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, mating systems and the extinction risk in populations with two sexes

The Allee effect is one of the population consequences of sexual reproduction that has received increased attention in recent years. Due to its impact on small population dynamics, it is commonly accepted that Allee effects should render populations more extinction prone. In particular, monogamous species are considered more susceptible to the Allee effect and hence, more extinction prone, than polygamous species. Although this hypothesis has received theoretical support, there is little empirical evidence. In this study, we investigate (1) how variation in tertiary sex ratio affects the presence and intensity of the Allee effect induced by mating system, as well as (2) how this effect contributes to extinction risk. In contrast with previous predictions, we show that all mating systems are likely to experience a strong Allee effect when the operational sex ratio (OSR) is balanced. This strong Allee effect does not imply being exceptionally extinction prone because it is associated with an OSR that result in a relatively small extinction risk. As a consequence, the impact of Allee effects on overall extinction risk is buffered. Moreover, the OSR of natural populations appears to be often male biased, thus making it unlikely that they will suffer from an Allee effect induced by mating system.     

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.     

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.     

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.     

Plant population dynamics, pollinator foraging, and the selection of self-fertilization

Many flowering plants rely on pollinators, self-fertilization, or both for reproduction. We model the consequences of these features for plant population dynamics and mating system evolution. Our mating systems-based population dynamics model includes an Allee effect. This often leads to an extinction threshold, defined as a density below which population densities decrease. Reliance on generalist pollinators who primarily visit higher density plant species increases the extinction threshold, whereas autonomous modes of selfing decrease and can eliminate the threshold. Generalist pollinators visiting higher density plant species coupled with autonomous selfing may introduce an effect where populations decreasing in density below the extinction threshold may nonetheless persist through selfing. The extinction threshold and selfing at low density result in populations where individuals adopting a single reproductive strategy exhibit mating systems that depend on population density. The ecological and evolutionary analyses provide a mechanism where prior selfing evolves even though inbreeding depression is greater than one-half. Simultaneous consideration of ecological and evolutionary dynamics confirms unusual features (e.g., evolution into extinction or abrupt increases in population density) implicit in our separate consideration of ecological and evolutionary scenarios. Our analysis has consequences for understanding pollen limitation, reproductive assurance, and the evolution of mating systems.     

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.     

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.     

Culturally transmitted paternity beliefs and the evolution of human mating behaviour

Recent anthropological findings document how certain lowland South American societies hold beliefs in 'partible paternity', which allow children to have more than one 'biological' father. This contrasts with Western beliefs in 'singular paternity', and biological reality, where children have just one father. Here, mathematical models are used to explore the coevolution of paternity beliefs and the genetic variation underlying human mating behaviour. A gene-culture coevolutionary model found that populations exposed to a range of selection regimes typically converge on one of two simultaneously stable equilibria; one where the population is monogamous and believes in singular paternity, and the other where the population is polygamous and believes in partible paternity. A second agent-based model, with alternative assumptions regarding the formation of mating consortships, broadly replicated this finding in populations with a strongly female-biased sex ratio, consistent with evidence for high adult male mortality in the region. This supports an evolutionary scenario in which ancestral South American populations with differing paternity beliefs were subject to divergent selection on genetically influenced mating behaviour, facilitated by a female-biased sex ratio, leading to the present-day associations of female control, partible paternity and polygamy in some societies, and male control, singular paternity and monogamy in others.     

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.     

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.     

Density dependent dispersal decisions and the Allee effect

The decision to move between patches in the environment is among the most important life history choices an organism can make. I derive a new density dependent dispersal rule, and examine how dispersal decisions based on avoiding fitness loss associated with an Allee effect or competitive effects impact upon population dynamics in spatially structured populations with qualitatively different dynamics. I also investigate the effects of the number of patches in the system and a limit to the patch sampling time available to dispersers. Dispersing to avoid competitive pressures can destabilise otherwise stable population dynamics, and stabilise chaotic dynamics. Dispersing to avoid an Allee effect does not qualitatively change local population dynamics until eventually driving unstable populations to global extinction with a sufficiently high fitness threshold. A time limit for sampling can stabilise dynamics if dispersal is based on escaping the Allee effect, and rescue populations from global extinction. The results are sensitive to the number of patches available in the environment and suggest that dispersal to avoid an Allee effect will only arise under biologically plausible conditions, i.e. where there is a limit to the number of dispersal attempts that can be made between generations.     

Turtle mating patterns buffer against disruptive effects of climate change

For organisms with temperature-dependent sex determination (TSD), skewed offspring sex ratios are common. However, climate warming poses the unique threat of producing extreme sex ratio biases that could ultimately lead to population extinctions. In marine turtles, highly female-skewed hatchling sex ratios already occur and predicted increases in global temperatures are expected to exacerbate this trend, unless species can adapt. However, it is not known whether offspring sex ratios persist into adulthood, or whether variation in male mating success intensifies the impact of a shortage of males on effective population size. Here, we use parentage analysis to show that in a rookery of the endangered green turtle (Chelonia mydas), despite an offspring sex ratio of 95 per cent females, there were at least 1.4 reproductive males to every breeding female. Our results suggest that male reproductive intervals may be shorter than the 2-4 years typical for females, and/or that males move between aggregations of receptive females, an inference supported by our satellite tracking, which shows that male turtles may visit multiple rookeries. We suggest that male mating patterns have the potential to buffer the disruptive effects of climate change on marine turtle populations, many of which are already seriously threatened.     

Sex-role reversal of a monogamous pipefish without higher potential reproductive rate in females

In monogamous animals, males are usually the predominant competitors for mates. However, a strictly monogamous pipefish Corythoichthys haematopterus exceptionally exhibits a reversed sex role. To understand why its sex role is reversed, we measured the adult sex ratio and the potential reproductive rate (PRR), two principal factors influencing the operational sex ratio (OSR), in a natural population of southern Japan. The adult sex ratio was biased towards females throughout the breeding season, but the PRR, which increased with water temperature, did not show sexual difference. We found that an alternative index of the OSR (Sf/Sm: sex ratio of 'time in') calculated from the monthly data was consistently biased towards females. The female-biased OSR associated with sex-role reversal has been reported in some polyandrous or promiscuous pipefish, but factors biasing the OSR differed between these pipefish and C. haematopterus. We concluded that the similar PRR between the sexes in C. haematopterus does not confer reproductive benefit of polygamous mating on either sex, resulting in strict monogamous mating, and its female-biased adult sex ratio promotes female-female competition for a mate, resulting in sex-role reversal.     

Variable reproductive success in fragmented populations

Marine habitats are naturally patchy and anthropogenic disturbance can further fragment them. Many marine animals are sessile as adults or obligate inhabitants of particular habitats, so populations living in isolated patches of habitat are linked largely by dispersal of planktonic larvae. Theoretically, larvae are more likely to find and settle into large patches of habitat than small patches, thus small habitat patches may experience a more discontinuous supply of recruits resulting in small populations with unusual size- or age-structures or odd sex ratios — conditions where Allee effects on reproductive success are likely. We tested this hypothesis for the Caribbean spotted spiny lobster (Panulirus guttatus), an obligate inhabitant of coral patch reefs whose mating dynamics are size-dependent. We found that P. guttatus were less abundant on small reefs where their size structure and per capita reproductive success were significantly more variable, particularly among large females that are susceptible to sperm limitation that diminishes fertilization rates. These results are indicative of Allee effects and provide a mechanistic understanding of how size-dependent mating dynamics influence reproductive success in ways that alter population dynamics in patchy habitats.     

Polyploidy influences sexual system and mating patterns in the moss Atrichum undulatum sensu lato

Background and Aims

Evolutionary transitions between separate and combined sexes have frequently occurred across various plant lineages. In mosses, which are haploid-dominant, evolutionary transitions from separate to combined sexes are often associated with genome doubling. Polyploidy and hermaphroditism have strong effects on the inbreeding depression of a population, and are subsequently predicted to affect the mating system.

Methods

We tested the association between ploidy (haploid, diploid or triploid gametophytes) and mating system in 21 populations of Atrichum undulatum sensu lato, where sex ratios vary widely. For each population, we measured the sex ratio, estimated selfing rates using allozyme markers and determined the level of ploidy through flow cytometry.

Key Results

Hermaphrodites in A. undulatum were either diploid or triploid. However, many diploid populations were strictly separate-sexed, suggesting that hermaphroditism is not a necessary result of genome doubling. Levels of selfing were strongly supported as being greater than zero in one population with strictly separate-sexed individuals, and one-third of populations with hermaphrodites.

Conclusions

Although hermaphrodites are associated with triploidy, hermaphroditism is not a necessary outcome of genome duplication. Hermaphroditism, but not genome duplication alone, increased estimated selfing rates, probably due to the occurrence of selfing within a gametophyte. Thus, genome duplication can influence the mating system and the associated evolution and maintenance of reproductive traits.     

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.     

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.     

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.     

Critical patch size generated by Allee effect in gypsy moth, Lymantria dispar (L.)

Allee effects are important dynamical mechanisms in small-density populations in which per capita population growth rate increases with density. When positive density dependence is sufficiently severe (a 'strong' Allee effect), a critical density arises below which populations do not persist. For spatially distributed populations subject to dispersal, theory predicts that the occupied area also exhibits a critical threshold for population persistence, but this result has not been confirmed in nature. We tested this prediction in patterns of population persistence across the invasion front of the European gypsy moth (Lymantria dispar) in the United States in data collected between 1996 and 2008. Our analysis consistently provided evidence for effects of both population area and density on persistence, as predicted by the general theory, and confirmed here using a mechanistic model developed for the gypsy moth system. We believe this study to be the first empirical documentation of critical patch size induced by an Allee effect.