee effects in invading populations by adopting as a case study the gypsy moth (Lymantria dispar L.), an important insect defoliator for which considerable demographic information is available. 4. We used release-recapture experiments to parameterize a model that describes probabilities that males locate females along various spatial and temporal offsets between male and female adult emergence. 5. Based on these experimental results, we developed a generalized model of mating success that demonstrates the existence of an Allee threshold, below which introduced gypsy moth populations are likely to go extinct without any management intervention.     

Genetic Diversity in Introduced Populations with an Allee Effect

A phenomenon that strongly influences the demography of small introduced populations and thereby potentially their genetic diversity is the demographic Allee effect, a reduction in population growth rates at small population sizes. We take a stochastic modeling approach to investigate levels of genetic diversity in populations that successfully overcame either a strong Allee effect, in which populations smaller than a certain critical size are expected to decline, or a weak Allee effect, in which the population growth rate is reduced at small sizes but not negative. Our results indicate that compared to successful populations without an Allee effect, successful populations with a strong Allee effect tend to (1) derive from larger founder population sizes and thus have a higher initial amount of genetic variation, (2) spend fewer generations at small population sizes where genetic drift is particularly strong, and (3) spend more time around the critical population size and thus experience more genetic drift there. In the case of multiple introduction events, there is an additional increase in diversity because Allee-effect populations tend to derive from a larger number of introduction events than other populations. Altogether, a strong Allee effect can either increase or decrease genetic diversity, depending on the average founder population size. By contrast, a weak Allee effect tends to decrease genetic diversity across the entire range of founder population sizes. Finally, we show that it is possible in principle to infer critical population sizes from genetic data, although this would require information from many independently introduced populations.     

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.     

Changing skewness: an early warning signal of regime shifts in ecosystems

Empirical evidence for large-scale abrupt changes in ecosystems such as lakes and vegetation of semi-arid regions is growing. Such changes, called regime shifts, can lead to degradation of ecological services. We study simple ecological models that show a catastrophic transition as a control parameter is varied and propose a novel early warning signal that exploits two ubiquitous features of ecological systems: nonlinearity and large external fluctuations. Either reduced resilience or increased external fluctuations can tip ecosystems to an alternative stable state. It is shown that changes in asymmetry in the distribution of time series data, quantified by changing skewness, is a model-independent and reliable early warning signal for both routes to regime shifts. Furthermore, using model simulations that mimic field measurements and a simple analysis of real data from abrupt climate change in the Sahara, we study the feasibility of skewness calculations using data available from routine monitoring.