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.     

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.     

Genetic rescue in interconnected populations of small and large size of the self-incompatible Ranunculus reptans

Small populations of our study species Ranunculus reptans have reduced fitness because of inbreeding, genetic load, and reduced mate availability; that is, they suffer from a three-fold genetic Allee effect. Here, we investigate how the effect of interpopulation outbreeding on offspring fitness depends on population size. We performed within- and between-population crosses with plants originating from 15 populations, and measured offspring performance in a common environment. Interpopulation outbreeding led to an increase in population means of clonal performance, which was defined as the number of rooted offspring rosettes produced per maternal ovule. This fitness gain mainly occurred at the life stage of seed set. It was especially pronounced for populations with a long-term history of small size inferred from their low genetic diversity, estimated from eight allozyme loci. We conclude that in a self-incompatible plant such as R. reptans, interpopulation outbreeding can lead to an immediate genetic rescue effect due to increased cross-compatibility and heterosis, and that this rescue effect is increased as population size decreases.     

The presence of heterospecific males causes an Allee effect

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

Establishing a beachhead: a stochastic population model with an Allee effect applied to species invasion

We formulated a spatially explicit stochastic population model with an Allee effect in order to explore how invasive species may become established. In our model, we varied the degree of migration between local populations and used an Allee effect with variable birth and death rates. Because of the stochastic component, population sizes below the Allee effect threshold may still have a positive probability for successful invasion. The larger the network of populations, the greater the probability of an invasion occurring when initial population sizes are close to or above the Allee threshold. Furthermore, if migration rates are low, one or more than one patch may be successfully invaded, while if migration rates are high all patches are invaded.     

Allee effects, immigration, and the evolution of species' niches

Theoretical studies of adaptation to sink environments (with conditions outside the niche requirements of a species) have shown that immigration from source habitats can either facilitate or inhibit local adaptation. Here, we examine the influence of immigration on the evolution of local adaptation, given an Allee effect (i.e., at low densities, absolute fitness increases with population density). We consider a deterministic model for evolution at a haploid locus, and a stochastic individual-based model for evolution of a quantitative trait, and several kinds of Allee effects. We demonstrate that increased immigration can greatly facilitate adaptive evolution in the sink; with greater immigration, local population sizes rise, and because of the Allee effect, there is a positive indirect effect of immigration on local fitness. This makes it easier for alleles of modest effect to be captured by natural selection, transforming the sink into a locally adapted population that can persist without immigration.     

Life cycles of the freshwater, planktonic copepod Cyclops scutifer Sars on a north–south gradient in North America

A seasonal cycle longer than one year is exceptional both among marine and freshwater zooplankton, but occurs widely in the freshwater cyclopoid Cyclops scutiferSars. This paper presents the life cycle of this species in eight North American populations in lakes from 42° to 69° N. Five of the populations had life cycles showing a combination of a one and two year cycle. Two populations had a 1-year cycle, and the southernmost had a combination of a 0.5-year and a 1-year cycle. Lengths of cycles increased with latitude and thus inversely with the temperature. Populations in all lakes were divided into a direct and a delayed line of development. Diapause was not documented in any of the lakes.     

Genetic Diversity in Introduced Populations with an Allee Effect

A phenomenon that strongly influences the demography of small introduced populations and thereby potentially their genetic diversity is the demographic Allee effect, a reduction in population growth rates at small population sizes. We take a stochastic modeling approach to investigate levels of genetic diversity in populations that successfully overcame either a strong Allee effect, in which populations smaller than a certain critical size are expected to decline, or a weak Allee effect, in which the population growth rate is reduced at small sizes but not negative. Our results indicate that compared to successful populations without an Allee effect, successful populations with a strong Allee effect tend to (1) derive from larger founder population sizes and thus have a higher initial amount of genetic variation, (2) spend fewer generations at small population sizes where genetic drift is particularly strong, and (3) spend more time around the critical population size and thus experience more genetic drift there. In the case of multiple introduction events, there is an additional increase in diversity because Allee-effect populations tend to derive from a larger number of introduction events than other populations. Altogether, a strong Allee effect can either increase or decrease genetic diversity, depending on the average founder population size. By contrast, a weak Allee effect tends to decrease genetic diversity across the entire range of founder population sizes. Finally, we show that it is possible in principle to infer critical population sizes from genetic data, although this would require information from many independently introduced populations.     

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.     

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.     

Population cycles produce periodic range boundary pulses

Classical theories of biological invasions predict constant rates of spread that can be estimated from measurable life history parameters, but such outcomes depend strongly on assumptions that are often unmet in nature. Subsequent advances have demonstrated how relaxing assumptions of these foundational models results in other spread patterns seen in nature, including invasions that accelerate through time, or that alternate among periods of expansion, retraction, and stasis of range boundaries. In this paper, we examine how periodic population fluctuations affect temporal patterns of range expansion by coupling empirical data on the gypsy moth invasion in North America with insights from a model incorporating population cycles, Allee effects, and stratified diffusion. In an analysis of field data, we found that gypsy moth spread exhibits pulses with a period of 6 yr, which field data and model simulations suggest is the result of a 6‐yr population cycle in established populations near the invasion front. Model simulations show that the development of periodic behavior in range expansion depends primarily on the period length of population cycles. The period length of invasion pulses corresponded to the population cycle length, and the regularity of invasion pulses tended to decline with increases in population cycle length. A key insight of this research is that dynamics of established populations, behind the invasion front, can have strong effects on spread. Our findings suggest that coordination between separate management programs targeting low‐density spreading and established outbreaking populations, respectively, could increase the efficacy of efforts to mitigate gypsy moth impacts. Given the variety of species experiencing population fluctuations, Allee effects, and stratified diffusion, insights from this study are potentially important to understanding how the range boundaries of many species change.     

Geographic body size variation in the periodical cicadas Magicicada: implications for life cycle divergence and local adaptation

Seven species in three species groups (Decim, Cassini and Decula) of periodical cicadas (Magicicada) occupy a wide latitudinal range in the eastern United States. To clarify how adult body size, a key trait affecting fitness, varies geographically with climate conditions and life cycle, we analysed the relationships of population mean head width to geographic variables (latitude, longitude, altitude), habitat annual mean temperature (AMT), life cycle and species differences. Within species, body size was larger in females than males and decreased with increasing latitude (and decreasing habitat AMT), following the converse Bergmann's rule. For the pair of recently diverged 13‐ and 17‐year species in each group, 13‐year cicadas were equal in size or slightly smaller on average than their 17‐year counterparts despite their shorter developmental time. This fact suggests that, under the same climatic conditions, 17‐year cicadas have lowered growth rates compared to their 13‐years counterparts, allowing 13‐year cicadas with faster growth rates to achieve body sizes equivalent to those of their 17‐year counterparts at the same locations. However, in the Decim group, which includes two 13‐year species, the more southerly, anciently diverged 13‐year species (Magicicada tredecim) was characterized by a larger body size than the other, more northerly 13‐ and 17‐year species, suggesting that local adaptation in warmer habitats may ultimately lead to evolution of larger body sizes. Our results demonstrate how geographic clines in body size may be maintained in sister species possessing different life cycles.     

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.     

Environmental conditions during early life determine the consequences of inbreeding in Agrostemma githago (Caryophyllaceae)

In an inbred population, selection may reduce the frequency of deleterious recessive alleles through a process known as purging. Empirical studies suggest, however, that the efficacy of purging in natural populations is highly variable. This variation may be due, in part, to variation in the expression of inbreeding depression available for selection to act on. This experiment investigates the roles of life stage and early‐life environment in determining the expression of inbreeding depression in Agrostemma githago. Four population‐level crosses (‘self’, ‘within’, ‘near’ and ‘far’) were conducted on 20 maternal plants from a focal population. Siblings were planted into one of three early environmental treatments with varying stress levels. Within the focal population, evidence for purging of deleterious recessive alleles, as well as for variation in the expression of inbreeding depression across the life cycle was examined. In addition, the effect of early environment on the expression of inbreeding depression and the interaction with cross‐type was measured. We find that deleterious recessive alleles have not been effectively purged from our focal population, the expression of inbreeding depression decreases over the course of the life cycle, and a stressful early environment reduces the variance in inbreeding depression expressed later in life, but does not consistently influence the relative fitness of inbred versus outcrossed individuals.     

SELECTION FOR PRIME-NUMBER INTERVALS IN A NUMERICAL MODEL OF PERIODICAL CICADA EVOLUTION

Periodical cicadas are known for unusually long and prime-numbered life cycles (13 and 17 years) for insects. To explain the evolution of prime-numbered reproductive intervals (life cycles), the hybridization hypothesis claims that prime numbers greatly reduce the chance of hybridization with other life cycles. We investigate the hybridization hypothesis using a simulation model. This model is a deterministic, discrete population model with three parameters: larval survival per year, clutch size, and emergence success. Reproductive intervals from 10 years to 20 years compete for survival in the simulations. The model makes three key assumptions: a Mendelian genetic system, random mating among broods of different life-cycle lengths, and integer population sizes. Longer life cycles have larger clutch sizes but suffer higher total mortality than shorter life cycles. Our results show that (1) nonprime-numbered reproductive intervals disappear rapidly in comparison to the selection among the various prime-numbered life cycles, (2) the selection of prime-numbered intervals happens only when populations are at the verge of extinction, and (3) the 13- and 17-year prime phenotypes evolve under certain conditions of the model and may coexist. The hybridization hypothesis is discussed in light of other hypotheses for the evolution of periodical cicada life cycles.     

Variation in Allee effects: evidence,unknowns, and directions forward

Allee effects, positive effects of population size or density on per-capita fitness, are of broad interest in ecology and conservation due to their importance to the persistence of small populations and to range boundary dynamics. A number of recent studies have highlighted the importance of spatiotemporal variation in Allee effects and the resulting impacts on population dynamics. These advances challenge conventional understanding of Allee effects by reframing them as a dynamic factor affecting populations instead of a static condition. First, we synthesize evidence for variation in Allee effects and highlight potential mechanisms. Second, we emphasize the “Allee slope,” i.e., the magnitude of the positive effect of density on the per-capita growth rate, as a metric for demographic Allee effects. The more commonly used quantitative metric, the Allee threshold, provides only a partial picture of the underlying forces acting on population growth despite its implications for population extinction. Third, we identify remaining unknowns and strategies for addressing them. Outstanding questions about variation in Allee effects fall broadly under three categories: (1) characterizing patterns of natural variability; (2) understanding mechanisms of variation; and (3) implications for populations, including applications to conservation and management. Future insights are best achieved through robust interactions between theory and empiricism, especially through mechanistic models. Understanding spatiotemporal variation in the demographic processes contributing to the dynamics of small populations is a critical step in the advancement of population ecology.     

Population models with Allee effect: a new model

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

Population models with Allee effect: a new model

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

Synchrony's double edge: transient dynamics and the Allee effect in stage structured populations

In populations subject to positive density dependence, individuals can increase their fitness by synchronizing the timing of key life history events. However, phenological synchrony represents a perturbation from a population's stable stage structure and the ensuing transient dynamics create troughs of low abundance that can promote extinction. Using an ecophysiological model of a mass-attacking pest insect, we show that the effect of synchrony on local population persistence depends on population size and adult lifespan. Results are consistent with a strong empirical pattern of increased extinction risk with decreasing initial population size. Mortality factors such as predation on adults can also affect transient dynamics. Throughout the species range, the seasonal niche for persistence increases with the asynchrony of oviposition. Exposure to the Allee effect after establishment may be most likely at northern range limits, where cold winters tend to synchronize spring colonization, suggesting a role for transient dynamics in the determination of species distributions.     

Allee effects, aggregation, and invasion success

Understanding the factors that influence successful colonization can help inform ecological theory and aid in the management of invasive species. When founder populations are small, individual fitness may be negatively impacted by component Allee effects through positive density dependence (e.g., mate limitation). Reproductive and survival mechanisms that suffer due to a shortage of conspecifics may scale up to be manifest in a decreased per-capita population growth rate (i.e., a demographic Allee effect). Mean-field population level models are limited in representing how component Allee effects scale up to demographic Allee effects when heterogeneous spatial structure influences conspecific availability. Thus, such models may not adequately characterize the probability of establishment. In order to better assess how individual level processes influence population establishment and spread, we developed a spatially explicit individual-based stochastic simulation of a small founder population. We found that increased aggregation can affect individual fitness and subsequently impact population growth; however, relatively slow dispersal—in addition to initial spatial structure—is required for establishment, ultimately creating a tradeoff between probability of initial establishment and rate of subsequent spread. Since this result is sensitive to the scaling up of component Allee effects, details of individual dispersal and interaction kernels are key factors influencing population level processes. Overall, we demonstrate the importance of considering both spatial structure and individual level traits in assessing the consequences of Allee effects in biological invasions.