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.     

捕食者具有厌食性反应且食饵具有Allee效应的捕食系统

在Dubis动力系统的基础上,建立了具有Allee效应的捕食系统模型。对系统的稳定性进行了分析,受Allee效应的影响,食饵种群可能因为种群大小处于临界点以下而趋于灭绝。通过对系统进行模拟,结果表明:不受Allee效应的影响,系统的演化属于一种理想化的情形系统到达P(平衡)点的时间较不受Allee效应影响时系统到达P点的时间短,不利于生物的进化,而在Allee效应的影响下,系统的演化将达到一个平衡状态。由此,说明Allee效应为濒临灭绝物种的管理提供了重要的理论依据,对管理部门的决策有参考指导作用。     

Oscillations in a size-structured prey-predator model

This article introduces a predator–prey model with the prey structured by body size, based on reports in the literature that predation rates are prey-size specific. The model is built on the foundation of the one-species physiologically structured models studied earlier. Three types of equilibria are found: extinction, multiple prey-only equilibria and possibly multiple predator–prey coexistence equilibria. The stabilities of the equilibria are investigated. Comparison is made with the underlying ODE Lotka–Volterra model. It turns out that the ODE model can exhibit sustain oscillations if there is an Allee effect in the net reproduction rate, that is the net reproduction rate grows for some range of the prey’s population size. In contrast, it is shown that the structured PDE model can exhibit sustain oscillations even if the net reproductive rate is strictly declining with prey population size. We find that predation, even size-non-specific linear predation can destabilize a stable prey-only equilibrium, if reproduction is size specific and limited to individuals of large enough size. Furthermore, we show that size-specific predation can also destabilize the predator–prey equilibrium in the PDE model. We surmise that size-specific predation allows for temporary prey escape which is responsible for destabilization in the predator–prey dynamics.     

High reproductive rates result in high predation risks: a mechanism promoting the coexistence of competing prey in spatially structured populations

I tested the hypothesis that spatial structure provides a trade-off between reproduction and predation risk and thereby facilitates predator-mediated coexistence of competing prey species. I compared a cellular automata model to a mean-field model of two prey species and their common predator. In the mean-field model, the prey species with the higher reproductive rate (the superior competitor) always outcompeted the other species (the inferior competitor), both in the presence of and the absence of the predator. In the cellular automata model, both prey species, which differed only in their reproductive rates, coexisted for a long time in the presence of their common predator at intermediate levels of predation. At low predation rates, the superior competitor dominated, while high predation rates favored the inferior competitor. This discrepancy in the results of the different models was due to a trade-off that spontaneously emerged in spatially structured populations; that is, the more clustered distribution of the superior competitor made it more susceptible to predation. In addition, coexistence of competing prey species declined with increasing dispersal ranges of either prey or predator, which suggests that the trade-off that results from spatial structure becomes less important as either prey or predator disperse over a broader range.     

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.     

Life-history plasticity and inbreeding depression under mate limitation and predation risk: cumulative lifetime fitness dissected with a life table response experiment

Environmental effects on the evolution of mating systems are increasingly discussed, but we lack many examples of how environmental conditions affect the expression and consequences of alternative mating systems. Variation in mate availability sets up a trade-off between reproductive assurance and inbreeding depression, but the consequences of both mate limitation and inbreeding may depend on other environmental conditions. Predation risk is common under natural conditions, and known to affect allocation to reproduction, but we know little about the effects of isolation and inbreeding under predation risk. We reared selfed and outcrossed hermaphroditic freshwater snails (Physa acuta) in four environments (predator cues present or absent crossed with mating partners available or not) and quantified life-history traits and cumulative lifetime fitness. Our results confirm that isolation from mates can increase longevity and growth, resulting in higher lifetime fecundity. Thus, we observed no evidence for mate limitation of reproduction. However, reproduction under isolation (i.e., selfing) resulted in inbreeding depression, which should counteract the benefits of selfing. Inbreeding depression in fitness occurred in both predator and no-predator environments, but there was no overall change in inbreeding depression with predator cues. This represents, to our knowledge, the first empirical estimate of the effect of predation risk on inbreeding depression in an animal. Cumulative fitness was most influenced by early survival and especially early fecundity. As predation risk and inbreeding (both ancestral and due to a lack of mates) reduced early fecundity, these effect are predicted to have important contributions to population growth under natural conditions. Therefore life-history plasticity (e.g., delayed reproduction) is likely to be very important to overall fitness.     

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

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

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.     

Bistability and an Allee effect as emergent consequences of stage-specific predation

The Allee effect, a reduction of individual fitness at low population density that can lead to sudden and unannounced extinctions, has been shown to come about through a number of mechanisms, usually associated with group behavior or mate search. Recent papers show that it may arise through size-selective predation, without explicit assumptions relating individual fitness to population density. It arises from the shift that a predator induces in the population stage distribution of its prey. We study the parameter conditions that lead to such an emergent Allee effect. The emergent Allee effect occurs under fairly broad conditions. We show that stage-specific predation can also induce bistability between alternative states where both prey and predator are present. A perturbation analysis on the equilibria shows that all equilibria are highly robust to changes in predator density. Our work shows that when size-specific interactions are taken into account, bistabilities and catastrophic collapses are possible even in purely exploitative food webs, which has substantial implications for questions related to food web theory and conservation issues.     

Spatial dynamics of specialist seed predators on synchronized and intermittent seed production of host plants

Masting, the synchronized and intermittent seed production by plant populations, provides highly variable food resources for specialist seed predators. Such a reproductive mode helps minimize seed losses through predator satiation and extinction of seed predator populations. The seed predators can buffer the resource variation through dispersal or extended diapause. We developed a spatially explicit resource-consumer model to understand the effect of masting on specialist seed predators. The masting dynamics were assumed to follow a resource-based model for plant reproduction, and the population dynamics of the predator were represented by a spatially extended Nicholson-Bailey model. The resultant model demonstrated that when host plants reproduce intermittently, seed predator populations go locally extinct, but global persistence of the predator is facilitated by dispersal or extended diapause. Global extinction of the predator resulted when the intermittent reproduction is highly synchronized among plants. An approximate invasion criterion for the predators showed that negative lag-1 autocorrelation in seeding reduces invasibility, and positive lag-1 cross-correlation enhances invasibility. Spatial synchronization in seeding at local scale caused by pollen coupling (or climate forcing) further prevented invasion of the predators. If the predators employed extended diapause, extremely high temporal variability in reproduction was required for plants to evade the predators.     

Predator-driven component Allee effects in a wild ungulate

Negative density dependence is an important driver of population dynamics of large vertebrates. Allee effects (positive density dependence), however, can affect small populations. Allee effects can be generated by predation and recent research has revealed potentially important indirect effects of predation on population dynamics. For wild populations, however, quantification of both Allee effects and indirect effects of predation remains scarce. We monitored for 27 years a bighorn sheep (Ovis canadensis) population that declined dramatically as episodes of cougar (Puma concolor) predation depressed survival. Predation led to a positive relationship between lamb survival and population size below a threshold, and to an overall positive relationship between yearling and adult ewe survival and population size. During years of high predation, lambs also suffer mortality through reduced growth, contributing a third of the total impact of predation on lamb survival. There was no positive association between population growth and population size, probably because growth was affected by several factors other than predation, including disease. Our results support the contention that predator-driven component Allee effects may exacerbate the effects of other environmental drivers and increase the risk of extinction of small populations.     

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.     

Impact of natural enemies on obligately cooperative breeders

Obligately cooperative breeders (cooperators) display a negative growth rate once they fall below a minimum density. Constraints imposed by natural enemies, such as predators or competitors, may push cooperator groups closer to this threshold, thus increasing the risk that stochastic fluctuations will drive them below it. This may indirectly drive these groups to extinction, thereby increasing the risk of population extinction. In this paper, we construct mathematical models of the dynamics of groups of cooperators and non-cooperators in the presence of two types of enemies: enemies whose dynamics do not depend on the dynamics of their victim (e.g., amensal competitor, generalist predator) and those whose dynamics do. In the latter case, we distinguish positive (e.g., specialist predator) and negative (e.g., bilateral competitor) reciprocal effects. These models correspond to the classical amensal, predation and competition models, in the presence of an Allee effect. We then develop the models to study consequences at the population level. By comparing models with or without an Allee effect, we show that enemies decrease the group size of cooperators more than that of non-cooperators, and this increases their group extinction risk. We also demonstrate how an Allee effect at a lower dynamical level can have consequences at a higher level: inverse density dependence at the group level generated lower population sizes and higher risks of population extinction. Our results also suggest that demographic compensation can be achieved by cooperators through an increased intrinsic growth rate, or by decreasing the enemy constraint. Both of these types of compensation have been observed in empirical studies of cooperators.     

Potential Component Allee Effects and Their Impact on Wetland Management in the Conservation of Endangered Anurans

Effective management of wetland quantity and quality is crucial for effective conservation of declining amphibian populations. In particular, frogs and toads that employ aggregative breeding strategies may suffer negative population impacts in response to changes in availability of aquatic breeding habitat, including overabundance of suitable habitat, if density of conspecifics attending aggregations is positively correlated with reproductive success. Here we document such a positive relationship, potentially the first example of a component Allee effect in an anuran, in the critically endangered Houston toad (Bufo houstonensis). We assessed the relationship between mean yearly chorus size and reproductive success of males at the pond level using an information theoretic model selection approach and a two-sample t-test. The chosen model contained the single variable of mean yearly chorus size to predict probability of reproduction, as selected using the Akaike Information Criterion corrected for small sample size and Akaike weight. Mean chorus sizes were significantly higher among ponds exhibiting evidence of reproduction than in those that showed no evidence of reproduction. Our results suggest that chorusing alone is a poor proxy for inference of population stability and highlight a need for reassessment of widely-used amphibian monitoring protocols. Further, amphibian conservation efforts should account for potential Allee effects in order to optimize benefits and avoid underestimating critical population thresholds, particularly in species exhibiting rapid population declines.     

Modelling the genetics and demography of step cline formation: gastropod populations preyed on by experimentally introduced crabs

Whether a prey population goes extinct or adapts in response to an invading predator may depend on the number of contiguous populations that experience increased predation. We created invaded snail populations by building shelters for predatory shore crabs on a rocky intertidal bench. The crabs preyed selectively on thin-shelled snails tethered next to the shelters but did not prey on those more than 2 m away. This caused strong directional selection for increased shell thickness in populations close to the shelters but did not change selection in those farther away. The field experiment was used to parameterize a new individual-based quantitative genetic model that included demography. In the model a detectable step cline in shell thickness evolved rapidly even though the region of increased predation was shorter than Slatkin's characteristic length. The cline's step size in the model was similar to that measured in the field 10 years after the experiment began.     

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.     

Harvesting and predation of a sex- and age-structured population

We study a discrete-time system of equations for a structured ungulate population exploited by human harvesting or a dynamic predator. The population is divided into juveniles, and female and male adults. Harvesting is concentrated on adults (trophy hunting of males or population control measures on females), whereas predation occurs in juveniles. Though the model consists of four nonlinear equations, we find explicit expressions for the steady states. We use these explicit expressions to investigate harvesting rates that allow population persistence, rates that ensure population control, and optimal harvesting efforts. Several reductions of complexity allow for a detailed analysis of the dynamics of the model. Most notably, we find that even compensatory density dependence can lead to a period doubling bifurcation, that the model does not support consumer–resource cycles, and that an Allee effect can emerge from the interplay of stage-specific predation and density-dependent prey reproduction.     

Differentiation of reproductive and competitive ability in the invaded range of Senecio inaequidens: the role of genetic Allee effects, adaptive and nonadaptive evolution

Genetic differentiation in the competitive and reproductive ability of invading populations can result from genetic Allee effects or r/K selection at the local or range-wide scale. However, the neutral relatedness of populations may either mask or falsely suggest adaptation and genetic Allee effects. In a common-garden experiment, we investigated the competitive and reproductive ability of invasive Senecio inaequidens populations that vary in neutral genetic diversity, population age and field vegetation cover. To account for population relatedness, we analysed the experimental results with 'animal models' adopted from quantitative genetics. Consistent with adaptive r/K differentiation at local scales, we found that genotypes from low-competition environments invest more in reproduction and are more sensitive to competition. By contrast, apparent effects of large-scale r/K differentiation and apparent genetic Allee effects can largely be explained by neutral population relatedness. Invading populations should not be treated as homogeneous groups, as they may adapt quickly to small-scale environmental variation in the invaded range. Furthermore, neutral population differentiation may strongly influence invasion dynamics and should be accounted for in analyses of common-garden experiments.     

Allee effects limit population viability of an annual plant

ABSTRACT Allee effects may be experienced by plants when populations are too small or isolated to receive sufficient pollinator services to replace themselves. This article reports experimental data from an annual herb, Clarkia concinna, documenting that small patches suffered reproductive failure due to lack of effective pollination when critical thresholds of isolation were exceeded. In contrast, sufficiently large patches attracted pollinators regardless of their degree of isolation. These data accord with data on patch extinctions showing that small and isolated patches have a higher extinction rate than do large patches and with observations showing chronically low reproductive success in such patches prior to extinction. While not conclusively demonstrating that Allee effects cause extinction in small and isolated patches, the data are suggestive. Although threshold effects have been postulated in several mathematical models of population viability, this is the first report of data from natural populations that display the occurrence of such thresholds. These results have implications for the management of endangered plants, which often are restricted to isolated, small populations, as well as suggesting a potential limit to spatial spread in plant populations dependent on animal vectors for reproduction.     

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.