A spatially explicit model for an Allee effect: why wolves recolonize so slowly in Greater Yellowstone

A reduced probability of finding mates at low densities is a frequently hypothesized mechanism for a component Allee effect. At low densities dispersers are less likely to find mates and establish new breeding units. However, many mathematical models for an Allee effect do not make a distinction between breeding group establishment and subsequent population growth. Our objective is to derive a spatially explicit mathematical model, where dispersers have a reduced probability of finding mates at low densities, and parameterize the model for wolf recolonization in the Greater Yellowstone Ecosystem (GYE). In this model, only the probability of establishing new breeding units is influenced by the reduced probability of finding mates at low densities. We analytically and numerically solve the model to determine the effect of a decreased probability in finding mates at low densities on population spread rate and density. Our results suggest that a reduced probability of finding mates at low densities may slow recolonization rate.     

Allee effect and self-fertilization in hermaphrodites: reproductive assurance in a structured metapopulation

Reproductive assurance through selfing during colonization events or when population densities are low has often been put forward as a mechanism selecting for the evolution of self-fertilization. Such arguments emphasize on the role of both local demography and metapopulation processes. We developed a model for the evolution of self-fertilization in a structured metapopulation in which local densities are not steady because of population growth. Reproduction by selfing is density-independent (reproductive assurance) but selfed seeds endure inbreeding depression, whereas reproduction by outcrossing is density-dependent (Allee effect). First, we derived an analytical criterion for metapopulation viability as a function of the selfing rate and metapopulation parameters. We show that outcrossers can develop a viable metapopulation when they produce a high amount of dispersal seeds that counterbalances their incapacity to found new populations from low densities. Second, the model shows there is a positive feedback between demography and outcrossing rates, leading to either complete outcrossing or selfing. Specifically, we illustrate that inbreeding depression can paradoxically favor the evolution of selfing because of its negative effect on density. Also, complete outcrossing can be selected despite pollen limitation, although it does not provide a full seed set. This model underlines the influence of the mating system both on demography and gene dynamics in a metapopulation context.     

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.     

集合种群的理论框架与应用研究进展

集合种群的研究是当今国际生态学的重要前沿与热点。随着全球范围的生境破坏和破碎化,集合种群的研究方法已成为数学生态学、理论生态学和保护生物学的重要手段。由于其迅速的发展,集合种群的概念与理论得到迅速扩展与丰富。为了能总观集合种群进展的全局并开展进一步的工作．首先对集合种群的已有概念、理论和模型做了全面的分析和总结;其次对集合种群的发展和概念进行了探讨,以集合种群模型的中心框架：Levins的斑块占据模型为基础,展开对其它原理、效应和机制的探讨;主要包括了Levins原理．即当生境遭到破坏时,空斑块比例在集合种群灭绝前保持不变,然后还分析了Allee效应(集合种群的Allee效应主要是由于建群困难和扩散损失造成的);第三,分析了援救效应：迁入个体可以降低斑块中现有局域种群的灭绝风险。援救效应会增强集合种群的生存力,使空斑块比例下降。第四,探讨了两竞争集合种群的共存机制,即竞争,侵占妥协,其共存机制为空间生境中物种共存提供了有力的理论解释。最后,对集合种群群落中的灭绝债务进行了讨论。并给出了2种最为主要的集合种群空间模拟方法。     

Postmetamorphic dispersal and recruitment of first breeders in a Bufo calamita metapopulation

The migratory capacity of metamorphs and their later integration into local breeding populations was studied in a Bufo calamita metapopulation inhabiting the northern Rhineland, Germany (1991–1994). A new tagging method allowed unequivocal distinction among metamorphs (n = 1,709), which originated either from the spring-breeding cohort in 1991 and 1992 or from the summer-breeding cohort in 1991. The life history of batch-marked individuals was followed until adulthood, focusing on postmetamorphic dispersal, choice of breeding locality and timing of adult reproduction. Almost all metamorphs left the patches used for breeding within a few weeks after emergence. They were found at about the same density in all favourable microhabitats in a radius of up to 2 km around their native patch. This demonstrates that even small toadlets (6–11 mm snout-to-vent length) migrate distances of several hundred metres within a short time and that they significantly help prevent local extinction in sink populations (rescue effect). Recaptures of reproductive adults originating from tagged metamorphs were limited to males belonging to the 1997 spring and summer cohorts. The absence of recaptured females was probably due to their greater vagility and later sexual maturation. First-breeding males (2 years old) without exception joined the breeding cohort from which they originated: early breeders originated from spawn of the spring cohort, mid-summer breeders from spawn of the summer cohort. This result is in accordance with the hypothesis that the integration of first breeders to a specific breeding cohort is genetically controlled in central European natterjacks. The exchange of individuals between different cohorts in the same patch is considerably less than the migratory exchange between the same breeding cohorts among local populations, as predicted from the allozyme data. Received: 28 October1996 / Accepted: 4 June 1997     

Hierarchical competition models with the Allee effect III: multispecies

A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.     

人类活动影响下具有Allee效应的非自治种群演化模式的研制及其应用——以丹顶鹤为例

提出“中国的丹顶鹤是否是一活着的灭绝物种(活死者) ?”这一重大科学问题。要回答这个问题,首先必须建立有关丹顶鹤种群演化与人类活动、栖息地斑块平均面积和斑块数关系的动力学模式,其次必须对丹顶鹤种群的大小进行动力学预测。前者涉及到所谓的“相互作用的标度理论”,后者则属于“物种多样性动力学预测”这一崭新的研究领域。首次应用标度理论,阐述了单个物种环境容量(K)与斑块数(P)的标度性质,并用实测的小三江平原的丹顶鹤资料进行了验证发现,K∝P0 .7。同时,在对logistic模式改进的基础上,引进人类活动累积效应及其作用的时间因素,首次提出了人类活动影响下具有Allee效应的非自治种群演化模式。并以丹顶鹤为例,模拟了其种群演化特征,预测了其灭绝时间。模拟结果发现:对于我国的珍稀物种丹顶鹤,其繁殖率的相对高低对物种灭绝的影响并不显著,但Allee效应对其物种灭绝的影响却是明显的,Allee效应越弱,物种灭绝时间越长。如果小三江平原湿地的生境质量得不到有效的恢复和提高,该区丹顶鹤将有可能会在330～4 2 8a后走向灭绝,即丹顶鹤的灭绝对现有栖息地毁坏的响应具有330～4 2 8a的时间滞后性。因此,认为丹顶鹤是一种典型的“活着的灭绝物种”,这一点必须引起政府、科学家和公众的高度重视。     

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.     

Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect

Invasion of an exotic species initiated by its local introduction is considered subject to predator-prey interactions and the Allee effect when the prey growth becomes negative for small values of the prey density. Mathematically, the system dynamics is described by two nonlinear diffusion-reaction equations in two spatial dimensions. Regimes of invasion are studied by means of extensive numerical simulations. We show that, in this system, along with well-known scenarios of species spread via propagation of continuous population fronts, there exists an essentially different invasion regime which we call a patchy invasion. In this regime, the species spreads over space via irregular motion and interaction of separate population patches without formation of any continuous front, the population density between the patches being nearly zero. We show that this type of the system dynamics corresponds to spatiotemporal chaos and calculate the dominant Lyapunov exponent. We then show that, surprisingly, in the regime of patchy invasion the spatially average prey density appears to be below the survival threshold. We also show that a variation of parameters can destroy this regime and either restore the usual invasion scenario via propagation of continuous fronts or brings the species to extinction; thus, the patchy spread can be qualified as the invasion at the edge of extinction. Finally, we discuss the implications of this phenomenon for invasive species management and control.     

The importance of census times in discrete-time growth-dispersal models

Dispersal has been the focus of spatial ecology for a few decades. What should be a proper theoretical framework for understanding and modelling of dispersal processes remains a controversial issue though. Integrodifference equations (IDE) model the spatial dynamics of a population with distinct growth and dispersal stages in their life cycle. Depending on the stage observed, the equations take on different forms, only one of which is usually studied in the literature. Here we reveal that while these different forms are mathematically equivalent, the biological conclusions drawn from the different forms may differ considerably. We provide a summary of similarities and differences and point out the greatest potential caveats when applying IDE.     

Random models of Menzerath-Altmann law in genomes

Recently, a random breakage model has been proposed to explain the negative correlation between mean chromosome length and chromosome number that is found in many groups of species and is consistent with Menzerath-Altmann law, a statistical law that defines the dependency between the mean size of the whole and the number of parts in quantitative linguistics. Here, the central assumption of the model, namely that genome size is independent from chromosome number is reviewed. This assumption is shown to be unrealistic from the perspective of chromosome structure and the statistical analysis of real genomes. A general class of random models, including that random breakage model, is analyzed. For any model within this class, a power law with an exponent of -1 is predicted for the expectation of the mean chromosome size as a function of chromosome length, a functional dependency that is not supported by real genomes. The random breakage and variants keeping genome size and chromosome number independent raise no serious objection to the relevance of correlations consistent with Menzerath-Altmann law across taxonomic groups and the possibility of a connection between human language and genomes through that law.     

Range size in mid-domain models of species diversity

Geographical patterns of species diversity have been examined using mid-domain null models, in which the ranges of individual species are simulated by randomly arranging them on a bounded one- or two-dimensional continent. These models have shown that structured patterns in the geographical distribution of biodiversity can arise even under a fully stochastic procedure. In particular, mid-domain models have demonstrated that the random generation of ranges of different sizes and locations can produce a gradient of species diversity similar to the one found in real assemblages, with a peak at the middle of a continent. A less explored feature of mid-domain models is the pattern of range-size frequency distribution. Numerical simulations have provided some insights about the geographic pattern of average range size, but no exploration of the shape of range-size frequency distributions has been carried out. Here I present analytical and numerical models that generate explicit predictions for patterns of range size under the assumptions of mid-domain models of species diversity. Some generalizations include: (1) Mid-domain models predict no geographic gradient of average range size; the mean range size of species occurring at any point on a continent is constant (0.5 of the extent of the continent in the one-dimensional model, 0.25 of the area of the continent in the two-dimensional case); (2) Variance in range size is lowest at the middle of a continent and highest near the corners of a square-shaped continent; (3) The range-size frequency distribution is highly right-skewed at any point of a continent, but the skewness is highest near the corners. Despite their alleged weaknesses, mid-domain models are adequate null models against which real-world patterns can be contrasted.     

A spatially explicit model of sex ratio evolution in response to sex-biased dispersal

Sex-biased dispersal occurs in all seed plants and many animal species. Theoretical models have shown that sex-biased dispersal can lead to evolutionarily stable biased sex ratios. Here, we use a spatially explicit chessboard model to simulate the evolution of sex ratio in response to sex-biased dispersal range and sex-biased dispersal rate. Two life cycles are represented in the model: one in which both sexes disperse before mating (DDM), the other in which males disperse before mating and mated females or zygotes disperse after mating (DMD). Model parameters include factors like dispersal rate, dispersal range, number of individuals per patch, and habitat heterogeneity.When dispersal range is sex biased, we find that, in a homogeneous environment, the sex ratio is generally biased towards the sex that disperses more widely (sex ratio range: 0.47–0.52). In a heterogeneous environment, the sex ratio is generally biased towards the more dispersive sex in good habitats, and towards the less dispersive sex in poor habitats (sex ratio range: 0–1). This is opposite to the effect of sex-biased dispersal rate, which favours the production of the more dispersive sex in poor habitats and the less dispersive sex in good habitats (sex ratio range: 0–1). To allow for a comparison with theoretical predictions, data concerning sex-biased dispersal and habitat-dependent sex ratios should thus incorporate information about the spatial scale of both dispersal and environmental heterogeneity.     

Digging through model complexity: using hierarchical models to uncover evolutionary processes in the wild

The growing interest for studying questions in the wild requires acknowledging that eco-evolutionary processes are complex, hierarchically structured and often partially observed or with measurement error. These issues have long been ignored in evolutionary biology, which might have led to flawed inference when addressing evolutionary questions. Hierarchical modelling (HM) has been proposed as a generic statistical framework to deal with complexity in ecological data and account for uncertainty. However, to date, HM has seldom been used to investigate evolutionary mechanisms possibly underlying observed patterns. Here, we contend the HM approach offers a relevant approach for the study of eco-evolutionary processes in the wild by confronting formal theories to empirical data through proper statistical inference. Studying eco-evolutionary processes requires considering the complete and often complex life histories of organisms. We show how this can be achieved by combining sequentially all life-history components and all available sources of information through HM. We demonstrate how eco-evolutionary processes may be poorly inferred or even missed without using the full potential of HM. As a case study, we use the Atlantic salmon and data on wild marked juveniles. We assess a reaction norm for migration and two potential trade-offs for survival. Overall, HM has a great potential to address evolutionary questions and investigate important processes that could not previously be assessed in laboratory or short time-scale studies.     

Invasion dynamics of the glassy-winged sharpshooter <Emphasis Type="Italic">Homalodisca vitripennis</Emphasis> (Germar) (Hemiptera: Cicadellidae) in French Polynesia

The glassy-winged sharpshooter, Homalodisca vitripennis (Germar) [formerly Homalodisca coagulata (Say)] (Hemiptera: Cicadellidae), has recently emerged as a serious invasive pest. From its natural range in the southeast USA and northeast Mexico, it invaded successively California (late 1980s), French Polynesia (1999), Hawaii (2004), and recently Easter Island (2005) inadvertently through the transportation of infested plants. In French Polynesia, H. vitripennis has reached impressive densities becoming an important pest threatening agriculture, native biodiversity, as well as being a major social nuisance. Since 1999, H. vitripennis spread rapidly from Tahiti to neighboring islands, colonizing most of the archipelagos of French Polynesia. In this paper, we present the results of surveys of H. vitripennis populations from 15 islands of French Polynesia and use these data to investigate the invasion dynamics and colonization processes of this pest in a tropical climate. We found H. vitripennis present in 10 islands with two new records confirmed. Our analyses suggest that: (1) H. vitripennis abundance is strongly associated with urbanization, with highest pest densities found in the most developed coastal areas of infested islands, (2) H. vitripennis may exhibit an Allee effect during the early phase of an invasion, and (3) the invasion dynamics of H. vitripennis conform to a stratified dispersal model marked by rapid long-distance human-mediated movement.     

Density dependence in resource exploitation: empirical test of Levins' metapopulation model

Levins' model of metapopulation dynamics is modified to incorporate variable degrees of density dependence in the per capita exploitation of resource patches. We demonstrate a simple means of testing for this density dependence in a sample of metapopulations, each at its equilibrium balance of local colonization to extinction. The fraction of habitable unoccupied patches equilibrates to a constant number under the null model of density independent colonization, and to a constant proportion under strong density dependence. We compare the null model to two density dependent alternatives, using data on exploitation of nest boxes by collared flycatchers Ficedula albicollis . The analysis shows how predicted trends in the equilibrium unoccupied fraction are similar for both spatial interference and net immigration. This needs to be recognized, since the null hypothesis of a constant unused resource applies also to the dynamics of consumable resources, where it is expressed in a constant stock of uneaten prey at the dynamic equilibrium of predators to prey.     

An explicit model for the inbreeding load in the evolutionary analysis of selfing

I present analytical predictions for the equilibrium inbreeding load expected in a population under mutation, selection, and a regular mating system for any population size and for any magnitude and recessivity of the deleterious effects. Using this prediction, I deduce the relative fitness of mutant alleles with small effect on selfing to explore the situations where selfing or outcrossing are expected to evolve. The results obtained are in agreement with previous literature, showing that natural selection is expected to lead to stable equilibria where populations show either complete outcrossing or complete selfing, and that selfing is promoted by large deleterious mutation rates. I find that the evolution of selfing is favored by a large recessivity of deleterious effects, while the magnitude of homozygous deleterious effects only becomes relevant in relatively small populations. This result contradicts the standard assumption that purging in large populations will only promote selfing when homozygous deleterious effects are large, and implies that previously published results obtained assuming lethal mutations in large populations can be extrapolated to nonlethal alleles of similar recessivity. This conclusion and the general approach used in this analysis can be useful in the study of the evolution of mating systems.     

The importance of explicit chain representation in protein folding models: an examination of Ising-like models

A class of models that represents a protein chain as a sequence of "folded" and "unfolded" residues has recently been used to correlate rates and mechanisms of protein folding with the protein native structure. In order to better understand the conditions under which these "Ising-like" models apply, we compare results from this model to those obtained from an off-lattice model which uses the same potential function. We find that Ising-like models by construction impose folding via a highly sequential nucleation-condensation mechanism, which in turn leads to more rugged energy landscapes, fewer "pathways" to the native state, and in the specific case examined here, the cold shock protein A from Escherichia coli, a qualitative difference in the most likely order of events in folding.