Modelling the mating system of polar bears: a mechanistic approach to the Allee effect

Allee effects may render exploited animal populations extinction prone, but empirical data are often lacking to describe the circumstances leading to an Allee effect. Arbitrary assumptions regarding Allee effects could lead to erroneous management decisions so that predictive modelling approaches are needed that identify the circumstances leading to an Allee effect before such a scenario occurs. We present a predictive approach of Allee effects for polar bears where low population densities, an unpredictable habitat and harvest-depleted male populations result in infrequent mating encounters. We develop a mechanistic model for the polar bear mating system that predicts the proportion of fertilized females at the end of the mating season given population density and operational sex ratio. The model is parametrized using pairing data from Lancaster Sound, Canada, and describes the observed pairing dynamics well. Female mating success is shown to be a nonlinear function of the operational sex ratio, so that a sudden and rapid reproductive collapse could occur if males are severely depleted. The operational sex ratio where an Allee effect is expected is dependent on population density. We focus on the prediction of Allee effects in polar bears but our approach is also applicable to other species.     

Demographic stochasticity, allee effects, and extinction: the influence of mating system and sex ratio

Demographic stochasticity has a substantial influence on the growth of small populations and consequently on their extinction risk. Mating system is one of several population characteristics that may affect this. We use a stochastic pair-formation model to investigate the combined effects of mating system, sex ratio, and population size on demographic stochasticity and thus on extinction risk. Our model is designed to accommodate a continuous range of mating systems and sex ratios as well as several levels of stochasticity. We show that it is not mating system alone but combinations of mating system and sex ratio that are important in shaping the stochastic dynamics of populations. Specifically, polygyny has the potential to give a high demographic variance and to lower the stochastic population growth rate substantially, thus also shortening the time to extinction, but the outcome is highly dependent on the sex ratio. In addition, population size is shown to be important. We find a stochastic Allee effect that is amplified by polygyny. Our results demonstrate that both mating system and sex ratio must be considered in conservation planning and that appreciating the role of stochasticity is key to understanding their effects.     

Neglecting uncertainty behind Allee effect estimation may generate false predictions of population extinction risk

Estimation of extinction thresholds arising from Allee effects (Allee thresholds) and related probabilities of population extinction is notoriously difficult. One way is to analyze adequately parameterized population models. Traditionally, a point estimate is substituted for the Allee effect strength in such models. However, each point estimate entails an underlying uncertainty. We explore how accounting for this uncertainty affects the probability of population extinction, and show that this probability decreases sigmoidally with increasing population density, even in the absence of any stochasticity. Deviations from when only a point estimate of the Allee effect strength is used can be significant, unless stochasticity is added and the stochastic noise intensity is high. Significant deviations from when only a point estimate is used also occur when the Allee threshold and the environmental carrying capacity of the species are close enough one to another. We also show that the impact of the uncertainty in the Allee effect strength estimate increases as the Allee effect strength itself increases and decreases as the species recovery potential increases. This is not a good news, since we would like to preferentially and efficiently manage slowly recovering populations prone to strong Allee effects. Still, there is a way to come up with relatively good Allee threshold estimates. Besides an obvious option of collecting as many data as possible, the impact of the uncertainty can be mitigated by diversifying Allee effect experiments such that we put more emphasis on larger size groups. This is somewhat surprising, given that frequent complaints on the (im)possibility of detecting Allee effects concern difficulties in locating, observing and experimenting on rare populations. Our results extend current theory surrounding Allee effects and have broad ramifications for applied ecology.     

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.     

局域种群的Allee效应和集合种群的同步性

从包含Allee效应的局域种群出发,建立了耦合映像格子模型,即集合种群模型.通过分析和计算机模拟表明:(1)当局域种群受到Allee效应强度较大时,集合种群同步灭绝;(2)而当Allee效应强度相对较弱时,通过稳定局域种群动态(减少混沌)使得集合种群发生同步波动,而这种同步波动能够增加集合种群的灭绝风险;(3)斑块间的连接程度对集合种群同步波动的发生有很大的影响,适当的破碎化有利于集合种群的续存.全局迁移和Allee效应结合起来增加了集合种群同步波动的可能,从而增加集合种群的灭绝风险.这些结果对理解同步性的机理、利用同步机理来制定物种保护策略和害虫防治都有重要的意义.     

General Allee effect in two-species population biology

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator–prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a ‘phase space core’ of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar systems. Finally, we show how the presence of semistable equilibria increases the number of possible Allee effect cores.     

General Allee effect in two-species population biology

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator-prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a 'phase space core' of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar systems. Finally, we show how the presence of semistable equilibria increases the number of possible Allee effect cores.     

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。     

Population dynamics under parasitic sex ratio distortion.

We analyse the population dynamic effects of sex ratio distortion by vertically transmitted, feminizing parasites. We show that, for diploid hosts, sex ratio distortion may lead to extinction as males become too rare to maintain the host population through reproduction. Feminizers can magnify Allee effects, broadening the range of conditions leading to extinction of small populations. Depending on male mating constraints and strength of density dependence, feminizers may either increase or decrease the equilibrium host density. Under conditions leading to deterministic host extinction, stochastic elimination of the parasite may allow the host population to recover. Hence, infection by parasitic sex ratio distorters may be transient in finite populations. We consider the implications of this process for parasite prevalence, host population regulation, and sex ratio evolution.     

Habitat destruction and the extinction debt revisited: The Allee effect

Habitat destruction, often caused by anthropogenic disturbance, can lead to the extinction of species at an unprecedented rate. It is important, therefore, to consider habitat destruction when assessing population viability. Another factor often ignored in population viability analysis, is the Allee effect that adds to the risk of populations already on the verge of extinction. Understanding the Allee effect on species dynamics and response to habitat destruction has intrinsic value in conservation prioritization. Here, the Allee effect was considered in a multi-species hierarchical competition model. Results showed that species persistence declines dramatically due to the Allee effect, and certain species become more susceptible to habitat destruction than others. Two extinction orders emerged under habitat destruction: either the best competitor becomes extinct first or the best colonizer first. The extinction debt and order, as well as the time lag between habitat destruction and species extinction, were found to be determined by species abundance and the intensity of the Allee effect.     

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.     

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.     

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.     

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.     

Single-species models of the Allee effect: extinction boundaries,sex ratios and mate encounters

We critically review and classify models of single-species population dynamics subject to the demographic Allee effect with emphasis on non-spatial, deterministic approach. Inclusion of spatial movement and stochastic phenomena does not substantially change the behaviour; stochasticity only "blurs" step-like character of the Allee effect into a sigmoidal form. The outcome of all non-spatial, deterministic models is either unconditional extinction, extinction-survival scenario (ES), or unconditional survival. Three major model classes are recognized: (1) one-dimensional heuristic models, (2) one-dimensional models with mating probability and fixed sex ratio, and (3) two-sex models with variable adult sex ratio. Each class is characterized by the shape of extinction boundary which separates extinction from survival in the ES scenario. The latter two classes may give better predictions of extinction thresholds than heuristic models but require specific information and are data intensive. In one-dimensional models with fixed sex ratio, population cannot survive if density/number of males decreases below some threshold while there is no such restriction on females. Individual-based models seem to be most capable of explaining mechanisms leading to the Allee effect.     

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.     

Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.     

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.     

Pollen limitation and Allee effect related to population size and sex ratio in the endangered Ottelia acuminata (Hydrocharitaceae): implications for conservation and reintroduction

Small populations may suffer more severe pollen limitation and result in Allee effects. Sex ratio may also affect pollination and reproduction success in dioecious species, which is always overlooked when performing conservation and reintroduction tasks. In this study, we investigated whether and how population size and sex ratio affected pollen limitation and reproduction in the endangered Ottelia acuminata, a dioecious submerged species. We established experimental plots with increasing population size and male sex ratio. We observed insect visitation, estimated pollen limitation by hand‐pollinations and counted fruit set and seed production per fruit. Fruit set and seed production decreased significantly in small populations due to pollinator scarcity and thus suffered more severe pollen limitation. Although frequently visited, female‐biased larger populations also suffered severe pollen limitation due to few effective visits and insufficient pollen availability. Rising male ratio enhanced pollination service and hence reproduction. Unexpectedly, pollinator preferences did not cause reduced reproduction in male‐biased populations because of high pollen availability. However, reproductive outputs showed more variability in severe male‐biased populations. Our results revealed two component Allee effects in fruit set and seed production, mediated by pollen limitation in O. acuminata. Moreover, reproduction decreased significantly in larger female‐biased populations, increasing the risk of an Allee effect.     

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

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