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.     

Variation in developmental time affects mating success and Allee effects

A fundamental question in biological conservation and invasion biology is why do some populations go extinct? Allee effects, notably those caused by mate location failure, are potentially key factors leading to the extinction of sparse populations. Several previous studies have focused on the inability of males and females to locate each other in space when populations are at low densities but here we investigate the effects of differences in the timing of male and female maturation on mating success. We develop a generalized model to clarify the role of protandry (the appearance of males before female emergence) and variability in adult maturation times. We show that temporal asynchrony can substantially reduce the probability of successful mating. We then apply this generalized model to estimate mating success in invading populations of the gypsy moth in North America in relation to local climate and its associated seasonality. Considerable geographic heterogeneity was observed in simulated mating success and this variability was not correlated with previous evaluations of bioclimatic requirements and habitat suitability. Furthermore, we found that the generalized model of temporal asynchrony provided reliable predictions and that detailed modeling of gypsy moth development was not necessary.     

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     

Speeds of invasion in a model with strong or weak Allee effects

We study an invasion model based on a reaction-diffusion equation with an Allee effect. We use a special, piecewise-linear, population growth rate. This function allows us to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effect. We show that the minimum speed for a sufficiently weak Allee may, in fact, be the same as that derived by means of linearization.     

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.     

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.     

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.     

Allee effects, immigration, and the evolution of species' niches

Theoretical studies of adaptation to sink environments (with conditions outside the niche requirements of a species) have shown that immigration from source habitats can either facilitate or inhibit local adaptation. Here, we examine the influence of immigration on the evolution of local adaptation, given an Allee effect (i.e., at low densities, absolute fitness increases with population density). We consider a deterministic model for evolution at a haploid locus, and a stochastic individual-based model for evolution of a quantitative trait, and several kinds of Allee effects. We demonstrate that increased immigration can greatly facilitate adaptive evolution in the sink; with greater immigration, local population sizes rise, and because of the Allee effect, there is a positive indirect effect of immigration on local fitness. This makes it easier for alleles of modest effect to be captured by natural selection, transforming the sink into a locally adapted population that can persist without immigration.     

The evidence for Allee effects

Allee effects are an important dynamic phenomenon believed to be manifested in several population processes, notably extinction and invasion. Though widely cited in these contexts, the evidence for their strength and prevalence has not been critically evaluated. We review results from 91 studies on Allee effects in natural animal populations. We focus on empirical signatures that are used or might be used to detect Allee effects, the types of data in which Allee effects are evident, the empirical support for the occurrence of critical densities in natural populations, and differences among taxa both in the presence of Allee effects and primary causal mechanisms. We find that conclusive examples are known from Mollusca, Arthropoda, and Chordata, including three classes of vertebrates, and are most commonly documented to result from mate limitation in invertebrates and from predator–prey interactions in vertebrates. More than half of studies failed to distinguish component and demographic Allee effects in data, although the distinction is crucial to most of the population-level dynamic implications associated with Allee effects (e.g., the existence of an unstable critical density associated with strong Allee effects). Thus, although we find conclusive evidence for Allee effects due to a variety of mechanisms in natural populations of 59 animal species, we also find that existing data addressing the strength and commonness of Allee effects across species and populations is limited; evidence for a critical density for most populations is lacking. We suggest that current studies, mainly observational in nature, should be supplemented by population-scale experiments and approaches connecting component and demographic effects. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

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 in biological invasions

Understanding the dynamics of small populations is obviously important for declining or rare species but is also particularly important for invading species. The Allee effect, where fitness is reduced when conspecific density is low, can dramatically affect the dynamics of biological invasions. Here, we summarize the literature of Allee effects in biological invasions, revealing an extensive theory of the consequences of the Allee effect in invading species and some empirical support for the theory. Allee effects cause longer lag times, slower spread and decreased establishment likelihood of invasive species. Expected spatial ranges, distributions and patterns of species may be altered when an Allee effect is present. We examine how the theory can and has been used to detect Allee effects in invasive species and we discuss how the presence of an Allee effect and its successful or unsuccessful detection may affect management of invasives. The Allee effect has been shown to change optimal control decisions, costs of control and the estimation of the risk posed by potentially invasive species. Numerous ways in which the Allee effect can influence the efficacy of biological control are discussed.     

Allee effects under climate change

Understanding how climate change affects population dynamics is crucial for assessing future of biodiversity. Here I ask how can Allee effects, occurring when mean individual fitness is reduced in rare populations, respond to increasing temperature. Despite the role Allee effects play in ecology of invasive, threatened and harvested populations, impacts of climate change on Allee effects are practically unknown. Analysis of two population models reveals that whereas the Allee effect driven by predation generally weakens as temperature increases, the Allee effect due to need of finding mates is predicted to become stronger when warming occurs. For the former model, the metabolic theory suggests that with increasing temperature prey growth rate should increase faster than predator attack rate. Increasing temperature thus weakens the Allee effect. In the latter, gypsy moth population model, mating rate increases with warming due to enhanced female?male encounter rate and temperature‐induced modifications in female and male adult emergence distributions. However, male and female mortality rates increase, too and the net effect is strengthening of the Allee effect. These results have repercussions also for pest control, indicating that augmentation of biocontrol agents may perhaps be not as effective as using pesticides or disrupting mating.     

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.     

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.