Allee Threshold Research Articles

Predator-induced and mating limitation-induced Allee effects in a discrete-time [formula omitted] epidemic model

We use a Susceptible–Infected–Immune–Susceptible (SIMS) discrete-time epidemic model that incorporates both predator-induced and mating limitation-induced Allee effects in the demographic dynamics to study the impact of an infectious disease on population persistence or extinction. When the Allee effect is strong, with or without the disease, the total population goes extinct whenever the initial total population size is smaller than the Allee threshold. However, when the Allee effect is weak or the initial population sizes are high and the total population is either asymptotically constant or under geometric growth, we use the next generation method to compute the basic reproductive number, ℜ0, and use it to predict the (uniform) persistence or extinction of the infective and immune populations.

Allee threshold and stochasticity in biological invasions: Colonization time at low propagule pressure

We consider the problem of estimating the time needed for species colonization. The analysis is based upon the known population dynamic models by Dennis with minor modification to the Allee effect description, which allows us to obtain an analytical expression for the colonization time. For the stochastic counterpart of the models in diffusion approximation, we (1) propose the description of immigration stochasticity, (2) provide the estimates of time required for the population to overcome strong demographic Allee effect, and (3) consider the numerical results for mean colonization time and its uncertainty. Strong Allee effect strictly disallows populations at small immigration rates to colonize new habitats, unless the stochasticity in immigration, environment, or demography is present, or incorporated into the model. Immigration stochasticity, complementing with environmental and demographic stochasticity, enables the populations to overcome the Allee threshold even at low values of propagule pressure.

On the evidence of an Allee effect in herring populations and consequences for population survival: A model-based study

We propose a deterministic model for a population subject to a strong Allee effect and undertake a model based study on commercially valuable herring fish population. We analyze the time series of two herring populations from the Icelandic and Canadian regions from the Global Population Dynamics Database with GPDD Id 1765, 1759. The parameters for the proposed models are estimated using Nonlinear Least Squares and Grid Search procedures. Confidence intervals for the parameters are computed using both Nonlinear Least Squares and regression bootstrap estimates. In the stochastic counterpart of the model we consider demographic noise to estimate different extinction measures viz. probability of extinction and expected time to extinction. The data histogram of population size is well approximated through the quasi-stationary distributions of the proposed stochastic model. The hypothesis of the presence of a strong Allee effect is prominent in both of the herring populations. The presence of a strong Allee effect in these two populations makes them more vulnerable to extinction. External perturbation or uncontrolled harvesting may drive the populations below the Allee threshold where the probability of extinction is high. We suggest that, our analysis can have a huge impact on understanding extinction patterns and enable us to identify demographic threats and guide decision making in conservation management. In addition, a similar analysis can be used in understanding the conservation status for other species.

Mate‐finding allee effect in spruce budworm population dynamics

AbstractAllee effects can cause populations to decline due to decreasing population growth rates with decreasing density and play a major role in population dynamics. Mate‐finding failure, a common mechanism contributing to demographic Allee effects, is usually difficult to demonstrate because of the arduous nature of sampling individuals at very low densities. In a rising outbreak of the eastern spruce budworm, Choristoneura fumiferana (Clemens) (Lepidoptera: Tortricidae), we used caged and tethered virgin females in traps to measure mating success over population densities ranging from deep endemic to outbreak conditions. We found that mating success increased with increasing population density, and that at endemic population densities, females experienced difficulties attracting males and mating, demonstrating for the first time a mate‐finding Allee effect in the spruce budworm. The relationship between population density and mating success is nonlinear. As population density increased, the proportion of mated females eventually reached a plateau and mating success was not 100% even at the highest moth densities, probably due to female reluctance to mate and perhaps interference competition by males for access to females. Both laboratory‐reared and wild females were equally effective in synthesizing pheromone, attracting males, and mating. Our results strongly suggest that a mate‐finding Allee effect is involved in maintaining low‐density spruce budworm populations below an Allee threshold where they fail to grow. Factors such as changes in predation pressure and immigration could help populations overcome this Allee threshold.

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.

Competition, facilitation and the Allee effect

Individuals of different species may interact in many different ways, such as competition, mutualism, or predation, to name but a few. Recent theory and experiments reveal that whether an interaction is beneficial or detrimental to the dynamics of a population often depends on species densities and other environmental factors. Here, we explore how, for suitable densities, facilitation may arise between two competing species with an Allee effect. We consider two different mechanisms for the Allee effect: 1) plant species with obligate insect pollination, and 2) generalist predation. In the first case, a second plant species, competing for nutrients, may have a facilitative effect by attracting more pollinators. In the second case, another potentially competing species may serve to satiate the same generalist predator and thereby have a facilitative effect. We explore three aspects of facilitation in each of the two systems. The focal species may benefit from the presence of a ‘competitor’ if it experiences 1) the removal of the Allee threshold, 2) a lowering of the Allee threshold, or 3) an increase in carrying capacity. We find that the latter two effects occur in both study systems whereas the first only occurs for the generalist predation system but not for the plant‐pollination system. We give precise conditions on when such a facilitative effect can be expected. We also demonstrate several unexpected outcomes of these two‐species interactions with multiple steady states, such as obligate co‐occurence; we draw parallels to the dynamics of species known as ‘ecosystem engineers’, and we discuss implications for conservation and management.

Bioeconomic synergy between tactics for insect eradication in the presence of Allee effects

Preventing the establishment of invading pest species can be beneficial with respect to averting future environmental and economic impacts and also in preventing the accumulation of control costs. Allee effects play an important role in the dynamics of newly established, low-density populations by driving small populations into self-extinction, making Allee effects critical in influencing outcomes of eradication efforts. We consider interactions between management tactics in the presence of Allee effects to determine cost-effective and time-efficient combinations to achieve eradication by developing a model that considers pesticide application, predator augmentation and mating disruption as control tactics, using the gypsy moth as a case study. Our findings indicate that given a range of constant expenditure levels, applying moderate levels of pesticides in conjunction with mating disruption increases the Allee threshold which simultaneously substantially decreases the time to eradication relative to either tactic alone. In contrast, increasing predation in conjunction with other tactics requires larger economic expenditures to achieve similar outcomes for the use of pesticide application or mating disruption alone. These results demonstrate the beneficial synergy that may arise from nonlinearities associated with the simultaneous application of multiple eradication tactics and offer new prospects for preventing the establishment of damaging non-native species.

Evolutionary dynamics and strong Allee effects

We describe the dynamics of an evolutionary model for a population subject to a strong Allee effect. The model assumes that the carrying capacity k(u), inherent growth rate r(u), and Allee threshold a(u) are functions of a mean phenotypic trait u subject to evolution. The model is a plane autonomous system that describes the coupled population and mean trait dynamics. We show bounded orbits equilibrate and that the Allee basin shrinks (and can even disappear) as a result of evolution. We also show that stable non-extinction equilibria occur at the local maxima of k(u) and that stable extinction equilibria occur at local minima of r(u). We give examples that illustrate these results and demonstrate other consequences of an Allee threshold in an evolutionary setting. These include the existence of multiple evolutionarily stable, non-extinction equilibria, and the possibility of evolving to a non-evolutionary stable strategy (ESS) trait from an initial trait near an ESS.

Combining Tactics to Exploit Allee Effects for Eradication of Alien Insect Populations

Invasive species increasingly threaten ecosystems, food production, and human welfare worldwide. Hundreds of eradication programs have targeted a wide range of nonnative insect species to mitigate the economic and ecological impacts of biological invasions. Many such programs used multiple tactics to achieve this goal, but interactions between tactics have received little formal consideration, specifically as they interact with Allee dynamics. If a population can be driven below an Allee threshold, extinction becomes more probable because of factors such as the failure to find mates, satiate natural enemies, or successfully exploit food resources, as well as demographic and environmental stochasticity. A key implication of an Allee threshold is that the population can be eradicated without the need and expense of killing the last individuals. Some combinations of control tactics could interact with Allee dynamics to increase the probability of successful eradication. Combinations of tactics can be considered to have synergistic (greater efficiency in achieving extinction from the combination), additive (no improvement over single tactics alone), or antagonistic (reduced efficiency from the combination) effects on Allee dynamics. We highlight examples of combinations of tactics likely to act synergistically, additively, or antagonistically on pest populations. By exploiting the interacting effects of multiple tactics on Allee dynamics, the success and cost-effectiveness of eradication programs can be enhanced.

Host Demographic Allee Effect, Fatal Disease, and Migration: Persistence or Extinction

In this paper, we focus on biodiversity, a major problem for ecosystem resilience. We use extensions of the suspectible-infected (SI) epidemic model of Hilker et al. to study how population persistence or extinction of a vulnerable species relates to habitat dependent Allee thresholds, fatal disease dynamics, and migration rates in both discrete and continuum sets of compartments. We analyze the migration-linked models and establish verifiable conditions that guarantee host population persistence (with or without infected individuals) or extinction.

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible–infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.

Evolution of plant-pollinator mutualisms in response to climate change.

Climate change has the potential to desynchronize the phenologies of interdependent species, with potentially catastrophic effects on mutualist populations. Phenologies can evolve, but the role of evolution in the response of mutualisms to climate change is poorly understood. We developed a model that explicitly considers both the evolution and the population dynamics of a plant–pollinator mutualism under climate change. How the populations evolve, and thus whether the populations and the mutualism persist, depends not only on the rate of climate change but also on the densities and phenologies of other species in the community. Abundant alternative mutualist partners with broad temporal distributions can make a mutualism more robust to climate change, while abundant alternative partners with narrow temporal distributions can make a mutualism less robust. How community composition and the rate of climate change affect the persistence of mutualisms is mediated by two-species Allee thresholds. Understanding these thresholds will help researchers to identify those mutualisms at highest risk owing to climate change.

Effect of environmental fluctuations on invasion fronts

We determine the density profile and velocity of invasion fronts in one-dimensional infinite habitats in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction–diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain analytical expressions for the front profile and velocity by constructing a variational principle. The effect of the noise differs, depending on whether it affects the density-independent growth rate, the intraspecific competition term or the Allee threshold. Fluctuations in the density-independent growth rate increase the invasion velocity and the population density of the invaded area. Fluctuations in the competition term also change the population density of the invaded area, but modify the invasion velocity only for certain initial conditions. Fluctuations in the Allee threshold can induce pulled or pushed invasion fronts as well as invasion failure. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction–diffusion equations with multiplicative noise.

Discrete-time host–parasitoid models with Allee effects: Density dependence versus parasitism

We present two general discrete-time host–parasitoid models with Allee effects on the host. In the first model, it is assumed that parasitism occurs prior to density dependence, while in the second model we assume that density dependence operates first followed by parasitism. It is shown that both models have similar asymptotic behaviour. The parasitoid population will definitely go extinct if the maximal growth rate of the host population is less than or equal to one, independent of whether density dependence or parasitism occurs first. The fate of the population is initial condition dependent if this maximal growth rate exceeds one. In particular, there exists a host population threshold, the Allee threshold, below which the host population goes extinct and so does the parasitoid. This threshold is the same for both models. Numerical examples with different functions are simulated to illustrate our analytical results.

Allee effects in an iteroparous host population and in host-parasitoid interactions

We investigate a stage-structured model of an iteroparouspopulation with two age classes. The population is assumed toexhibit Allee effects through reproduction. The asymptoticdynamics of the model depend on the maximal reproductive number ofthe population. The population may persist if the maximalreproductive number is greater than one. There exists a populationthreshold in terms of the unstable interior equilibrium. The hostpopulation will become extinct if its initial distribution liesbelow the threshold and the host population can persistindefinitely if its initial distribution lies above thethreshold. In addition, if the unstable equilibrium is a saddlepoint and the system has no $2$-cycles, then the stable manifoldof the saddle point provides the Allee threshold for the host.Based on this host population system, we construct ahost-parasitoid model to study the impact of Allee effects uponthe population interaction. The parasitoid population may drivethe host to below the Allee threshold so that both populationsbecome extinct. On the other hand, under some conditions on theparameters, the host-parasitoid system may possess an interiorequilibrium and the populations may coexist as an interiorequilibrium.

Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.

Adaptive dynamics of cooperation may prevent the coexistence of defectors and cooperators and even cause extinction.

It has recently been demonstrated that ecological feedback mechanisms can facilitate the emergence and maintenance of cooperation in public goods interactions: the replicator dynamics of defectors and cooperators can result, for example, in the ecological coexistence of cooperators and defectors. Here we show that these results change dramatically if cooperation strategy is not fixed but instead is a continuously varying trait under natural selection. For low values of the factor with which the value of resources is multiplied before they are shared among all participants, evolution will always favour lower cooperation strategies until the population falls below an Allee threshold and goes extinct, thus evolutionary suicide occurs. For higher values of the factor, there exists a unique evolutionarily singular strategy, which is convergence stable. Because the fitness function is linear with respect to the strategy of the mutant, this singular strategy is neutral against mutant invasions. This neutrality disappears if a nonlinear functional response in receiving benefits is assumed. For strictly concave functional responses, singular strategies become uninvadable. Evolutionary branching, which could result in the evolutionary emergence of cooperators and defectors, can occur only with locally convex functional responses, but we illustrate that it can also result in coevolutionary extinction.

On stochastic population model with the Allee effect

This paper discusses the existence, uniqueness and asymptotic stability of the solution to the stochastic population model with the Allee effect, which is represented by the stochastic differential equation d N t = N t ( 1 − N t T ) ( N t K − 1 ) ( r d t + α d W t ) . We are focused on the correlation between the growth rate r and population size N t , which is shown to be positive if the population size N t is above the Allee threshold T , and negative if N t < T .

Synthesizing ecology and evolution for the study of invasive species.

This special volume is a compilation of papers from a workshop on Synthesizing Ecology and Evolution for the Study of Invasive Species, held in March 2009 at Lake Tahoe, CA, USA. The goal of this workshop was to bring together a diverse group of scientists to discuss gaps in research pertaining to invasion biology and outline research objectives for the future, with an emphasis on synthesizing the ecology and evolution of invasive species. Discussion among the participants exposed several areas that warrant additional data, experimentation, model development, and theoretical integration. For example, a substantial portion of the discussion was devoted toward examining the implications of theoretical models of invasive populations that couple ecological and evolutionary dynamics, including themes such as gene ‘surfing’ tracking waves of invasion, the evolution of Allee thresholds, and the importance of details of genetic architectures underlying traits undergoing selection during invasions. Discussion also focused on factors that have received inadequate attention in the field, such as the role of trans-generational plasticity (TGP) in promoting invasive success, coevolution between invasive and native species, and evolution of host–pathogen interactions during invasions. During the discussion, the possibility was raised that epigenetic changes might be important in promoting rapid evolution during invasions. Participants also discussed methodological advances that are enhancing the types of data available to analyse invasions, such as genomic approaches that could identify loci underlying critical traits that limit invasions. Several investigators noted the paucity of data from the native range of invasive populations, which is unfortunate, given that such information can illuminate the specific habitats that foster potentially invasive populations, and provide insights into the evolutionary history of such populations. Finally, information from the literature on biological control could be mined more intensively to extract predictions on colonization success of invasive species, as the determinants of success in deliberate introductions should provide insight into comparable factors that accompany successful accidental introductions. The wide-ranging discussions that took place at this workshop will continue into the future and pave the way for a deepening of our understanding of the coupled ecological and evolutionary dimensions of invasions. In this special issue, we provide a sampler of the discussions on the theoretical, conceptual, and empirical integration of evolution and ecology that permeated the Lake Tahoe workshop, as well as some lucid and compelling case studies of rapid evolution during invasions.

Discrete host-parasitoid models with Allee effects and age structure in the host

We study a stage-structured single species population model with Allee effects. The asymptotic dynamics of the model depend on the maximal growth rate of the population as well as on its initial population size. We also investigate two models of host-parasitoid interaction with stage-structure and Allee effects in the host. The parasitoid population may drive the host population to extinction in both models even if the initial host population is beyond the Allee threshold.