Allee effects, invasion pinning, and species' borders

All species' ranges are the result of successful past invasions. Thus, models of species' invasions and their failure can provide insight into the formation of a species' geographic range. Here, we study the properties of invasion models when a species cannot persist below a critical population density known as an "Allee threshold." In both spatially continuous reaction-diffusion models and spatially discrete coupled ordinary-differential-equation models, the Allee effect can cause an invasion to fail. In patchy landscapes (with dynamics described by the spatially discrete model), range limits caused by propagation failure (pinning) are stable over a wide range of parameters, whereas, in an uninterrupted habitat (with dynamics described by a spatially continuous model), the zero velocity solution is structurally unstable and thus unlikely to persist in nature. We derive conditions under which invasion waves are pinned in the discrete space model and discuss their implications for spatially complex dynamics, including critical phenomena, in ecological landscapes. Our results suggest caution when interpreting abrupt range limits as stemming either from competition between species or a hard environmental limit that cannot be crossed: under a wide range of plausible ecological conditions, species' ranges may be limited by an Allee effect. Several example systems appear to fit our general model.     

Integrodifference equations, Allee effects, and invasions

 Models that describe the spread of invading organisms often assume no Allee effect. In contrast, abundant observational data provide evidence for Allee effects. We study an invasion model based on an integrodifference equation with an Allee effect. We derive a general result for the sign of the speed of invasion. We then examine a special, linear–constant, Allee function and introduce a numerical scheme that allows us to estimate the speed of traveling wave solutions. Received: 14 April 2000 / Revised version: 23 December 2000 / Published online: 8 February 2002     

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.     

Hierarchical competition models with the Allee effect II: the case of immigration

ABSTRACT

This is part II of an earlier paper that dealt with hierarchical models with the Allee effect but with no immigration. In this paper, we greatly simplify the proofs in part I and provide a proof of the global dynamics of the non-hyperbolic cases that were previously conjectured. Then, we show how immigration to one of the species or to both would, drastically, change the dynamics of the system. It is shown that if the level of immigration to one or to both species is above a specified level, then there will be no extinction region where both species go to extinction.     

Functional traits,convergent evolution,and periodic tables of niches

Ecology is often said to lack general theories sufficiently predictive for applications. Here, we examine the concept of a periodic table of niches and feasibility of niche classification schemes from functional trait and performance data. Niche differences and their influence on ecological patterns and processes could be revealed effectively by first performing data reduction/ordination analyses separately on matrices of trait and performance data compiled according to logical associations with five basic niche ‘dimensions’, or aspects: habitat, life history, trophic, defence and metabolic. Resultant patterns then are integrated to produce interpretable niche gradients, ordinations and classifications. Degree of scheme periodicity would depend on degrees of niche conservatism and convergence causing species clustering across multiple niche dimensions. We analysed a sample data set containing trait and performance data to contrast two approaches for producing niche schemes: species ordination within niche gradient space, and niche categorisation according to trait‐value thresholds. Creation of niche schemes useful for advancing ecological knowledge and its applications will depend on research that produces functional trait and performance datasets directly related to niche dimensions along with criteria for data standardisation and quality. As larger databases are compiled, opportunities will emerge to explore new methods for data reduction, ordination and classification.     

具有Allee效应的合作进化

Allee效应对物种的续存是潜在的干扰因素,在很大程度上将增加种群局部甚至全局灭绝的可能性。对许多物种,尤其是濒临物种更容易受其影响。将Allee效应引入囚徒困境博弈模型,通过理论分析与数值模拟相结合的方法分析讨论了Allee效应对合作进化的影响。研究结果表明:在恶劣的环境条件下,Allee效应极易使物种灭绝,不利于合作进化;在相对优越的环境条件下(死亡率较低),Allee效应促进合作进化,且Allee效应强度越强,更有利于合作进化,不过种群的空间斑块占有率也会随着Allee效应强度的增强而降低,使物种最终灭绝。     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Multiple Allee effects and population management

Allee effects, strongly related to the extinction vulnerability of populations and gradually becoming acknowledged by both theoretically oriented and applied ecologists, have already been shown to have important roles in the dynamics of many populations. Although not yet widely recognized, two or more Allee effects can occur simultaneously in the same population. Here, we review the evidence for multiple Allee effects and show that their interactions can take several forms, many of which are far from inconsequential. We suggest that more research is needed to assess the prevalence and interactions of multiple Allee effects, as failing to take them into account could have adverse consequences for the management of threatened or exploited populations.     

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.     

Pushed beyond the brink: Allee effects,environmental stochasticity,and extinction

To understand the interplay between environmental stochasticity and Allee effects, we analyse persistence, asymptotic extinction, and conditional persistence for stochastic difference equations. Our analysis reveals that persistence requires that the geometric mean of fitness at low densities is greater than one. When this geometric mean is less than one, asymptotic extinction occurs with high probability for low initial population densities. Additionally, if the population only experiences positive density-dependent feedbacks, conditional persistence occurs provided the geometric mean of fitness at high population densities is greater than one. However, if the population experiences both positive and negative density-dependent feedbacks, conditional persistence only occurs if environmental fluctuations are sufficiently small. We illustrate counter-intuitively that environmental fluctuations can increase the probability of persistence when populations are initially at low densities, and can cause asymptotic extinction of populations experiencing intermediate predation rates despite conditional persistence occurring at higher predation rates.     

Overcoming Allee effects through evolutionary,genetic, and demographic rescue

Despite the amplified threats of extinction facing small founder populations, successful colonization sometimes occurs, bringing devastating ecological and economic consequences. One explanation may be rapid evolution, which can increase mean fitness in populations declining towards extinction, permitting persistence and subsequent expansion. Such evolutionary rescue may be particularly important, given Allee effects. When a population is introduced at low density, individuals often experience a reduction in one or more components of fitness due to novel selection pressures that arise from diminished intraspecific interactions and positive density dependence (i.e. component Allee effects). A population can avoid extinction if it can adapt and recover on its own (i.e. evolutionary rescue), or if additional immigration sustains the population (i.e. demographic rescue) or boosts its genetic variation that facilitates adaptation (i.e. genetic rescue). These various forms of rescue have often been invoked as possible mechanisms for specific invasions, but their relative importance to invasion is not generally understood. Within a spatially explicit modelling framework, we consider the relative impact of each type of rescue on the probability of successful colonization, when there is evolution of a multi-locus quantitative trait that influences the strength of component Allee effects. We demonstrate that when Allee effects are important, the effect of demographic rescue via recurrent immigration overall provides the greatest opportunity for success. While highlighting the role of evolution in the invasion process, we underscore the importance of the ecological context influencing the persistence of small founder populations.     

Combined impacts of Allee effects and parasitism

Despite their individual importance for population dynamics and conservation biology, the combined impacts of Allee effects and parasitism have received little attention. We built a mathematical model to compare the dynamics of populations with or without Allee effects when infected by microparasites. We show that the influence of an Allee effect takes the form of a tradeoff. The presence of an Allee effect in host populations may protect them, by reducing the range of population sizes that allow parasite spread. Yet if infection spreads, the Allee effect weakens host populations by reducing their size and by widening the range of parasite species that lead them to extinction. These results have important implications for predicting the survival of threatened populations or the success of reintroductions, and may help define size ranges within which given populations should be maintained to prevent both epidemics and Allee effects driven extinctions.     

集合种群的Allee效应研究进展

随着全球环境破坏的加剧,物种丧失的速度加快,人们日益关注生物多样性的保护。种群生物学和自然保护生物学的一些研究表明,如果一个局域种群受到Allee效应的影响,最终可能走向灭绝。从物种保护的角度考虑,分别介绍了集合种群水平上的Allee效应的和似Allee效应,比较了集合种群的Allee效应和似Allee效应产生的原因,以及集合种群的Allee效应和局域种群的Allee效应之间的关系、集合种群的似Allee效应和局域种群的Allee效应之间的关系,并提出集合种群的Allee效应还需要进一步的研究。     

With Allee effects,life for the social carnivore is complicated

Anthropogenic modification of the landscape, resultant habitat loss, and decades of persecution have resulted in severe decline and fragmentation of large carnivore populations worldwide. Infectious disease is also identified as a primary threat to many carnivores. In wildlife species, population demography and group persistence are strongly influenced by group or population size. This is referred to as the Allee effect, in which a population or group is at an increased risk of extinction when the number or density of individuals falls below some threshold due to ecological and/or genetic factors. However, in social mammalian species, the relationship between the number of individuals and the risk of extinction is complicated because aggregation may enhance pathogen exposure and transmission. Although theoretical studies of the interaction between infectious disease transmission and Allee effects reveal important implications for carnivore management and population extinction risk, information about the interaction has yet to be synthesized. In this paper, we assess life history strategies of medium to large carnivore species (≥2.4 kg) and their influence on population dynamics, with a special focus on infectious disease. While declining population trends are observed in 73 % of all carnivores (both social and solitary species), infectious disease is identified as a significant cause of population decline in 45 % of social carnivores and 3 % of solitary carnivores. Furthermore, where carnivores suffer a combination of rapid population decline and infectious disease, Allee effects may be more likely to impact social as compared to solitary carnivore populations. These potentially additive interactions may strongly influence disease transmission dynamics and population persistence potential. Understanding the mechanisms that can result in Allee effects in endangered carnivore populations and the manner in which infectious disease interfaces at this nexus may define the outcome of developed conservation strategies.     

Allee effects,extinctions, and chaotic transients in simple population models

Discrete time single species models with overcompensating density dependence and an Allee effect due to predator satiation and mating limitation are investigated. The models exhibit four behaviors: persistence for all initial population densities, bistability in which a population persists for intermediate initial densities and otherwise goes extinct, extinction for all initial densities, and essential extinction in which "almost every" initial density leads to extinction. For fast-growing populations, these models show populations can persist at high levels of predation even though lower levels of predation lead to essential extinction. Alternatively, increasing the predator's handling time, the population's carrying capacity, or the likelihood of mating success may lead to essential extinction. In each of these cases, the mechanism behind these disappearances are chaotic dynamics driving populations below a critical threshold determined by the Allee effect. These disappearances are proceeded by chaotic transients that are proven to be approximately exponentially distributed in length and highly sensitive to initial population densities.     

Variation in Allee effects: evidence,unknowns, and directions forward

Allee effects, positive effects of population size or density on per-capita fitness, are of broad interest in ecology and conservation due to their importance to the persistence of small populations and to range boundary dynamics. A number of recent studies have highlighted the importance of spatiotemporal variation in Allee effects and the resulting impacts on population dynamics. These advances challenge conventional understanding of Allee effects by reframing them as a dynamic factor affecting populations instead of a static condition. First, we synthesize evidence for variation in Allee effects and highlight potential mechanisms. Second, we emphasize the “Allee slope,” i.e., the magnitude of the positive effect of density on the per-capita growth rate, as a metric for demographic Allee effects. The more commonly used quantitative metric, the Allee threshold, provides only a partial picture of the underlying forces acting on population growth despite its implications for population extinction. Third, we identify remaining unknowns and strategies for addressing them. Outstanding questions about variation in Allee effects fall broadly under three categories: (1) characterizing patterns of natural variability; (2) understanding mechanisms of variation; and (3) implications for populations, including applications to conservation and management. Future insights are best achieved through robust interactions between theory and empiricism, especially through mechanistic models. Understanding spatiotemporal variation in the demographic processes contributing to the dynamics of small populations is a critical step in the advancement of population ecology.     

Catastrophes,connectivity and Allee effects in the design of marine reserve networks

Catastrophic events, like oil spills and hurricanes, occur in many marine systems. One potential role of marine reserves is buffering populations against disturbances, including the potential for disturbance-driven population collapses under Allee effects. This buffering capacity depends on reserves in a network providing rescue effects, setting up a tradeoff where reserves need to be connected to facilitate rescue, but also distributed in space to prevent simultaneous extinction. We use a set of population models to examine how dispersal ability and the disturbance regime interact to determine the optimal reserve spacing. We incorporate fishing in a spatially-explicit model to understand the effect of objective choice (e.g. conservation versus fisheries yield) on the optimal reserve spacing. We show that the optimal spacing between reserves increases when accounting for catastrophes with larger spacing needed when Allee effects interact with catastrophes to increase the probability of extinction. We also show that classic tradeoffs between conservation and fishing objectives disappear in the presence of catastrophes. Specifically, we found that at intermediate levels of disturbance, it is optimal to spread out reserves in order to increase both population persistence and to maximize spillover into non-reserve areas.