Dangerously few liaisons: a review of mate-finding Allee effects

In this paper, we review mate-finding Allee effects from ecological and evolutionary points of view. We define ‘mate-finding’ as mate searching in mobile animals, and also as the meeting of gametes for sessile animals and plants (pollination). We consider related issues such as mate quality and choice, sperm limitation and physiological stimulation of reproduction by conspecifics, as well as discussing the role of demographic stochasticity in generating mate-finding Allee effects. We consider the role of component Allee effects due to mate-finding in generating demographic Allee effects (at the population level). Compelling evidence for demographic Allee effects due to mate-finding (as well as via other mechanisms) is still limited, due to difficulties in censusing rare populations or a failure to identify underlying mechanisms, but also because of fitness trade-offs, population spatial structure and metapopulation dynamics, and because the strength of component Allee effects may vary in time and space. Mate-finding Allee effects act on individual fitness and are thus susceptible to change via natural selection. We believe it is useful to distinguish two routes by which evolution can act to mitigate mate-finding Allee effects. The first is evolution of characteristics such as calls, pheromones, hermaphroditism, etc. which make mate-finding more efficient at low density, thus eliminating the Allee effect. Such adaptations are very abundant in the natural world, and may have arisen to avoid Allee effects, although other hypotheses are also possible. The second route is to avoid low density via adaptations such as permanent or periodic aggregation. In this case, the Allee effect is still present, but its effects are avoided. These two strategies may have different consequences in a world where many populations are being artificially reduced to low density: in the first case, population growth rate can be maintained, while in the second case, the mechanism to avoid Allee effects has been destroyed. It is therefore in these latter populations that we predict the greatest evidence for mate-finding Allee effects and associated demographic consequences. This idea is supported by the existing empirical evidence for demographic Allee effects. Given a strong effect that mate-finding appears to have on individual fitness, we support the continuing quest to find connections between component mate-finding Allee effects (individual reproductive fitness) and the demographic consequences. There are many reasons why such studies are difficult, but it is important, particularly given the increasing number of populations and species of conservation concern, that the ecological community understands more about how widespread demographic Allee effects really are, and why.     

Mate search and mate-finding Allee effect: on modeling mating in sex-structured population models

Many demographic and other factors are sex-specific. To assess their impacts on population dynamics, we need sex-structured models. Such models have been shown to produce results different from those predicted by asexual models, yet need to explicitly consider mating dynamics. Modeling mating is challenging and no generally accepted formulation exists. Mating is often impaired at low densities due to difficulties of individuals in locating mates, a phenomenon termed a mate-finding Allee effect. Widely applied models of this Allee effect assume either that only male density determines the rate at which females mate or that male and female densities are equal. Contrarily, when detailed models of mating dynamics are sometimes developed, the female mating rate is rarely reported, making quantification of the mate-finding Allee effect difficult. Here, we develop an individual-based model of mating dynamics that accounts for spatial search of one sex for another, and quantify the rate at which females mate, depending on male and female densities and under a number of reasonable mating scenarios. We find that this rate increases with male and female densities (hence observing a mate-finding Allee effect), in a decelerating or sigmoid way, that mating can be most efficient at either low or high female densities, and that the mate search rate may undergo density-dependent selection. We also show that mate search trajectories evolve to be as straight as possible when targets are sedentary, yet that when targets move the search can be less straight without seriously affecting the female mating rate. Some recommendations for modeling two-sex population dynamics are also provided.     

Estimating Allee Dynamics before They Can Be Observed: Polar Bears as a Case Study

Allee effects are an important component in the population dynamics of numerous species. Accounting for these Allee effects in population viability analyses generally requires estimates of low-density population growth rates, but such data are unavailable for most species and particularly difficult to obtain for large mammals. Here, we present a mechanistic modeling framework that allows estimating the expected low-density growth rates under a mate-finding Allee effect before the Allee effect occurs or can be observed. The approach relies on representing the mechanisms causing the Allee effect in a process-based model, which can be parameterized and validated from data on the mechanisms rather than data on population growth. We illustrate the approach using polar bears (Ursus maritimus), and estimate their expected low-density growth by linking a mating dynamics model to a matrix projection model. The Allee threshold, defined as the population density below which growth becomes negative, is shown to depend on age-structure, sex ratio, and the life history parameters determining reproduction and survival. The Allee threshold is thus both density- and frequency-dependent. Sensitivity analyses of the Allee threshold show that different combinations of the parameters determining reproduction and survival can lead to differing Allee thresholds, even if these differing combinations imply the same stable-stage population growth rate. The approach further shows how mate-limitation can induce long transient dynamics, even in populations that eventually grow to carrying capacity. Applying the models to the overharvested low-density polar bear population of Viscount Melville Sound, Canada, shows that a mate-finding Allee effect is a plausible mechanism for slow recovery of this population. Our approach is generalizable to any mating system and life cycle, and could aid proactive management and conservation strategies, for example, by providing a priori estimates of minimum conservation targets for rare species or minimum eradication targets for pests and invasive species.     

Spatially implicit approaches to understand the manipulation of mating success for insect invasion management

Recent work indicates that Allee effects (the positive relationship between population size and per capita growth rate) are critical in determining the successful establishment of invading species. Allee effects may create population thresholds, and failure to establish is likely if invading populations fall below these thresholds. There are many mechanisms that may contribute to Allee effects, but mate-location failure is a common cause in sexually reproducing insects. Consequently, mate-location failure represents a type of “weak link” that may be enhanced in order to achieve eradication of insect populations during the early stages of invasion. In this paper, spatially implicit models that account for mating behavior of both sexes are used to explore the enhancement of mate-location failure in pest eradication programs. Distinct from the previous studies, the Allee effect emerges from a mechanistic representation of mate-location failure in our model. Three specific eradication strategies, sterile insect release, mass-trapping, and mating disruption, are incorporated into the model and tested for their ability to depress population growth during the early stages of invasions. We conducted simulations with the model parameterized to represent two types of insects: Coleopteran-like insects which are long-lived and capable of multiple matings, but have low daily reproductive rates, and Lepidopteran-like insects which are capable of mating only once per generation, have an ephemeral reproductive stage, and have high reproductive rates. Simulations indicated that: (1) many insect pests are more likely to be eradicated than had been previously predicted by classic models which do not account for mate-finding difficulties, (2) for Lepidopteran-like insects, mass-trapping has the greatest potential for eradication among the three methods when a large number of traps can be installed, although mating-disruption will be the most effective if we can anticipate confusion or trail-masking mechanisms of disruption, and (3) populations of Coleopteran-like insects may be most effectively eradicated using the sterile male release method. Though more detailed models should be tailored for individual species, we expect that the spatially implicit approaches outlined in this paper can be widely adapted to study the efficiency of various eradication approaches in sparse conditions. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The role of Allee effects in gypsy moth, <Emphasis Type="Italic">Lymantria dispar</Emphasis> (L.), invasions

Allee effects have been applied historically in efforts to understand the low-density population dynamics of rare and endangered species. Many biological invasions likewise experience the phenomenon of decreasing population growth rates at low population densities because most founding populations of introduced nonnative species occur at low densities. In range expansion of established species, the initial colonizers of habitat beyond the organism’s current range are usually at low density, and thus could be subject to Allee dynamics. There has been consistent empirical and theoretical evidence demonstrating, and in some cases quantifying, the role of Allee dynamics in the gypsy moth, Lymantria dispar (L.), invasion of North America. In this review, we examine the potential causes of the Allee effect in the gypsy moth and highlight the importance of mate-finding failure as a primary mechanism behind an Allee effect, while the degree to which generalist predators induce an Allee effect remains unclear. We then explore the role of Allee effects in the establishment and spread dynamics of the gypsy moth system, which conceptually could serve as a model system for understanding how Allee effects manifest themselves in the dynamics of biological invasions.     

Host-Parasitoid Dynamics and the Success of Biological Control When Parasitoids Are Prone to Allee Effects

In sexual organisms, low population density can result in mating failures and subsequently yields a low population growth rate and high chance of extinction. For species that are in tight interaction, as in host-parasitoid systems, population dynamics are primarily constrained by demographic interdependences, so that mating failures may have much more intricate consequences. Our main objective is to study the demographic consequences of parasitoid mating failures at low density and its consequences on the success of biological control. For this, we developed a deterministic host-parasitoid model with a mate-finding Allee effect, allowing to tackle interactions between the Allee effect and key determinants of host-parasitoid demography such as the distribution of parasitoid attacks and host competition. Our study shows that parasitoid mating failures at low density result in an extinction threshold and increase the domain of parasitoid deterministic extinction. When proned to mate finding difficulties, parasitoids with cyclic dynamics or low searching efficiency go extinct; parasitoids with high searching efficiency may either persist or go extinct, depending on host intraspecific competition. We show that parasitoids suitable as biocontrol agents for their ability to reduce host populations are particularly likely to suffer from mate-finding Allee effects. This study highlights novel perspectives for understanding of the dynamics observed in natural host-parasitoid systems and improving the success of parasitoid introductions.     

Dynamics of a small re-introduced population of wild dogs over 25 years: Allee effects and the implications of sociality for endangered species’ recovery

We analysed 25 years (1980–2004) of demographic data on a small re-introduced population of endangered African wild dogs (Lycaon pictus) in Hluhluwe-iMfolozi Park (HiP), South Africa, to describe population and pack dynamics. As small populations of cooperative breeders may be particularly prone to Allee effects, this extensive data set was used to test the prediction that, if Allee effects occur, aspects of reproductive success, individual survival and population growth should increase with pack and population size. The results suggest that behavioural aspects of wild dogs rather than ecological factors (i.e. competitors, prey and rainfall) primarily have been limiting the HiP wild dog population, particularly a low probability of finding suitable mates upon dispersal at low pack number (i.e. a mate-finding Allee effect). Wild dogs in HiP were not subject to component Allee effects at the pack level, most likely due to low interspecific competition and high prey availability. This suggests that aspects of the environment can mediate the strength of Allee effects. There was also no demographic Allee effect in the HiP wild dog population, as the population growth rate was significantly negatively related to population size, despite no apparent ecological resource limitation. Such negative density dependence at low numbers indicates that behavioural studies of the causal mechanisms potentially generating Allee effects in small populations can provide a key to understanding their dynamics. This study demonstrates how aspects of a species’ social behaviour can influence the vulnerability of small populations to extinction and illustrates the profound implications of sociality for endangered species’ recovery. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Demographic and Component Allee Effects in Southern Lake Superior Gray Wolves

Recovering populations of carnivores suffering Allee effects risk extinction because positive population growth requires a minimum number of cooperating individuals. Conservationists seldom consider these issues in planning for carnivore recovery because of data limitations, but ignoring Allee effects could lead to overly optimistic predictions for growth and underestimates of extinction risk. We used Bayesian splines to document a demographic Allee effect in the time series of gray wolf (Canis lupus) population counts (1980–2011) in the southern Lake Superior region (SLS, Wisconsin and the upper peninsula of Michigan, USA) in each of four measures of population growth. We estimated that the population crossed the Allee threshold at roughly 20 wolves in four to five packs. Maximum per-capita population growth occurred in the mid-1990s when there were approximately 135 wolves in the SLS population. To infer mechanisms behind the demographic Allee effect, we evaluated a potential component Allee effect using an individual-based spatially explicit model for gray wolves in the SLS region. Our simulations varied the perception neighborhoods for mate-finding and the mean dispersal distances of wolves. Simulation of wolves with long-distance dispersals and reduced perception neighborhoods were most likely to go extinct or experience Allee effects. These phenomena likely restricted population growth in early years of SLS wolf population recovery.     

Caught between two Allee effects: Trade-off between reproduction and predation risk

Reproductive activities are often associated with conspicuous morphology or behaviour that could be exploited by predators. Individuals can therefore face a trade-off between reproduction and predation risk. Here we use simple models to explore population-dynamical consequences of such a trade-off for populations subject to a mate-finding Allee effect and an Allee effect due to predation. We present our results in the light of populations that belong to endangered species or pests and study their viability and resilience. We distinguish several qualitative scenarios characterized by the shape and strength of the trade-off and, in particular, identify conditions for which the populations survive or go extinct. Reproduction can be so costly that the population always goes extinct. In other cases, the population goes extinct only over a certain range of low, intermediate or high levels of reproductive activities. Moreover, we show that predator removal (e.g. in an attempt to save an endangered prey species) has the least effect on populations with low cost of reproduction in terms of predation and, conversely, predator addition (e.g. to eradicate a pest) is most effective for populations with high predation cost of reproduction. Our results indicate that a detailed knowledge of the trade-off can be crucial in applications: for some trade-off shapes, only intermediate levels of reproductive activities might guarantee population survival, while they can lead to extinction for others. We therefore suggest that the fate of populations subject to the two antagonistic Allee effects should be evaluated on a case-by-case basis. Although the literature offers no quantitative data on possible trade-off shapes in any taxa, indirect evidence suggests that the trade-off and both Allee effects can occur simultaneously, e.g. in the golden egg bug Phyllomorpha laciniata.     

A ‘Goldilocks’ hypothesis for dispersal of biological control agents

The rate at which biological control agents disperse from release sites has important implications for their establishment and spread. Low rates of dispersal can yield spread that is too slow and may necessitate redistribution efforts for importation biological control and a high density of release sites for augmentation. Low dispersal rates may also lead to inbreeding at the site of release. On the other hand, high rates of dispersal can lead to Allee effects at the leading edge of the invasion front, potentially reducing the likelihood of establishment. Given these disadvantages associated with both low and high dispersal rates, we argue that intermediate rates of dispersal are likely to maximize the probability of establishment and appropriate spread for biological control agents released in the context of either importation or augmentative biological control. We consider this putative relationship a ‘Goldilocks hypothesis’ since it posits an optimum at intermediate values. In this review paper we begin by discussing the rationale for the Goldilocks hypothesis and then provide a case study from our work on importation biological control of the soybean aphid, Aphis glycines. Work on the soybean aphid parasitoid Binodoxys communis has shown that long-distance dispersal of immature parasitoids within winged migrating aphids is unlikely. This is likely good news for importation biological control because parasitoids dispersed in this manner would likely encounter crippling Allee effects. On the other hand, results from a field release study also suggest that female B. communis females (but not males) disperse actively from release sites. This female-biased dispersal may lead to strong mate-finding Allee effects and therefore may make establishment less likely.     

Allee效应对具有生境恢复的集合种群的影响

Allee效应与种群的灭绝密切相关,其研究对生态保护和管理至关重要。Allee效应对物种续存是潜在的干扰因素,濒危物种更容易受其影响,可能会增加生存于生境破碎化斑块的濒危物种的死亡风险,因此研究Allee效应对种群的动态和续存的影响是必要的。从包含由生物有机体对环境的修复产生的Allee效应的集合种群模型出发,引入由其他机制形成的Allee效应,建立了常微分动力系统模型和基于网格模型的元胞自动机模型。通过理论分析和计算机模拟表明:(1)强Allee效应不利于具有生境恢复的集合种群的续存;(2)生境恢复有利于种群续存;(3)局部扩散影响了集合种群的空间结构、动态行为和稳定性,生境斑块之间的局部作用将会减缓或消除集合种群的Allee效应,有利于集合种群的续存。     

French wasps in the New World: experimental biological control introductions reveal a demographic Allee effect

Many populations introduced into a novel environment fail to establish. One underlying process is the Allee effect, i.e., the difficulty of individuals to survive and reproduce when rare, and the consequently low or negative population growth. Although observations showing a positive relation between initial population size and establishment probability suggest that the Allee effect could be widespread in biological invasions, experimental tests are scarce. Here, we used a biological control program against Diuraphis noxia (Mordvilko) (Hemiptera: Aphididae) in the United States to manipulate initial population size of the introduced parasitoid Aphelinus asychis Walker (Hymenoptera: Aphelinidae) originating from France. For eight populations and three generations after introduction, we studied spatial distribution and spread, density, mate-finding, and population growth. Dispersal was lower in small populations during the first generation. Smaller initial population size nonetheless resulted in lower density during the three generations studied. The proportion of mated females and the population sex ratio were not affected by initial population size or population density. Net reproductive rate decreased with density within each generation, suggesting negative density-dependence. But for a given density, net reproductive rate was smaller in populations initiated with few individuals than in populations initiated with many individuals. Hence, our results demonstrate a demographic Allee effect. Mate-finding is excluded as an underlying mechanism, and other component Allee effects may have been overwhelmed by negative density-dependence in reproduction. Impact of generalist predators could provide one potential explanation for the relationship between initial population size and net reproductive rate. However, the continuing effect of initial population size on population growth suggests genetic processes may have been involved in the observed demographic Allee effect.     

Modelling mating success of saproxylic beetles in relation to search behaviour, population density and substrate abundance

We compared the efficiency of two mate-finding strategies exploited by representatives of the beetle families Cisidae and Anobiidae (genus Dorcatoma) that live inside fruiting bodies of wood-decaying fungi. In the Cisidae both sexes are attracted to host odour, but no pheromones seem to be present (nonpheromone strategy). In the Dorcatoma species only the females are attracted to host odour, but having found a host they attract males with a sexual pheromone (pheromone strategy). With a simulation model, we compared the efficiency of the two strategies at four densities of trees hosting fungal fruiting bodies and at three relative densities of insects. We found only small differences in efficiency between the two strategies at high relative densities of conspecific individuals, regardless of host tree density. The pheromone strategy was relatively more efficient when the relative density of insects or the density of host trees decreased. Thus, species adopting the nonpheromone strategy are probably more sensitive to habitat fragmentation and more likely to decline and go extinct at low population densities (because of Allee effects) than species using the pheromone strategy. Copyright 2003 Published by Elsevier Science Ltd on behalf of The Association for the Study of Animal Behaviour.      

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261-269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.     

Linking the allee effect, sexual reproduction, and temperature-dependent sex determination via spatial dynamics

We develop a spatially explicit, two-sex, individual-based model (IBM) and a derived spatially homogeneous model (SHM) to describe the Allee effect due to scarcity of mating possibilities at low population sizes or densities. The SHM, based on coupled difference equations, represents the first spatially homogeneous approach to this phenomenon, which differentiates between sexes and relies only on measurable population parameters. The IBM reinforces the findings of the SHM by adopting more realistic mate search strategies of diffusive movement and active search. Both models are characterized by a hyperbolic-shaped extinction boundary in the male-female state space, which contrasts with a linear boundary in one-dimensional models of the Allee effect. We examine how the position of the extinction boundary depends on population demography (primary sex ratio, reproduction and mortality probabilities) and adopted mate search strategies. The investigation of different phases in the IBM dynamics emphasizes the differences between local and global densities and shows the importance of scale when assessing the Allee effect. To demonstrate the potential application of our models, we combine the SHM and available data to predict the impact of environmental temperature changes on two turtle species with temperature-dependent sex determination.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.     

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.     

What do simple models reveal about the population dynamics of a cooperatively breeding species?

Research on cooperatively breeding species has shown that their population dynamics differ from those of conventional breeders. Populations of cooperators are structured into groups, and group‐level Allee effects are likely common. We assess the ability of phenomenological models, lacking explicit group structure, to describe population dynamics in cooperative meerkats Suricata suricatta, and we assess potential Allee effects at the population level. Using maximum likelihood model fitting and information theoretic model selection, applied to time series data from a wild meerkat population, we find simple models that incorporate rainfall and conventional density dependence to be the most parsimonious of the models considered. Detecting no population‐level Allee effect, we conclude that explicit consideration of population structure will be key to understanding the mechanisms behind population dynamics in cooperatively breeding species.     

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.     

Landmarking and strong Allee thresholds

Mate-finding difficulties in small populations are often postulated to create strong demographic Allee effects that increase the probability of extinction of native species or, similarly, decrease the probability that non-native species will successfully invade. Many species make use of a restricted number of mating locations, detectable from long-distance, that are not selected for habitat reasons (e.g., hilltopping in butterflies). This ‘landmarking’ strategy may specifically address the problem of overcoming mate-finding difficulties. Using a variant of the birthday problem, we demonstrate that populations which locate a restricted number of mate-finding sites using landmark features may have high probability of successful mating even at very low population densities. Therefore, a strong Allee threshold, if it exists, may be very small, and non-native species that make use of this strategy may have a very good chance of population establishment at low density.