Predation and the evolutionary dynamics of species ranges

Gene flow that hampers local adaptation can constrain species distributions and slow invasions. Predation as an ecological factor mainly limits prey species ranges, but a richer array of possibilities arises once one accounts for how predation alters the interplay of gene flow and selection. We extend previous single-species theory on the interplay of demography, gene flow, and selection by investigating how predation modifies the coupled demographic-evolutionary dynamics of the range and habitat use of prey. We consider a model for two discrete patches and a complementary model for species along continuous environmental gradients. We show that predation can strongly influence the evolutionary stability of prey habitat specialization and range limits. Predators can permit prey to expand in habitat or geographical range or, conversely, cause range collapses. Transient increases in predation can induce shifts in prey ranges that persist even if the predator itself later becomes extinct. Whether a predator tightens or loosens evolutionary constraints on the invasion speed and ultimate size of a prey range depends on the predator effectiveness, its mobility relative to its prey, and the prey's intraspecific density dependence, as well as the magnitude of environmental heterogeneity. Our results potentially provide a novel explanation for lags and reversals in invasions.     

Theoretical models of species' borders: single species approaches

The range of potential mechanisms limiting species' distributions in space is nearly as varied and complex as the diversity of life itself. Yet viewed abstractly, a species' border is a geographic manifestation of a species' demographic responses to a spatially and temporally varying world. Population dynamic models provide insight into the different routes by which range limits can arise owing to gradients in demographic rates. In a metapopulation context, for example, range limits may be caused by gradients in extinction rates, colonization rates or habitat availability. We have consider invasion models in uniform and heterogeneous environments as a framework for understanding non-equilibrium range limits, and explore conditions under which invasions may cease to spread leaving behind a stationary range limit. We conclude that non-equilibrial range dynamics need further theoretical and empirical attention.     

The relation of density regulation to habitat specialization, evolution of a species' range, and the dynamics of biological invasions

Prior studies of the evolution of species' niches and ranges have identified the importance of within-population genetic variance, migration rate, and environmental heterogeneity in determining evolutionarily stable patterns of species' range and habitat use. Different combinations of these variables can produce either habitat specialists or generalists and cause either stable range limits or unbounded expansion. We examine the effect of density regulation on a species' range and habitat use within a landscape comprised of two discrete habitats and along continuous environmental gradients. Using the theta-logistic formulation, we demonstrate the following. (1) Spatially uniform density regulation generally weakens gene swamping and opposes habitat specialization and range limitation. (2) The form of density regulation should play an important role in determining whether the equilibrium species' range is limited by gene flow. (3) Even when no long-term limited-range equilibrium occurs, quasi-stable (or even contracting) range limits may be maintained for a long period during the initial phases of an invasion; the length of this period depends on the form of density regulation. (4) The steady state invasion speed in heterogeneous environments depends on the form of density regulation. Implications for the study of biological invasions are discussed, and directions for further exploration are sketched.     

The role of Allee effects in gypsy moth, <Emphasis Type="Italic">Lymantria dispar</Emphasis> (L.), invasions

Allee effects have been applied historically in efforts to understand the low-density population dynamics of rare and endangered species. Many biological invasions likewise experience the phenomenon of decreasing population growth rates at low population densities because most founding populations of introduced nonnative species occur at low densities. In range expansion of established species, the initial colonizers of habitat beyond the organism’s current range are usually at low density, and thus could be subject to Allee dynamics. There has been consistent empirical and theoretical evidence demonstrating, and in some cases quantifying, the role of Allee dynamics in the gypsy moth, Lymantria dispar (L.), invasion of North America. In this review, we examine the potential causes of the Allee effect in the gypsy moth and highlight the importance of mate-finding failure as a primary mechanism behind an Allee effect, while the degree to which generalist predators induce an Allee effect remains unclear. We then explore the role of Allee effects in the establishment and spread dynamics of the gypsy moth system, which conceptually could serve as a model system for understanding how Allee effects manifest themselves in the dynamics of biological invasions.     

Species' borders: a unifying theme in ecology

Biologists have long been fascinated by species' borders, and with good reason. Understanding the ecological and evolutionary dynamics of species' borders may prove to be the key that unlocks new understanding across a wide range of biological phenomena. After all, geographic range limits are a point of entry into understanding the ecological niche and threshold responses to environmental change. Elucidating patterns of gene flow to, and returning from, peripheral populations can provide important insights into the nature of adaptation, speciation and coevolution. Species' borders form natural laboratories for the study of the spatial structure of species interactions. Comparative studies from the center to the margin of species' ranges allow us to explore species' demographic responses along gradients of increasing environmental stress. Range dynamics further permit investigation into invasion dynamics and represent bellwethers for a changing climate. This set of papers explores ecological and evolutionary dynamics of species' borders from diverse empirical and theoretical perspectives.     

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.     

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.     

Initial conditions and their effect on invasion velocity across heterogeneous landscapes

Accurate, time dependent control options are required to halt biological invasions prior to equilibrium establishment, beyond which control efforts are often impractical. Although invasions have been successfully modeled using diffusion theory, diffusion models are typically confined to providing simple range expansion estimates. In this work, we use a Susceptible/Infected cellular automaton (CA) to simulate diffusion. The CA model is coupled with a network model to track the speed and direction of simulated invasions across heterogeneous landscapes, allowing for identification of locations for targeted control in both time and space. We evaluated the role of the location of initial establishment insofar as it affected the pattern and rate of spread and how these are influenced by patch attributes such as size. Our results show that the location of initial establishment can significantly affect the temporal dynamics of an invasion. Traditional network metrics such as degree and measures of topological distance were insufficient for predicting the direction and speed of the invasion. Our coupled models allow the dynamic tracking of invasions across fragmented landscapes for both theoretical and practical applications.     

Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect

Understanding dynamical complexity of alien species invasion is an important and timely issue as it is believed that identification of possible invasion scenarios may result in a more effective invasive species management and control. To address this issue, mathematical modeling is widely recognized as a convenient theoretical tool as it often helps to reveal generic tendencies in a situation when it would hardly be possible otherwise. However, although much attention has been paid to spatially continuous models of biological invasion, spatially discrete models have remained outside of the mainstream. Meanwhile, species habitat is often distinctly patchy or even fragmented, and then spatially discrete models seems to be the most appropriate modeling framework. In this paper, we consider invasion scenarios in a space- and time-discrete system described by a coupled map lattice. The alien species is assumed to be affected by the strong Allee effect and by a specialist predator. We first consider the stage of species introduction and obtain analytical conditions to distinguish between invasion success and invasion failure. We then focus on species spatial spread. By means of extensive computer simulations, we identify the main scenarios of species spread (e.g. patchy invasion and multiple traveling bands) and reveal the corresponding structure of the parameter space. Counter-intuitively, we have found that alien species can invade over the whole domain even in the case when its local persistence would not be possible.     

An evolutionary tolerance model explaining spatial patterns in species richness under environmental gradients and geometric constraints

In this study, we developed a simulation model based on the ecological and evolutionary dynamics of geographical ranges, to understand the role of species' environmental tolerances and the strength of the environmental gradient in determining spatial patterns in species richness. Using an one-dimensional space, we present the model and dissect its parameters. Also, we test the ability of the model to simulate richness in complex two-dimensional domains and to fit real patterns in species richness, using South American Tyrannidae as an example. We found that a mid-spatial peak in species richness arises spontaneously under conditions of high environmental tolerances and/or a weak environmental gradient, since this condition causes wide species' geographic ranges, which are constrained by domain's boundary and tend to overlap in the middle. Our model was also a good predictor of real patterns in species richness, especially under conditions of high environmental strength and small species' tolerance. We conclude that this kind of spatial simulation models based on species' physiological tolerance may be an important tool to understand the evolutionary dynamics of species' geographic ranges and in spatial patterns of species richness.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

Extraordinary sex ratios: cultural effects on ecological consequences

We model sex-structured population dynamics to analyze pairwise competition between groups differing both genetically and culturally. A sex-ratio allele is expressed in the heterogametic sex only, so that assumptions of Fisher's analysis do not apply. Sex-ratio evolution drives cultural evolution of a group-associated trait governing mortality in the homogametic sex. The two-sex dynamics under resource limitation induces a strong Allee effect that depends on both sex ratio and cultural trait values. We describe the resulting threshold, separating extinction from positive growth, as a function of female and male densities. When initial conditions avoid extinction due to the Allee effect, different sex ratios cannot coexist; in our model, greater female allocation always invades and excludes a lesser allocation. But the culturally transmitted trait interacts with the sex ratio to determine the ecological consequences of successful invasion. The invading female allocation may permit population persistence at self-regulated equilibrium. For this case, the resident culture may be excluded, or may coexist with the invader culture. That is, a single sex-ratio allele in females and a cultural dimorphism in male mortality can persist; a low-mortality resident trait is maintained by father-to-son cultural transmission. Otherwise, the successfully invading female allocation excludes the resident allele and culture and then drives the population to extinction via a shortage of males. Finally, we show that the results obtained under homogeneous mixing hold, with caveats, in a spatially explicit model with local mating and diffusive dispersal in both sexes.     

A theorem for the invasion triangle and its applicability for invasion biology

The triangle conceptual model is a construct that is foundational across several fields of the natural sciences including the study of diseases, invasive species, and fire. The invasion triangle incorporates the complex ecological and evolutionary interactions between the qualities of the abiotic environment, the invader, and the biotic interactions that describes or predicts the impacts of the invasive species. Although the triangle concept is widely used among fields, to date there has not been an analytical implementation of the model. Current modelling in invasion biology often only considers the effects of one or two factors on the outcomes of species introductions. A mathematical implementation of the triangle model will allow a more comprehensive consideration of the various ecological factors. Here, we provide the first mathematical theorem for an interpretation of the invasion triangle that allows for the consideration of time. This new analytical development of the triangle is flexible, and can be used to model the spatial and temporal population dynamics observed in invasions. We also describe the conditions under which invasion is maintained when factors change with opposing effects. In this interpretation, the lower limits for invasion are explicitly defined and each component can move independently. The complexity of the interactions between factors contributing to invasions is integrated into the single model, such as those suggested by major invasion hypotheses. We briefly describe how the theorem can be applied to account for various phenomena in range dynamics using rapid range expansion and the time lag in invasions as examples. Future work can explicitly define the interdependence among components to suit more specific questions.     

利用元胞自动机研究一类捕食食饵模型中的斑块扩散现象

利用概率元胞自动机模型对空间隐式的、食饵具Allee效应的一类捕食食饵模型进行模拟,发现随着相关参数的变化,种群的空间扩散前沿由连续的扩散波逐渐转变为一种相互隔离的斑块向外扩散,这种斑块扩散现象与以往的扩散模式有所不同。研究结果表明：(1)在斑块扩散的情况下,相关参数的微小变化会导致种群灭绝或者形成连续的扩散波,即斑块扩散发生在种群趋于灭绝和连续扩散之间;(2)当种群的空间扩散方式为斑块扩散时,种群的扩散速度会变慢,与其他扩散方式下的速度有着明显的区别。该研究结果对生物入侵控制和外来物种监测有重要的启发和指导作用。     

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.     

Spatially implicit approaches to understand the manipulation of mating success for insect invasion management

Recent work indicates that Allee effects (the positive relationship between population size and per capita growth rate) are critical in determining the successful establishment of invading species. Allee effects may create population thresholds, and failure to establish is likely if invading populations fall below these thresholds. There are many mechanisms that may contribute to Allee effects, but mate-location failure is a common cause in sexually reproducing insects. Consequently, mate-location failure represents a type of “weak link” that may be enhanced in order to achieve eradication of insect populations during the early stages of invasion. In this paper, spatially implicit models that account for mating behavior of both sexes are used to explore the enhancement of mate-location failure in pest eradication programs. Distinct from the previous studies, the Allee effect emerges from a mechanistic representation of mate-location failure in our model. Three specific eradication strategies, sterile insect release, mass-trapping, and mating disruption, are incorporated into the model and tested for their ability to depress population growth during the early stages of invasions. We conducted simulations with the model parameterized to represent two types of insects: Coleopteran-like insects which are long-lived and capable of multiple matings, but have low daily reproductive rates, and Lepidopteran-like insects which are capable of mating only once per generation, have an ephemeral reproductive stage, and have high reproductive rates. Simulations indicated that: (1) many insect pests are more likely to be eradicated than had been previously predicted by classic models which do not account for mate-finding difficulties, (2) for Lepidopteran-like insects, mass-trapping has the greatest potential for eradication among the three methods when a large number of traps can be installed, although mating-disruption will be the most effective if we can anticipate confusion or trail-masking mechanisms of disruption, and (3) populations of Coleopteran-like insects may be most effectively eradicated using the sterile male release method. Though more detailed models should be tailored for individual species, we expect that the spatially implicit approaches outlined in this paper can be widely adapted to study the efficiency of various eradication approaches in sparse conditions. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Overcoming Allee effects through evolutionary,genetic, and demographic rescue

Despite the amplified threats of extinction facing small founder populations, successful colonization sometimes occurs, bringing devastating ecological and economic consequences. One explanation may be rapid evolution, which can increase mean fitness in populations declining towards extinction, permitting persistence and subsequent expansion. Such evolutionary rescue may be particularly important, given Allee effects. When a population is introduced at low density, individuals often experience a reduction in one or more components of fitness due to novel selection pressures that arise from diminished intraspecific interactions and positive density dependence (i.e. component Allee effects). A population can avoid extinction if it can adapt and recover on its own (i.e. evolutionary rescue), or if additional immigration sustains the population (i.e. demographic rescue) or boosts its genetic variation that facilitates adaptation (i.e. genetic rescue). These various forms of rescue have often been invoked as possible mechanisms for specific invasions, but their relative importance to invasion is not generally understood. Within a spatially explicit modelling framework, we consider the relative impact of each type of rescue on the probability of successful colonization, when there is evolution of a multi-locus quantitative trait that influences the strength of component Allee effects. We demonstrate that when Allee effects are important, the effect of demographic rescue via recurrent immigration overall provides the greatest opportunity for success. While highlighting the role of evolution in the invasion process, we underscore the importance of the ecological context influencing the persistence of small founder populations.     

Allee effects in biological invasions

Understanding the dynamics of small populations is obviously important for declining or rare species but is also particularly important for invading species. The Allee effect, where fitness is reduced when conspecific density is low, can dramatically affect the dynamics of biological invasions. Here, we summarize the literature of Allee effects in biological invasions, revealing an extensive theory of the consequences of the Allee effect in invading species and some empirical support for the theory. Allee effects cause longer lag times, slower spread and decreased establishment likelihood of invasive species. Expected spatial ranges, distributions and patterns of species may be altered when an Allee effect is present. We examine how the theory can and has been used to detect Allee effects in invasive species and we discuss how the presence of an Allee effect and its successful or unsuccessful detection may affect management of invasives. The Allee effect has been shown to change optimal control decisions, costs of control and the estimation of the risk posed by potentially invasive species. Numerous ways in which the Allee effect can influence the efficacy of biological control are discussed.     

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.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.