Population models with Allee effect: a new model

In this paper, we develop several population models with Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model, which is the focus of our study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.     

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.     

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.     

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.     

Evidence of a component Allee effect driven by predispersal seed predation in a plant (Pedicularis rex,Orobanchaceae)

A small or sparse population may suffer a reduction in fitness owing to Allee effects. Here, we explored effects of plant density on pollination, reproduction and predation in the alpine herb Pedicularis rex over two years. We did not detect a significant difference in the pollination rate or fecundity (fruit set and the initial seed set) before predation between sparse and dense patches in either year, indicating no pollination-driven Allee effect. However, dense patches experienced significantly fewer attacks by predispersal seed predators in both years, resulting in a significantly decreased realized fecundity (final seed set), suggesting a component Allee effect driven by predispersal seed predation. Predation-driven Allee effects have been predicted by many models and demonstrated for a range of animals, but there is scant evidence for such effects in plants. Our study provides strong evidence of a component Allee effect driven by predation in a plant species.     

Scramble competitions can rescue endangered species subject to strong Allee effects

In this article, we study population dynamics of a general two-species discrete-time competition model where each species suffers from both strong Allee effects and scramble intra-specific competitions. We focus on how the combinations of the scramble intra-specific and inter-specific competition affect the extinction and coexistence of these two competing species where each species is subject to strong Allee effects. We derive sufficient conditions on the extinction, essential-like extinction and coexistence for such models. One of the most interesting findings is that scramble competitions can promote coexistence of these two species at their high densities. This is supported by the outcome of single species models with strong Allee effects. In addition, we apply theoretical results to a symmetric competition model with strong Allee effects induced by predator saturations where we give a completed study of its possible equilibria and attractors. Numerical simulations are performed to support our results.     

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.     

Bottom-up derivation of discrete-time population models with the Allee effect

This paper presents a framework in which various single-species discrete-time population models exhibiting the Allee effect are derived from first principles. Here, the Allee effect means a reduction in individual fitness at low population sizes. The derivation is based on the distribution of female and male individuals among discrete resource sites, in addition to competitive and cooperative interaction among individuals. These derivations show how the derived population models depend on the type and the intensity of competition, and the degree of clustering of individuals. Along with these models exhibiting the Allee effect, this paper also presents first-principles derivation of population models without the Allee effect which include a parameter relating to the intensity of competition.     

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.     

Plant sex alters Allee effects in aggregating plant parasites

Species interactions are central to our understanding of population dynamics. While density typically strengthens competition, reducing absolute fitness, Allee effects can reverse this pattern, increasing fitness with density. Allee effects emerge in host–parasite systems when higher parasite densities dilute immune responses or increase resource‐mobilization. The optimal density of individuals in these systems should be influenced by how host quality alters the rates at which facilitative and competitive effects change across densities. We tested these ideas using sumac Rhus typhina and a gall‐forming parasite Melaphis rhois that attacks sumac leaves. Fitness peaked at intermediate densities, indicating an Allee effect, but the fitness peak depended on host sex. Patterns of abundance mirrored fitness patterns, with galls clustered on leaves and female hosts supporting greater numbers of galls. Within leaves, galls near the stem were more fit, and gall‐makers preferentially oviposited near to the stem. The patterns of fitness and abundance are consistent with Allee effects caused by increased resource mobilization at higher gall‐maker densities rather than diluted immune responses. Our results suggest that Allee effects in parasites can be described as the summative effects of competitive and facilitative processes and, because both are common, Allee effects are likely common in host–parasite systems.     

Modelling mate-finding Allee effects and populations dynamics,with applications in pest control

In this paper, we review how mate-finding Allee effects enter population dynamical models that consider both sexes, highlight possible limitations of the more widely used “one-sex” models, and outline the links between the different model classes. We further explore interactions between the mate-finding Allee effect and other mechanisms relevant to pest-control strategies: release of natural enemies, sterile male release, and culling. Many of these strategies impose an additional component Allee effect on the population, and we discuss which of them might be efficient in the control of pest species that also suffer from the failure to locate mates. We focus primarily on eradication thresholds; our simple models show that most of the strategies yield similar results, and depending on the costs, one strategy or a combination of several can lead to the most efficient control.     

General population growth models with Allee effects in a random environment

Allee effects on population growth are quite common in nature, usually studied through deterministic models with a specific growth rate function.In order to seek the qualitative behaviour of populations induced by such effects, one should avoid model-specific behaviours. So, we use as a basis a general deterministic model, i.e. a model with a general growth rate function, to which we add the effect on the growth rate of the random fluctuations in environmental conditions. The resulting model is the general stochastic differential equation (SDE) model that we propose here.We consider two possible cases, weak Allee effects and strong Allee effects, which lead to different qualitative behaviours of the model.We will study the model properties for both cases in terms of existence and uniqueness of the solution, extinction and stationary behaviour of the population. The two cases will be compared with each other and with the general density-dependent SDE model without Allee effects.We then consider as an example the particular case of the classic logistic model and an Allee effect version of it.     

Allee effects and conspecific cueing jointly lead to conspecific attraction

Conspecific attraction is the preferential settlement into habitat patches with conspecifics. To be a good proximate strategy, fitness gains from settling with conspecifics must outweigh the costs of higher conspecific densities, such as intraspecific competition. Two types of benefits have been proposed to explain conspecific attraction: Allee effects (i.e., positive density dependence) and conspecific cueing (using conspecifics as an indicator of habitat quality). I present empirical evidence for conspecific attraction in the settlement of the porcelain crab, Petrolisthes cinctipes Randall (Anomura: Porcellanidae). Previous work demonstrated that P. cinctipes experiences strong intraspecific competition and that both Allee effects and conspecific cueing are present in P. cinctipes life-history. I developed an empirically-based fitness model of the costs and benefits of settling with conspecifics. Based on this model, I simulated optimal settlement to habitat patches that varied in conspecific density and habitat quality, where the correlation between density and habitat quality determined the level of conspecific cueing. I tested whether Allee effects alone, conspecific cueing alone, or Allee effects and conspecific cueing together could provide an ultimate explanation for the proximate settlement behavior of P. cinctipes. The settlement simulation was consistent with empirical settlement only when Allee effects and conspecific cueing were both included. Three life-history features are critical to this conclusion: (1) fitness is maximized at intermediate density, (2) fitness depends on the decisions of previous settlers, and (3) conspecific density provides good information about habitat quality. The quality of information garnered from conspecifics determines whether conspecific attraction is a good proximate strategy for settlement. I present a graphical illustration demonstrating how Allee effects and conspecific cueing work together to influence fitness, providing a conceptual framework for other systems.Electronic Supplementary Material Supplementary material is available for this article at and is accessible for authorized users.     

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.     

Pollen limitation or mate search need not induce an Allee effect

When a process modelling the availability of gametes is included explicitly in population models a critical depensation or Allee effect usually results. Non-spatial models cannot describe clumping and so small populations must be assumed very diffuse. Consequently individuals in small populations experience low contact rates and so reproduction is limited. In Nature invasions into new territory are unlikely to be as diffuse as those described by non-spatial models. We develop pair approximations to a probabilistic cellular automata model with independent pollination and seed setting processes (equivalently mate search and reproduction processes). Each process can be either global (population-wide) or local (within a small neighbourhood) or a mixture of the two. When either process is global the resulting model recaptures the Allee effect found in non-spatial models. However, if both processes are at least partially local we obtain a model in which Allee effects can be avoided altogether if individuals are suitably strong pollinators and colonisers. The Allee effect disappears because small populations are dramatically more clumped when colonisation is local and less wasteful of pollen when pollination is local.     

The presence of heterospecific males causes an Allee effect

The Allee effect is a positive causal relationship between any component of fitness and population density or size. Allee effects strongly affect the persistence of small or sparse populations. Predicting Allee effects remains a challenge, possibly because not all causal mechanisms are known. We hypothesized that reproductive interference (an interspecific reproductive interaction that reduces the fitness of the species involved) can generate an Allee effect. If the density of the interfering species is constant, an increase in the population of the species receiving interference may dilute the per capita effect of reproductive interference and may generate an Allee effect. To test this hypothesis, we examined the effect of heterospecific males on the relationship between per capita fecundity and conspecific density in Callosobruchus chinensis and C. maculatus. Of the two species, only C. maculatus females suffer reproductive interference from heterospecific males. Only C. maculatus, the species susceptible to reproductive interference, demonstrated an Allee effect, and only when heterospecific males were present. In contrast, C. chinensis, the species not susceptible to reproductive interference, demonstrated no Allee effect regardless of the presence of heterospecific males. Our results show that reproductive interference in fact generated an Allee effect, suggesting the potential importance of interspecific sexual interactions especially in small or sparse populations, even in the absence of a shared resource. It may be possible to predict Allee effects produced by this mechanism a priori by testing reproductive interference between closely related species.     

Interplay between Allee effects and collective movement in metapopulations

Population growth can be positively or negatively dependent on density. Therefore, the distribution pattern of individuals in a patchy environment can greatly affect the growth of each subpopulation and thereby of the metapopulation. When population growth presents positive density‐dependence (Allee effect), the distribution pattern becomes crucial, as small populations have an increased extinction risk. The way in which individuals move between patches largely determines the distribution pattern and thereby the population dynamics. Collective movement, in particular, should be expected to increase the potential number of colonisers and therefore the probability of colonising success. Here, we use mathematical modelling (differential equations and stochastic simulations) to study how collective movement can influence metapopulation dynamics when Allee effects are at stake. The models are inspired by the two‐spotted spider mite, a phytophagous pest of recognised agricultural importance. This sub‐social mite displays trail laying/following behaviour that can provoke collective movement. Moreover, experimental evidence suggests that it is subject to Allee effects. In the first part of this study we present a single‐species population growth model incorporating Allee effects, and study its properties. In the second part, this growth model is integrated into a larger simulation model consisting of a set of interconnected patches, in which the individuals move from one patch to the other either independently or collectively. Our results show that collective movement is more advantageous than independent dispersal only when Allee effects are present and strong enough. Furthermore they provide a theoretical framework that allows the quantification of the interplay between Allee effects and collective movement.     

A technique for analysis of density dependence in population models

The concept of density-dependent population growth is fundamental to our understanding of how populations persist. While it is generally agreed that negative density dependence must occur at high densities, the direction of density dependence may be negative (pure negative density dependence) or positive (demographic Allee effect) at low densities. In this article, we present a technique to link the direction of density dependence to generic ecological factors. This technique involves exploiting the presence of a particular bifurcation, known as a saddle-node-transcritical interaction. We first provide a method to detect this bifurcation in a given model and then demonstrate its ecological relevance using several existing mechanistic models. With a mathematical framework in place, we are able to identify scenarios in which neither a weak Allee effect nor pure negative density dependence are possible. More generally, we find conditions on parameter values that are necessary for transitions between pure negative density dependence and demographic Allee effects to occur.     

Impact of Allee effect on an eco-epidemiological system

In this paper, we propose a general ratio-dependent prey-predator model with disease in predator subject to the strong Allee effect in prey. We obtain the complete dynamics of both models: (a) full model with Allee effect; (b) full model without Allee effect. Model (a) may have more than one interior equilibrium point, but model (b) has only one interior equilibrium point. Numerical results reveal that the coexistence of all the populations at the endemic state is possible for both the models. But for the model with Allee effect, the coexistence can be destroyed by an increased supply of alternative food for the predators. It can also be proved that for the full model with Allee effect, the disease can be suppressed under certain parametric conditions. Also by comparing models (a) and (b), we conclude that Allee effect can create or destroy the interior attractor. Finally, we have studied the disease free-submodel (prey and susceptible predator model) with and without Allee effect. The comparative study between these two submodels leads to the following conclusions: 1) In the presence of Allee effect, the number of interior equilibrium points can change from zero to two whereas the submodel without Allee effect has unique interior equilibrium point; 2) Both with and without Allee effect, initial conditions play an important role on the survival and extinction of prey as well as its corresponding predator; 3) In the presence of Allee effect, bi-stability occurs with stable or periodic coexistence of prey and susceptible predator and the extinction of prey and susceptible predator; 4) Allee effect can generate or destroy the interior equilibrium points.     

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.