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.     

Saddle-Point Approximations,Integrodifference Equations,and Invasions

Invasion, the growth in numbers and spatial spread of a population over time, is a fundamental process in ecology. Governments and businesses expend vast sums to prevent and control invasions of pests and pestilences and to promote invasions of endangered species and biological control agents. Many mathematical models of biological invasions use nonlinear integrodifference equations to describe the growth and dispersal processes and to predict the speed of invasion fronts. Linear models have received less attention, perhaps because they are difficult to simulate for large times. In this paper, we use the saddle-point method, alias the method of steepest descent, to derive asymptotic approximations for the solutions of linear integrodifference equations. We work through five examples, for Gaussian, Laplace, and uniform dispersal kernels in one dimension and for asymmetric Gaussian and radially symmetric Laplace kernels in two dimensions. Our approximations are extremely close to the exact solutions, even for intermediate times. We also employ an empirical saddle-point approximation to predict densities using dispersal data. We use our approximations to examine the effects of censored dispersal data on estimates of invasion speed and population density.     

Density dependence slows invader spread in fragmented landscapes

Patchiness is a defining characteristic of most natural and anthropogenic habitats, yet much of our understanding of how invasions spread has come from models of spatially homogeneous environments. Except for populations with Allee effects, an invader's growth rate when rare and dispersal determine its spread velocity; intraspecific competition has little to no influence. How this result might change with landscape patchiness, however, is poorly understood. We used simulation models and their analytical approximations to explore the effect of density dependence on the spread of annual plant invaders moving through heterogeneous landscapes with gaps in suitable habitat. We found that landscape patchiness and discrete invader population size interacted to generate a strong role for density dependence. Intraspecific competition greatly slowed the spread of invasions through patchy landscapes by regulating how rapidly a population could produce enough seeds to surpass habitat gaps. Populations with continuously varying density showed no such effect of density dependence. We adapted a stochastic dispersal model to approximate spread when gap sizes were small relative to the mean dispersal distance and a Markov chain approximation for landscapes with large gaps. Our work suggests that ecologists must consider reproduction at both low and high densities when predicting invader spread.     

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.     

Incorporating Allee effects into reintroduction strategies

Allee effects, the reduction of vital rates at low population densities, can occur through several mechanisms, all of which potentially apply to reintroduced populations. Reintroduced populations are initially at low densities, hence Allee effects can potentially lead to reintroduction failure despite habitat quality being sufficient to allow long-term persistence if the population survived the establishment phase. The probability of such failures can potentially be reduced by releasing large numbers of organisms, by reducing post-release dispersal or mortality through management, or by directly managing the Allee effects, e.g., by implementing predator control or food supplementation until population size increases. However, such measures incur costs, as large releases have a greater impact on source populations, and management actions require financial and other resources. It is therefore essential to compare the costs and benefits of attempting to reduce Allee effects in reintroduction programs. Here we advocate the use of structured decision-making frameworks whereby alternative strategies are nominated, probability distributions of outcomes obtained under different strategies, and utilities assigned to different outcomes. We illustrate the potential application of such decision frameworks using projections from a stochastic population model including Allee effects. As there will seldom be estimates of Allee effects available from the species or system involved, it will be necessary to predict these effects based on the biology of the species and data from other systems. In doing so, it is important to identify mechanisms for proposed Allee effects, and to avoid misleading inferences from correlations subject to confounds. In particular, naive interpretations of correlations between numbers released and reintroduction success may exaggerate the benefits of releasing large numbers.     

Is two company or a crowd: How does conspecific density affect the small-scale dispersal of a weed biocontrol agent?

To predict the growth and spread of an insect population introduced for the biological control of weeds, one must first understand the factors affecting the movement of individuals in the population. The purpose of this study was to determine how the dispersal rate of Aphthona lacertosa (Rosenhauer) (Chrysomelidae) was affected by conspecific density and by the characteristics of leafy spurge (Euphorbia esula L.: Euphorbiaceae) in patches where these beetles feed. In 2002 in Manitoba and in 2003 in Alberta, Canada, between 200 and 2500 insects were released in small patches (<10 m²) of spurge. The number and location of beetles within patches was monitored over subsequent days. In 1 m² plots within patches, spurge ramet density, the proportions of vegetative and reproductive ramets, and ramet height were measured. In both years, beetle movement within patches and emigration from patches, was not affected by conspecific density. In Manitoba in 2002, beetles aggregated non-randomly on either vegetative or reproductive ramets within plots, but plot characteristics were not related to the formation of aggregations. In Alberta in 2003, plots in which beetles aggregated had significantly higher spurge density but did not differ in either the proportion of vegetative ramets or in the amount of non-spurge vegetation. These results suggest that density-dependent dispersal does not limit the population's ability to reach densities up to 2500 beetles/m².     

Identification, Pathogenicity and Comparative Virulence of Fusarium spp. Associated with Diseased Euphorbia spp. in Europe

Fusarium spp. isolated from diseased Euphorbia spp. in Europe were assessed for pathogenicity to North American accessions of leafy spurge ( Euphorbia esula/virgata ). Of the nine strains of Fusarium spp. isolated from diseased E. stepposa or E. virgata in the Caucasus region of Russia and E. esula/virgata in southern France, all were pathogenic to leafy spurge. There were significant differences in virulence among strains. Four strains, including the two that were most virulent, were identified as F. oxysporum . Four of the five other strains were identified as F. solani and one was identified as F. proliferatum . Three of the four most virulent strains to leafy spurge were isolated from E. stepposa . The most virulent strain was associated with root damage caused by insect biological control agents, as found earlier with domestic strains of Fusarium spp. pathogenic to leafy spurge. Two strains identified as F. solani were vegetatively compatible. It was concluded that further screening of a larger set of strains of foreign Fusarium spp. under quarantine conditions in the US or in limited overseas facilities would be justified, and could yield promising biological control agents for leafy spurge.     

Relating dispersal and range expansion of California sea otters

Linking dispersal and range expansion of invasive species has long challenged theoretical and quantitative ecologists. Subtle differences in dispersal can yield large differences in geographic spread, with speeds ranging from constant to rapidly increasing. We developed a stage-structured integrodifference equation (IDE) model of the California sea otter range expansion that occurred between 1914 and 1986. The non-spatial model, a linear matrix population model, was coupled to a suite of candidate dispersal kernels to form stage-structured IDEs. Demographic and dispersal parameters were estimated independent of range expansion data. Using a single dispersal parameter, alpha, we examined how well these stage-structured IDEs related small scale demographic and dispersal processes with geographic population expansion. The parameter alpha was estimated by fitting the kernels to dispersal data and by fitting the IDE model to range expansion data. For all kernels, the alpha estimate from range expansion data fell within the 95% confidence intervals of the alpha estimate from dispersal data. The IDE models with exponentially bounded kernels predicted invasion velocities that were captured within the 95% confidence bounds on the observed northbound invasion velocity. However, the exponentially bounded kernels yielded range expansions that were in poor qualitative agreement with range expansion data. An IDE model with fat (exponentially unbounded) tails and accelerating spatial spread yielded the best qualitative match. This model explained 94% and 97% of the variation in northbound and southbound range expansions when fit to range expansion data. These otters may have been fat-tailed accelerating invaders or they may have followed a piece-wise linear spread first over kelp forests and then over sandy habitats. Further, habitat-specific dispersal data could resolve these explanations.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.     

Integration of biological control agents with other weed management technologies: Successes from the leafy spurge (Euphorbia esula) IPM program

An invasive weed can occupy a variety of environments and ecological niches and generally no single control method can be used across all areas the weed is found. Biological control agents integrated with other methods can increase and/or improve site-specific weed control, but such combinatorial approaches have not been widely utilized. The successful leafy spurge (Euphorbia esula L.) control program provides examples for future integrated weed programs that utilize biological control agents with traditional methods. Weed control methods can be used separately, such as when the leafy spurge gall midge (Spurgia esulae Gagné) reduced seed production in wooded areas while herbicides prevented further spread outside the tree line. Traditional methods also can be used directly with biological control agents. Incorporation of Aphthona spp. with herbicides has resulted in more rapid and complete leafy spurge control than either method used alone. Also, the insect population often increased rapidly following herbicide treatment, especially in areas where Aphthona spp. were established for several years but had been ineffective. Incorporation of Aphthona spp. with sheep or goat grazing has resulted in a larger decline in leafy spurge production than insects alone and in weed density than grazing alone. Controlled burns can aid establishment of biological control agents in marginally suitable environments, but timing of the fire must be coordinated to the insect’s life-cycle to ensure survival. Integration of biological control agents with revegetation programs required the agent to be the last method introduced because the cultivation and herbicide treatments necessary to establish desirable grasses and forbs were destructive to the insect. In a practical application, herbicides were combined with Aphthona spp. to help the insect establish and control leafy spurge in the habitat of the western prairie fringed orchid (Platanthera praeclara Sheviak and Bowles), an endangered species. Several experimental designs can be used to evaluate biological control agents with cultural, mechanical, and chemical control methods or with additional biological agents.     

Population spread in patchy landscapes under a strong Allee effect

Many species of invasive insects establish and spread in regions around the world, causing enormous economical and environmental damage, in particular in forests. Some of these insects are subject to an Allee effect whereby the population must surpass a certain threshold in order to establish. Recent studies have examined the possibility of exploiting an Allee effect to improve existing control strategies. Forests and most other ecosystems show natural spatial variation, and human activities frequently increase the degree of spatial heterogeneity. It is therefore imperative to understand how the interplay between this spatial variation and individual movement behavior affects the overall speed of spread of an invasion. To this end, we study an integrodifference equation model in a patchy landscape and with Allee growth dynamics. Movement behavior of individuals varies according to landscape quality. Our study focuses on how the speed of the resulting traveling periodic wave depends on the interaction between landscape fragmentation, patch-dependent dispersal, and Allee population dynamics.     

Invasion by extremes: population spread with variation in dispersal and reproduction

For populations having dispersal described by fat-tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat-tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment-generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment-generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat-tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest-forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as (1/T)(piuR)/2)(1/2) m yr(-1), where T is generation time, and u is a distance parameter (m2) of Clark et al.'s 2Dt model having shape parameter p = 1. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life-history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.     

Effect of Leafy Spurge (Euphorbia esula) Genotype on Feeding Damage and Reproduction of Aphthona spp.: Implications for Biological Weed Control

North American leafy spurge (Euphorbia esula L.) is genetically diverse and composed of multiple genotypes introduced from several areas of Europe and Asia. Five species of leafy spurge flea beetle (Aphthona spp.) have been introduced as biological control agents for leafy spurge, but were collected in a relatively small region of Europe. Greenhouse and field experiments were conducted to determine if observed variation in feeding preference and reproduction of Aphthona spp. on North American leafy spurge may be due in part to leafy spurge genotype. Leafy spurge genotypes were collected from Austria; Manitoba in Canada; and the states of Montana, Nebraska, North Dakota, South Dakota, and Wyoming in the United States. Leafy spurge genotype affected feeding but not egg laying by Aphthona spp. adults. Aphthona czwalinae/lacertosa fed slightly less in a free-choice test on a genotype from Manitoba (7%) compared to genotypes from Nebraska and North Dakota (14%). Aphthona flava tended to feed less on the Nebraska genotype than any other genotype evaluated. Reproduction of Aphthona spp. was greatly affected by leafy spurge genotype. For instance, A. czwalinae/lacertosa produced 72 adults per plant from a Nebraska genotype compared to 11 to 32 adults per plant from all other genotypes evaluated. Thus, some observed variation in establishment and reproduction of Aphthona spp. can be attributed to leafy spurge genotype.     

Role of Human-Mediated Dispersal in the Spread of the Pinewood Nematode in China

Background

Intensification of world trade is responsible for an increase in the number of alien species introductions. Human-mediated dispersal promotes not only introductions but also expansion of the species distribution via long-distance dispersal. Thus, understanding the role of anthropogenic pathways in the spread of invading species has become one of the most important challenges nowadays.

Methodology/Principal Findings

We analysed the invasion pattern of the pinewood nematode in China based on invasion data from 1982 to 2005 and monitoring data on 7 locations over 15 years. Short distance spread mediated by long-horned beetles was estimated at 7.5 km per year. Infested sites located further away represented more than 90% of observations and the mean long distance spread was estimated at 111–339 km. Railways, river ports, and lakes had significant effects on the spread pattern. Human population density levels explained 87% of the variation in the invasion probability (P<0.05). Since 2001, the number of new records of the nematode was multiplied by a factor of 5 and the spread distance by a factor of 2. We combined a diffusion model to describe the short distance spread with a stochastic, individual based model to describe the long distance jumps. This combined model generated an error of only 13% when used to predict the presence of the nematode. Under two climate scenarios (stable climate or moderate warming), projections of the invasion probability suggest that this pest could expand its distribution 40–55% by 2025.

Conclusions/Significance

This study provides evidence that human-induced dispersal plays a fundamental role in the spread of the pinewood nematode, and appropriate control measures should be taken to stop or slow its expansion. This model can be applied to Europe, where the nematode had been introduced later, and is currently expanding its distribution. Similar models could also be derived for other species that could be accidentally transported by humans.     

A ‘Goldilocks’ hypothesis for dispersal of biological control agents

The rate at which biological control agents disperse from release sites has important implications for their establishment and spread. Low rates of dispersal can yield spread that is too slow and may necessitate redistribution efforts for importation biological control and a high density of release sites for augmentation. Low dispersal rates may also lead to inbreeding at the site of release. On the other hand, high rates of dispersal can lead to Allee effects at the leading edge of the invasion front, potentially reducing the likelihood of establishment. Given these disadvantages associated with both low and high dispersal rates, we argue that intermediate rates of dispersal are likely to maximize the probability of establishment and appropriate spread for biological control agents released in the context of either importation or augmentative biological control. We consider this putative relationship a ‘Goldilocks hypothesis’ since it posits an optimum at intermediate values. In this review paper we begin by discussing the rationale for the Goldilocks hypothesis and then provide a case study from our work on importation biological control of the soybean aphid, Aphis glycines. Work on the soybean aphid parasitoid Binodoxys communis has shown that long-distance dispersal of immature parasitoids within winged migrating aphids is unlikely. This is likely good news for importation biological control because parasitoids dispersed in this manner would likely encounter crippling Allee effects. On the other hand, results from a field release study also suggest that female B. communis females (but not males) disperse actively from release sites. This female-biased dispersal may lead to strong mate-finding Allee effects and therefore may make establishment less likely.     

Modeling invasive species spread in Lake Champlain via evolutionary computations

We use a reaction diffusion equation, together with a genetic algorithm approach for model selection to develop a general modeling framework for biological invasions. The diffusion component of the reaction diffusion model is generalized to include dispersal and advection. The reaction component is generalized to include both linear and non-linear density dependence, and Allee effect. A combination of the reaction diffusion and genetic algorithm is able to evolve the most parsimonious model for invasive species spread. Zebra mussel data obtained from Lake Champlain, which demarcates the states of New York and Vermont, is used to test the appropriateness of the model. We estimate the minimum wave spread rate of Zebra mussels to be 22.5 km/year. In particular, the evolved models predict an average northward advection rate of 60.6 km/year (SD ± 1.9), which compares very well with the rate calculated from the known hydrologic residence time of 60 km/year. A combination of a reaction diffusion model and a genetic algorithm is, therefore, able to adequately describe some of the hydrodynamic features of Lake Champlain and the spread of a typical invasive species—Zebra mussels within the lake.     

Why trees migrate so fast: confronting theory with dispersal biology and the paleorecord

Reid's paradox describes the fact that classical models cannot account for the rapid (10(2)-10(3) m yr-1) spread of trees at the end of the Pleistocene. I use field estimates of seed dispersal with an integrodifference equation and simulation models of population growth to show that dispersal data are compatible with rapid spread. Dispersal estimates lay to rest the possibility that rapid spread occurred by diffusion. The integrodifference model predicts that, if the seed shadow has a long 'fat' tail, then rapid spread is possible, despite short average dispersal distances. It further predicts that velocity is more sensitive to life history than is classical diffusion. Application of such models is frustrated because the tail of the seed shadow cannot be fitted to data. However, the data can be used to test a 'long-distance' hypothesis against alternative ('local') models of dispersal using Akaike's Information Criterion and likelihood ratio tests. Tests show that data are consistent with >10% of seed dispersed as a long (10(2) m) fat-tailed kernel. Models based on such kernels predict spread as rapid as that inferred from the pollen record. If fat-tailed dispersal explains these rapid rates, then it is surprising not to see large differences in velocities among taxa with contrasting life histories. The inference of rapid spread, together with lack of obvious life-history effects, suggests velocities may have not reached their potentials, being stalled by rates of climate change, geography, or both.     

Comparing Allee effect-based and dispersal-based neutral models for species abundance distribution patterns

Dispersal is important for biodiversity maintenance in both neutral and niche theories. However, little is known about the potential role of Allee effect at the community level. In the present study, we developed neutral models for quantifying the separate and joint influences of the Allee effect and dispersal process, respectively, on species abundance distribution (SAD) patterns. Tree census data from Barro Colorado Island (BCI), Panama were used as the case to compare different neutral SAD models. Results showed that Allee effects were not detected in the BCI tree SAD curve. By contrast, the neutral models with the incorporation of dispersal process (including both immigration and emigration) can remarkably improve the fitting power of neutral models on the BCI tree SAD curve. Finally, even though the influence is not detectable, the Allee effect-based SAD models still might be alternative SAD models for model comparison and null hypothesis testing.     

Analytical pyrolysis-pattern recognition for the characterisation of leafy spurge (Euphorbia esula L.) biotypes

Curie-point pyrolysis-gas chromatography-pattern recognition was used to elucidate chemical variations within leafy spurge (Euphorbia spp.). Hierarchical cluster analysis (HCA) readily identified two major clusters corresponding to E. esula and E. cyparissias. The E. esula cluster further separated into three distinct subclusters. Results from principal components analysis (PCA) and fuzzy c-varieties (FCVPC) pattern recognition were similar, verifying the presence of three biotypes among the E. esula samples studied. It is suggested that analytical pyrolysis in combination with pattern recognition may predict the behaviour of biocontrol agents introduced into fields to control leafy spurge.     

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.