Bistability and an Allee effect as emergent consequences of stage-specific predation

The Allee effect, a reduction of individual fitness at low population density that can lead to sudden and unannounced extinctions, has been shown to come about through a number of mechanisms, usually associated with group behavior or mate search. Recent papers show that it may arise through size-selective predation, without explicit assumptions relating individual fitness to population density. It arises from the shift that a predator induces in the population stage distribution of its prey. We study the parameter conditions that lead to such an emergent Allee effect. The emergent Allee effect occurs under fairly broad conditions. We show that stage-specific predation can also induce bistability between alternative states where both prey and predator are present. A perturbation analysis on the equilibria shows that all equilibria are highly robust to changes in predator density. Our work shows that when size-specific interactions are taken into account, bistabilities and catastrophic collapses are possible even in purely exploitative food webs, which has substantial implications for questions related to food web theory and conservation issues.     

A discrete-time host–parasitoid model with an Allee effect

We introduce a discrete-time host–parasitoid model with a strong Allee effect on the host. We adapt the Nicholson–Bailey model to have a positive density dependent factor due to the presence of an Allee effect, and a negative density dependence factor due to intraspecific competition. It is shown that there are two scenarios, the first with no interior fixed points and the second with one interior fixed point. In the first scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and an exclusion region in which the host survives and tends to its carrying capacity. In the second scenario, we show that either both host and parasitoid will go to extinction or there are two regions, an extinction region where both species go to extinction and a coexistence region where both species survive.     

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.     

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.     

Global dynamics and bifurcation analysis of a host–parasitoid model with strong Allee effect

In this paper, we study the global dynamics and bifurcations of a two-dimensional discrete time host–parasitoid model with strong Allee effect. The existence of fixed points and their stability are analysed in all allowed parametric region. The bifurcation analysis shows that the model can undergo fold bifurcation and Neimark–Sacker bifurcation. As the parameters vary in a small neighbourhood of the Neimark–Sacker bifurcation condition, the unique positive fixed point changes its stability and an invariant closed circle bifurcates from the positive fixed point. From the viewpoint of biology, the invariant closed curve corresponds to the periodic or quasi-periodic oscillations between host and parasitoid populations. Furthermore, it is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor. These theoretical results reveal the complex dynamics of the present model.     

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.     

An eco-epidemiological system with infected prey and predator subject to the weak Allee effect

In this article, we propose a general prey–predator model with disease in prey and predator subject to the weak Allee effects. We make the following assumptions: (i) infected prey competes for resources but does not contribute to reproduction; and (ii) in comparison to the consumption of the susceptible prey, consumption of infected prey would contribute less or negatively to the growth of predator. Based on these assumptions, we provide basic dynamic properties for the full model and corresponding submodels with and without the Allee effects. By comparing the disease free submodels (susceptible prey–predator model) with and without the Allee effects, we conclude that the Allee effects can create or destroy the interior attractors. This enables us to obtain the complete dynamics of the full model and conclude that the model has only one attractor (only susceptible prey survives or susceptible-infected coexist), or two attractors (bi-stability with only susceptible prey and susceptible prey–predator coexist or susceptible prey-infected prey coexists and susceptible prey–predator coexist). This model does not support the coexistence of susceptible-infected-predator, which is caused by the assumption that infected population contributes less or are harmful to the growth of predator in comparison to the consumption of susceptible prey.     

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.     

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.     

在杀虫剂作用下的一类具有Allee效应的天敌-害虫模型

对一类具有Allee效应的天敌-害虫模型作了理论分析,同时对在杀虫剂作用下的此系统又作了理论分析,比较了二者之间的区别,从而从理论上获知利用杀虫剂控制虫害的利弊．     

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261-269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.     

An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect

We consider a single-species model of population dynamics allowing for migrations and the Allee effect. Two types of migration are taken into account: one caused by environmental factors (e.g., a passive transport with the wind or water current) and the other associated with biological mechanisms. While the first type is apparently density-independent, the speed of migration in the second one can depend on the population density. Mathematically, this model consists of a non-linear partial differential equation of advection-diffusion-reaction type. Using an appropriate change of variables, we obtain an exact solution of the equation describing propagation of travelling population fronts. We show that, depending on parameter values and thus on the relative intensity of density-dependent and density-independent factors, the direction of the propagation can be different thus describing either species invasion or species retreat.     

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。     

捕食者具有厌食性反应且食饵具有Allee效应的捕食系统

在Dubis动力系统的基础上,建立了具有Allee效应的捕食系统模型。对系统的稳定性进行了分析,受Allee效应的影响,食饵种群可能因为种群大小处于临界点以下而趋于灭绝。通过对系统进行模拟,结果表明:不受Allee效应的影响,系统的演化属于一种理想化的情形系统到达P(平衡)点的时间较不受Allee效应影响时系统到达P点的时间短,不利于生物的进化,而在Allee效应的影响下,系统的演化将达到一个平衡状态。由此,说明Allee效应为濒临灭绝物种的管理提供了重要的理论依据,对管理部门的决策有参考指导作用。     

Allee effect and self-fertilization in hermaphrodites: reproductive assurance in demographically stable populations

The fact that selfing increases seed set (reproductive assurance) has often been put forward as an important selective force for the evolution of selfing. However, the role of reproductive assurance in hermaphroditic populations is far from being clear because of a lack of theoretical work. Here, I propose a theoretical model that analyzes self-fertilization in the presence of reproductive assurance. Because reproductive assurance directly influences the per capita growth rate, I developed an explicit demographic model for partial selfers in the presence of reproductive assurance, specifically when outcrossing is limited by the possibility of pollen transfer (Allee effect). Mating system parameters are derived as a function of the underlying demographical parameters. The functional link between population demography and mating system parameters (reproductive assurance, selfing rate) can be characterized. The demographic model permits the analysis of the evolution of self-fertilization in stable populations when reproductive assurance occurs. The model reveals some counterintuitive results such as the fact that increasing the fraction of selfed ovules can, in certain circumstances, increase the fraction of outcrossed ovules. Moreover, I demonstrate that reproductive assurance per se cannot account for the evolution of stable mixed selfing rates. Also, the model reveals that the extinction of outcrossing populations depends on small changes in population density (ecological perturbations), while the transition from outcrossing to selfing can, in certain cases, lead the population to extinction (evolutionary suicide). More generally, this paper highlights the fact that self-fertilization affects both the dynamics of individuals and the dynamics of selfing genes in hermaphroditic populations.     

Allee effects, mating systems and the extinction risk in populations with two sexes

The Allee effect is one of the population consequences of sexual reproduction that has received increased attention in recent years. Due to its impact on small population dynamics, it is commonly accepted that Allee effects should render populations more extinction prone. In particular, monogamous species are considered more susceptible to the Allee effect and hence, more extinction prone, than polygamous species. Although this hypothesis has received theoretical support, there is little empirical evidence. In this study, we investigate (1) how variation in tertiary sex ratio affects the presence and intensity of the Allee effect induced by mating system, as well as (2) how this effect contributes to extinction risk. In contrast with previous predictions, we show that all mating systems are likely to experience a strong Allee effect when the operational sex ratio (OSR) is balanced. This strong Allee effect does not imply being exceptionally extinction prone because it is associated with an OSR that result in a relatively small extinction risk. As a consequence, the impact of Allee effects on overall extinction risk is buffered. Moreover, the OSR of natural populations appears to be often male biased, thus making it unlikely that they will suffer from an Allee effect induced by mating system.     

Modelling the mating system of polar bears: a mechanistic approach to the Allee effect

Allee effects may render exploited animal populations extinction prone, but empirical data are often lacking to describe the circumstances leading to an Allee effect. Arbitrary assumptions regarding Allee effects could lead to erroneous management decisions so that predictive modelling approaches are needed that identify the circumstances leading to an Allee effect before such a scenario occurs. We present a predictive approach of Allee effects for polar bears where low population densities, an unpredictable habitat and harvest-depleted male populations result in infrequent mating encounters. We develop a mechanistic model for the polar bear mating system that predicts the proportion of fertilized females at the end of the mating season given population density and operational sex ratio. The model is parametrized using pairing data from Lancaster Sound, Canada, and describes the observed pairing dynamics well. Female mating success is shown to be a nonlinear function of the operational sex ratio, so that a sudden and rapid reproductive collapse could occur if males are severely depleted. The operational sex ratio where an Allee effect is expected is dependent on population density. We focus on the prediction of Allee effects in polar bears but our approach is also applicable to other species.     

新型降水分布数学模型研究及其应用

在分布式水文模型中,单元栅格内的降水输入是准确模拟各种水文过程的关键因素,寻求产生分布式降水数据的方法是水文模型研究的热点之一.在对国内外降水模型分析基础上,认为流域面上实际降水分布是天气系统降水与下垫面地形影响共同作用的结果,如果不受地形影响,天气系统降水的降水量等值线在平面上的分布近似为一组同心椭圆.根据这一原理,建立了一种能够模拟天气系统降水分布,并利用牛顿插值法对模拟结果进行地形影响修正的新型降水分布数学模型,提出了对降水中心位置及其中心降水量的模型模拟.利用黄土高原西川河流域实测资料对模型进行了检验,结果表明,该模型具有较高精度.由于模型概念简单明晰,且能指明降水中心位置及其中心降水量,因此在流域暴雨分析和洪水预报中具有一定价值.     

A stochastic gompertz model with hereditary effect

Summary In this paper we have studied a stochastic version of the Gompertz model for population growth of a single species after incorporating the aspect of heredity. Various statistical characteristics-the mean-value function, covariance-kernel, etc.-are evaluated for a delta-correlated process and their asymptotic values obtained. The effect of the hereditary kernel on the various statistics is discussed and it is found that it is to shift the distribution towards the origin.