具有Allee效应的合作进化

Allee效应对物种的续存是潜在的干扰因素,在很大程度上将增加种群局部甚至全局灭绝的可能性。对许多物种,尤其是濒临物种更容易受其影响。将Allee效应引入囚徒困境博弈模型,通过理论分析与数值模拟相结合的方法分析讨论了Allee效应对合作进化的影响。研究结果表明:在恶劣的环境条件下,Allee效应极易使物种灭绝,不利于合作进化;在相对优越的环境条件下(死亡率较低),Allee效应促进合作进化,且Allee效应强度越强,更有利于合作进化,不过种群的空间斑块占有率也会随着Allee效应强度的增强而降低,使物种最终灭绝。     

Bottom-up derivation of discrete-time population models with the Allee effect

This paper presents a framework in which various single-species discrete-time population models exhibiting the Allee effect are derived from first principles. Here, the Allee effect means a reduction in individual fitness at low population sizes. The derivation is based on the distribution of female and male individuals among discrete resource sites, in addition to competitive and cooperative interaction among individuals. These derivations show how the derived population models depend on the type and the intensity of competition, and the degree of clustering of individuals. Along with these models exhibiting the Allee effect, this paper also presents first-principles derivation of population models without the Allee effect which include a parameter relating to the intensity of competition.     

Modelling the effect of temperature on the range expansion of species by reaction-diffusion equations

The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.     

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.     

The Effect of Chemical Information on the Spatial Distribution of Fruit Flies: I Model Results

Animal aggregation is a general phenomenon in ecological systems. Aggregations are generally considered as an evolutionary advantageous state in which members derive the benefits of protection and mate choice, balanced by the costs of limiting resources and competition. In insects, chemical information conveyance plays an important role in finding conspecifics and forming aggregations. In this study, we describe a spatio-temporal simulation model designed to explore and quantify the effects of these infochemicals, i.e., food odors and an aggregation pheromone, on the spatial distribution of a fruit fly (Drosophila melanogaster) population, where the lower and upper limit of local population size are controlled by an Allee effect and competition. We found that during the spatial expansion and strong growth of the population, the use of infochemicals had a positive effect on population size. The positive effects of reduced mortality at low population numbers outweighed the negative effects of increased mortality due to competition. At low resource densities, attraction toward infochemicals also had a positive effect on population size during recolonization of an area after a local population crash, by decreasing the mortality due to the Allee effect. However, when the whole area was colonized and the population was large, the negative effects of competition on population size were larger than the positive effects of the reduction in mortality due to the Allee effect. The use of infochemicals thus has mainly positive effects on population size and population persistence when the population is small and during the colonization of an area. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users.     

Competition and the Effect of Spatial Resource Heterogeneity on Evolutionary Diversification

A model is presented to explore how the form of selection arising from competition for resources is affected by spatial resource heterogeneity. The model consists of a single species occupying two patches connected by migration, where the two patches can differ in the type of resources that they contain. The main goal is to determine the conditions under which competition for resources results in disruptive selection (i.e., selection favoring a polymorphism) since it is this form of selection that will give rise to the evolutionary diversification of resource exploitation strategies. In particular, comparing the conditions giving rise to disruptive selection when the two patches are identical to the conditions when they contain different resources reveals the effect of spatial resource heterogeneity. Results show that when the patches are identical, the conditions giving rise to disruptive selection are identical to those that give rise to character displacement in previous models. When the patches are different, the conditions giving rise to disruptive selection can be either more or less stringent depending upon demographic parameters such as the intrinsic rate of increase and the migration rate. Surprisingly, spatial resource heterogeneity can actually make forms of evolutionary diversification such as character displacement less likely. It is also found that results are dependent on how the resource exploitation strategies and the spatial resource heterogeneity affect the population dynamics. One robust conclusion however, is that spatial resource heterogeneity always has a disruptive effect when the migration rate between patches is low.     

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.     

Competitive Asymmetry Reduces Spatial Effects on Size-Structure Dynamics in Plant Populations

The growth of each individual in plant populations was simulatedby a spatial competition model for five density levels and fourdifferent spatial distribution patterns of individuals, varyingfrom highly clumped to regular. The simulation results wereanalysed using the diffusion model for evaluating the effectsof density and distribution pattern on the size-structure dynamicsin relation to the degree of competitive asymmetry. At low densities,changes in statistics of plant weight over time such as mean,coefficient of variation, skewness, and Box-Cox-transformedkurtosis differed greatly among spatial patterns, irrespectiveof the degree of competitive asymmetry. In completely symmetriccompetition, the spatial effect on size-structure dynamics remainedrelatively large irrespective of densities, although mean plantweight became similar among the spatial patterns with increasingdensity. However, the spatial effect diminished with increaseddensity in strongly asymmetric competition, when similar sizedistributions were realized irrespective of the spatial patterns.Therefore, it was concluded that: (1) irrespective of the degreeof competitive asymmetry, spatial pattern is important for size-structuredynamics at low densities; (2) spatial pattern is nearly immaterialunder strongly asymmetric competition at high densities; and(3) under crowded conditions, neighbourhood effects are muchmore apparent at the population level in less asymmetric competition.These processes and outcomes are linked to the forms of thefunctions of mean growth rate of individuals [G(t,x) function]and variance in growth rate [D(t,x) function]. These functionsare variable depending on the spatial pattern under symmetriccompetition, but are rather stable under strongly asymmetriccompetition at high densities irrespective of the spatial patterns.Therefore, size structure under strongly asymmetric competitioncan be regarded as a stable system, whereas that under symmetriccompetition is regarded as a variable system in relation tothe spatial pattern and process. From this, it was inferredthat: (1) the goodness-of-fit of spatial competition modelsfor crowded plant populations is higher in less asymmetric competition;and (2) higher species diversity in plant communities is associatedwith the lower degree of competitive asymmetry.Copyright 1994,1999 Academic Press Asymmetric competition, diffusion model, neighbourhood effect, size-structure stability, spatial competition model, spatial distribution pattern, species diversity, symmetric competition     

Global Production Increased by Spatial Heterogeneity in a Population Dynamics Model

Spatial and temporal heterogeneity are often described as important factors having a strong impact on biodiversity. The effect of heterogeneity is in most cases analyzed by the response of biotic interactions such as competition of predation. It may also modify intrinsic population properties such as growth rate. Most of the studies are theoretic since it is often difficult to manipulate spatial heterogeneity in practice. Despite the large number of studies dealing with this topics, it is still difficult to understand how the heterogeneity affects populations dynamics. On the basis of a very simple model, this paper aims to explicitly provide a simple mechanism which can explain why spatial heterogeneity may be a favorable factor for production. We consider a two patch model and a logistic growth is assumed on each patch. A general condition on the migration rates and the local subpopulation growth rates is provided under which the total carrying capacity is higher than the sum of the local carrying capacities, which is not intuitive. As we illustrate, this result is robust under stochastic perturbations.     

Scramble competitions can rescue endangered species subject to strong Allee effects

In this article, we study population dynamics of a general two-species discrete-time competition model where each species suffers from both strong Allee effects and scramble intra-specific competitions. We focus on how the combinations of the scramble intra-specific and inter-specific competition affect the extinction and coexistence of these two competing species where each species is subject to strong Allee effects. We derive sufficient conditions on the extinction, essential-like extinction and coexistence for such models. One of the most interesting findings is that scramble competitions can promote coexistence of these two species at their high densities. This is supported by the outcome of single species models with strong Allee effects. In addition, we apply theoretical results to a symmetric competition model with strong Allee effects induced by predator saturations where we give a completed study of its possible equilibria and attractors. Numerical simulations are performed to support our results.     

Algae-herbivore interactions with Allee effect and chemical defense

Macroalgae exhibit a variety of characteristics that provide a degree of protection from herbivores. One characteristic is the production of chemicals that are toxic to herbivores. The toxic effect of macroalgae on herbivorous reef fish is studied by means of a spatiotemporal model of population dynamics with a nonmonotonic toxin-determined functional response of herbivores. It is assumed that the growth rate of macroalgae is mediated by Allee effect. We see that under certain conditions the system is uniformly persistent. Conditions for local stability of the system is obtained with weak and strong Allee effects. We observe that in presence of Allee effect on macroalgae, the system exhibits complex dynamics including Hopf bifurcation and saddle-node bifurcation. The obtained results show that the spatiotemporal system does not exhibit diffusion-driven instability. Computer simulations have been carried out to illustrate different analytical results.     

Habitat destruction and the extinction debt revisited: The Allee effect

Habitat destruction, often caused by anthropogenic disturbance, can lead to the extinction of species at an unprecedented rate. It is important, therefore, to consider habitat destruction when assessing population viability. Another factor often ignored in population viability analysis, is the Allee effect that adds to the risk of populations already on the verge of extinction. Understanding the Allee effect on species dynamics and response to habitat destruction has intrinsic value in conservation prioritization. Here, the Allee effect was considered in a multi-species hierarchical competition model. Results showed that species persistence declines dramatically due to the Allee effect, and certain species become more susceptible to habitat destruction than others. Two extinction orders emerged under habitat destruction: either the best competitor becomes extinct first or the best colonizer first. The extinction debt and order, as well as the time lag between habitat destruction and species extinction, were found to be determined by species abundance and the intensity of the Allee effect.     

The evolution of phenotypic traits in a predator-prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.     

Pushed to the edge: Spatial sorting can slow down invasions

Our ability to understand population spread dynamics is complicated by rapid evolution, which renders simple ecological models insufficient. If dispersal ability evolves, more highly dispersive individuals may arrive at the population edge than less dispersive individuals (spatial sorting), accelerating spread. If individuals at the low-density population edge benefit (escape competition), high dispersers have a selective advantage (spatial selection). These two processes are often described as forming a positive feedback loop; they reinforce each other, leading to faster spread. Although spatial sorting is close to universal, this form of spatial selection is not: low densities can be detrimental for organisms with Allee effects. Here, we present two conceptual models to explore the feedback loops that form between spatial sorting and spatial selection. We show that the presence of an Allee effect can reverse the positive feedback loop between spatial sorting and spatial selection, creating a negative feedback loop that slows population spread.     

Allee effect,spatial structure and species coexistence

Both positive and negative interactions among species are common in communities. Until recently, attention has focused on negative interactions such as competition. However, the importance of positive interactions such as the Allee effect has recently been recognized. We construct a single-patch model that incorporates both an Allee effect and competition between two species. A species that experiences an Allee effect cannot establish in a patch which is already occupied by a competitor unless its density is over a critical value. This effect, when translated into a metapopulation, makes migrants of a species unable to colonize patches where another species has established. This interaction between the Allee effect and inter-specific competition creates and stabilizes spatial segregation of species. Therefore, under circumstances in which competition would preclude local coexistence, the presence of an Allee effect can allow coexistence at a metapopulation scale. Furthermore, we found that a species can resist displacement if stronger competitors experience an Allee effect.     

Pollen limitation or mate search need not induce an Allee effect

When a process modelling the availability of gametes is included explicitly in population models a critical depensation or Allee effect usually results. Non-spatial models cannot describe clumping and so small populations must be assumed very diffuse. Consequently individuals in small populations experience low contact rates and so reproduction is limited. In Nature invasions into new territory are unlikely to be as diffuse as those described by non-spatial models. We develop pair approximations to a probabilistic cellular automata model with independent pollination and seed setting processes (equivalently mate search and reproduction processes). Each process can be either global (population-wide) or local (within a small neighbourhood) or a mixture of the two. When either process is global the resulting model recaptures the Allee effect found in non-spatial models. However, if both processes are at least partially local we obtain a model in which Allee effects can be avoided altogether if individuals are suitably strong pollinators and colonisers. The Allee effect disappears because small populations are dramatically more clumped when colonisation is local and less wasteful of pollen when pollination is local.     

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.     

Impact of Allee effect on an eco-epidemiological system

In this paper, we propose a general ratio-dependent prey-predator model with disease in predator subject to the strong Allee effect in prey. We obtain the complete dynamics of both models: (a) full model with Allee effect; (b) full model without Allee effect. Model (a) may have more than one interior equilibrium point, but model (b) has only one interior equilibrium point. Numerical results reveal that the coexistence of all the populations at the endemic state is possible for both the models. But for the model with Allee effect, the coexistence can be destroyed by an increased supply of alternative food for the predators. It can also be proved that for the full model with Allee effect, the disease can be suppressed under certain parametric conditions. Also by comparing models (a) and (b), we conclude that Allee effect can create or destroy the interior attractor. Finally, we have studied the disease free-submodel (prey and susceptible predator model) with and without Allee effect. The comparative study between these two submodels leads to the following conclusions: 1) In the presence of Allee effect, the number of interior equilibrium points can change from zero to two whereas the submodel without Allee effect has unique interior equilibrium point; 2) Both with and without Allee effect, initial conditions play an important role on the survival and extinction of prey as well as its corresponding predator; 3) In the presence of Allee effect, bi-stability occurs with stable or periodic coexistence of prey and susceptible predator and the extinction of prey and susceptible predator; 4) Allee effect can generate or destroy the interior equilibrium points.     

Linking the allee effect, sexual reproduction, and temperature-dependent sex determination via spatial dynamics

We develop a spatially explicit, two-sex, individual-based model (IBM) and a derived spatially homogeneous model (SHM) to describe the Allee effect due to scarcity of mating possibilities at low population sizes or densities. The SHM, based on coupled difference equations, represents the first spatially homogeneous approach to this phenomenon, which differentiates between sexes and relies only on measurable population parameters. The IBM reinforces the findings of the SHM by adopting more realistic mate search strategies of diffusive movement and active search. Both models are characterized by a hyperbolic-shaped extinction boundary in the male-female state space, which contrasts with a linear boundary in one-dimensional models of the Allee effect. We examine how the position of the extinction boundary depends on population demography (primary sex ratio, reproduction and mortality probabilities) and adopted mate search strategies. The investigation of different phases in the IBM dynamics emphasizes the differences between local and global densities and shows the importance of scale when assessing the Allee effect. To demonstrate the potential application of our models, we combine the SHM and available data to predict the impact of environmental temperature changes on two turtle species with temperature-dependent sex determination.     

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.