ALLEE EFFECTS: POPULATION GROWTH,CRITICAL DENSITY,AND THE CHANCE OF EXTINCTION

This paper develops mathematical models to describe the growth, critical density, and extinction probability in sparse populations experiencing Allee effects. An Allee effect (or depensation) is a situation at low population densities where the per-individual growth rate is an increasing function of population density. A potentially important mechanism causing Allee effects is a shortage of mating encounters in sparse populations. Stochastic models are proposed for predicting the probability of encounter or the frequency of encounter as a function of population density. A negative exponential function is derived as such an encounter function under very general biological assumptions, including random, regular, or aggregated spatial patterns. A rectangular hyperbola function, heretofore used in ecology as the functional response of predator feeding rate to prey density, arises from the negative exponential function when encounter probabilities are assumed heterogeneous among individuals. These encounter functions produce Allee effects when incorporated into population growth models as birth rates. Three types of population models with encounter-limited birth rates are compared: (1) deterministic differential equations, (2) stochastic discrete birth-death processes, and (3) stochastic continuous diffusion processes. The phenomenon of a critical density, a major consequence of Allee effects, manifests itself differently in the different types of models. The critical density is a lower unstable equilibrium in the deterministic differential equation models. For the stochastic discrete birth-death processes considered here, the critical density is an inflection point in the probability of extinction plotted as a function of initial population density. In the continuous diffusion processes, the critical density becomes a local minimum (antimode) in the stationary probability distribution for population density. For both types of stochastic models, a critical density appears as an inflection point in the probability of attaining a small population density (extinction) before attaining a large one. Multiplicative (“environmental”) stochastic noise amplifies Allee effects. Harvesting also amplifies those effects. Though Allee effects are difficult to detect or measure in natural populations, their presence would seriously impact exploitation, management, and preservation of biological resources.     

THE EFFECTS OF REDUCING MATING LIKELIHOOD ON POPULATION VIABILITY

Abstract The success a species may have invading a patch previously unoccupied is of considerable interest for pest managers and conservation ecologists. The purpose here is to present a mechanistic approach to analyze reproductive Allee effects appearing through the failure in the process of fertilization in a two‐sex population and observe how the survival in an invaded patch is affected. This is in contrast to the usually employed stochastic models with a deterministic skeleton that describe the presence of Allee effects. A Poisson–Ricker model, which includes stochastic demography and sex determination with females classified as successfully fertilized or not fertilized, is used. Numerical approximations to the probabilities of extinction and the mean time to extinction are presented, for fixed parameter values, suggesting how stochasticity in the mating process combined with random fluctuations in the male and female densities, at each generation, contribute to the risk of extinction of a population which started an invasion at a low density.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

Global qualitative analysis of a discrete host‐parasitoid model with refuge and strong Allee effects

In this paper, we discuss the qualitative behavior of a discrete host‐parasitoid model with the host subject to refuge and strong Allee effects. More precisely, we study the local and global asymptotic stability, stable manifolds and unstable manifolds of boundary equilibrium points, existence and unique positive equilibrium point, local and global behavior of the positive equilibrium point, and the uniform persistence for the model with the host subject to the refuge or both refuge and strong Allee effects. It is also proved that the model undergoes a transcritical bifurcation in a small neighborhood of the boundary equilibrium point. Some numerical simulations are given to support our theoretical results. We can obtain that the addition of the refuge may make the parasitoids go extinct while the hosts survive or may stabilize the host‐parasitoid interaction; the addition of both refuge and strong Allee effects has either a negative or positive impact on the coexistence of both populations.     

COMPONENT ALLEE EFFECTS AND STAGE‐SPECIFIC PREDATION: A BRIEF INSIGHT INTO CONSERVATION AND BIOLOGICAL CONTROL ISSUES

In this paper, we propose a two‐stage structured population model subject to component Allee effects in fecundity and maturation, and with two disturbances (predation only and harvest and predation) acting on both stages. It is shown that this combination leads to a demographic Allee effect—a characteristic that could be exploited in pest biological control, but on the other hand, it represents a bane in conservation biology. The analysis is performed for disturbances with functional responses type 2 and 3, and the models show that they yield qualitatively similar results. This characteristic is discussed from the species conservation and biological control point of view, together with possible extensions of this work.     

Decision analysis with indeterminate or incoherent probabilities

This paper presents a new method of modeling indeterminate and incoherent probability judgments in decision analysis problems. The decision maker's degree of beliefs in the occurrence of an event is represented by a unimodal (in fact, concave) function on the unit interval, whose parameters are elicited in terms of lower and upper probabilities with attached confidence weights. This is shown to provide a unified framework for performing sensitivity analysis, reconciling incoherence, and combining expert judgments.This material is based upon work supported by the National Science Foundation under grant SES-85-11675, and by the Business Associates Fund at the Fuqua School of Business.     

Weak Allee effects and species coexistence

In this article, we study the population dynamics of a two-species discrete-time competition model where each species suffers from either predator saturation induced Allee effects and/or mate limitation induced Allee effects. We focus on the following two possible outcomes of the competition: 1. one species goes to extinction; 2. the system is permanent. Our results indicate that, even if one species’ intra-specific competition is less than its inter-specific competition, weak Allee effects induced by predation saturation can promote coexistence of the two competing species. This is supported by the outcome of two-species competition models without Allee effects. Also, we discuss our results and future work on multiple attractors in competition models with Allee effects.     

INFLUENCES OF AVAILABILITY ON PARAMETER ESTIMATES FROM SITE OCCUPANCY MODELS WITH APPLICATION TO SUBMERSED AQUATIC VEGETATION

Site occupancy models are commonly used by ecologists to estimate the probabilities of species site occupancy and of species detection. This study addresses the influence on site occupancy and detection estimates of variation in species availability among surveys within sites. Such variation in availability may result from temporary emigration, nonavailability of the species for detection, and sampling sites spatially when species presence is not uniform within sites. We demonstrate, using Monte Carlo simulations and aquatic vegetation data, that variation in availability and heterogeneity in the probability of availability may yield biases in the expected values of the site occupancy and detection estimates that have traditionally been associated with low‐detection probabilities and heterogeneity in those probabilities. These findings confirm that the effects of availability may be important for ecologists and managers, and that where such effects are expected, modification of sampling designs and/or analytical methods should be considered. Failure to limit the effects of availability may preclude reliable estimation of the probability of site occupancy.     

MODELING THE EFFECT OF RARITY VALUE ON THE EXPLOITATION OF A WILDLIFE SPECIES SUBJECTED TO THE ALLEE EFFECT

The overexploitation of wildlife species is a serious problem in the field of biodiversity conservation. The species subjected to natural Allee effects are even more threatened by exploitation. Moreover, for many wildlife species, their rarity can fuel their exploitation by making them disproportionately desirable and consequently increasing their market price. In this paper, a mathematical model is proposed and analyzed to study how the value that consumers place on rarity can threaten the survival of a species subjected to natural Allee effects. It is assumed that the value of a species increases as its density declines. The analysis of model shows that the increase in the consumers' response to rarity can drive the system to admit Hopf‐bifurcation and heteroclinic bifurcation. The occurrence of the heteroclinic cycle indicates that the increase in consumers' response to rarity can cause the extinction of the species. It is found that an increase in the Allee threshold causes a decrease in the threshold value of consumers' response below which extinction is inevitable.     

Scenario-based portfolio selection of investment projects with incomplete probability and utility information

In the selection of investment projects, it is important to account for exogenous uncertainties (such as macroeconomic developments) which may impact the performance of projects. These uncertainties can be addressed by examining how the projects perform across several scenarios; but it may be difficult to assign well-founded probabilities to such scenarios, or to characterize the decision makers’ risk preferences through a uniquely defined utility function. Motivated by these considerations, we develop a portfolio selection framework which (i) uses set inclusion to capture incomplete information about scenario probabilities and utility functions, (ii) identifies all the non-dominated project portfolios in view of this information, and (iii) offers decision support for rejection and selection of projects. The proposed framework enables interactive decision support processes where the implications of additional probability and utility information or further risk constraints are shown in terms of corresponding decision recommendations.     

Mathematical models of restoration and control of a single species with Allee effect

According to the initial density of a single species with Allee effect and corresponding management strategy, three kinds of mathematical models are presented to describe the evolutionary process of the species by impulsive differential equations. When the initial density of the species is larger than economic injury level (EIL) (or economical threshold, ET), impulsive harvest control is considered in a finite time to decrease the population of the species. The feasibility of the impulsive harvest control in a finite time is given by the existence of solution of the model with initial and boundary value problem. When the initial density of the species is less than EIL (or ET), the model with state feedback control is formulated according to the state of the species. The existence and stability of periodic solution of the model with state feedback control are discussed. When the initial density of the species is less than the Allee threshold and the species tends to extinction, the model with impulsive release at fixed moments is presented to study the restoration of the species. The conditions for the feasibility of periodic restoration of the species are given. Finally, some discussions are given.     

基于Allee效应诱导的Filippov生态系统的动力学行为研究

该文建立了一类由Allee效应诱导的非光滑Filippov切换系统.运用Filippov系统的定性分析方法,从理论上研究了系统的滑动区域、滑动模态和各类平衡点的存在性.同时用数值方法研究了系统的滑动模态分支、边界焦点分支及全局动力学行为.研究发现:Allee效应的强度可使种群的动态不稳定,不利于濒危生物种群的管理.     

Dynamics of a single species evolutionary model with Allee effects

We investigate the evolutionary outcomes of a single species population subject to Allee effects within the framework of a continuous strategy evolutionary game theory (EGT) model. Our model assumes a single trait creates a phenotypic trade-off between carrying capacity (i.e., competition) and predator evasion ability following a Gaussian distribution. This assumption contributes to one of our interesting findings that evolution prevents extinction even when population exhibits strong Allee effects. However, the extinction equilibrium can be an ESS under some special distributions of anti-predation phenotypes. The ratio of variation in competition and anti-predation phenotypes plays an important role in determining global dynamics of our EGT model: (a) evolution may suppress strong Allee effects for large values of this ratio; (b) evolution may preserve strong Allee effects for small values of this ratio by generating a low density evolutionary stable strategy (ESS) equilibrium which can serve as a potential Allee threshold; and (c) intermediate values of this ratio can result in multiple ESS equilibria.     

The impact of Allee effect on a predator–prey system with Holling type II functional response

In this paper, the Allee effect is incorporated into a predator–prey model with Holling type II functional response. Compared with the predator–prey model without Allee effect, we find that the Allee effect of prey species increases the extinction risk of both predators and prey. When the handling time of predators is relatively short and the Allee effect of prey species becomes strong, both predators and prey may become extinct. Moreover, it is shown that the model with Allee effect undergoes the Hopf bifurcation and heteroclinic bifurcation. The Allee effect of prey species can lead to unstable periodical oscillation. It is also found that the positive equilibrium of the model could change from stable to unstable, and then to stable when the strength of Allee effect or the handling time of predators increases continuously from zero, that is, the model admits stability switches as a parameter changes. When the Allee effect of prey species becomes strong, longer handling time of predators may stabilize the coexistent steady state.     

A Better Confidence Interval for Kappa (κ) on Measuring Agreement between Two Raters with Binary Outcomes

Abstract

Although the kappa statistic is widely used in measuring interrater agreement, it is known that the standard confidence interval estimation behaves poorly in small samples and for nonzero kappas. Efforts have been made to improve the estimation through transformation and Edgeworth expansion (Flack 1987). The results remain unsatisfactory when kappa is far from 0, however, even with the sample size as large as 100. In this article we reparameterize the kappa statistic to reveal its relationship with the marginal probability of agreement. The reparameterization not only gives a more meaningful interpretation of kappa but also clearly demonstrates that the range of kappa depends on the marginal probabilities. Various two- and three-dimensional plots are shown to illustrate the relationship among these parameters. The immediate application is to construct a new confidence interval based on the profile variance and reparameterization. Extensive simulation studies show that the new confidence interval performs very well in almost all parameter settings even when other methods fail.     

How does innovation''''s tail risk determine marginal tail risk of a stationary financial time series?

We discuss the relationship between the marginal tail risk probability and theinnovation's tail risk probability for some stationary financial time series models. We firstgive the main results on the tail behavior of a class of infinite weighted sums of randomvariables with heavy-tailed probabilities. And then, the main results are applied to threeimportant types of time series models; infinite order moving averages, the simple bilineartime series and the solutions of stochastic difference equations. The explicit formulasare given to describe how the marginal tail probabilities come from the innovation's tailprobabilities for these time series. Our results can be applied to the tail estimation of timeseries and are useful for risk analysis in finance.     

Permanence of delay competitive systems with weak Allee effects

This study discusses a multispecies delay competitive system with weak Allee effects. In the situation where the model is a single species, the weak Allee effect represents a biological mechanism in which an increase in population is beneficial for low densities, but detrimental for high densities. In other words, the per-capita growth rate of each species is formulated by a sign-changing function of population density. In this paper, an existence theorem of positive equilibrium is established using the Brouwer degree theory. For cases without intraspecific delays, it is shown that the system has the property of permanence. Furthermore, a sufficient condition for a positive equilibrium to be globally attractive is obtained by means of the Lyapunov method.     

Conditional measures and operators

A detailed study of the structure of conditional expectations and conditional probability measures is presented. Some characterizations of conditional expectations as a subclass of projection operators on Banach function spaces, and similarly conditional probabilities as a subclass of vector valued measures on such spaces are included. As applications of these results, a representation of Reynolds operators and related unified formulation of ergodic-martingale theorems are given.     

An extended discrete Ricker population model with Allee effects

Based on the classical discrete Ricker population model, we incorporate Allee effects by assuming rectangular hyperbola, or Holling-II type functional form, for the birth or growth function and formulate an extended Ricker model. We explore the dynamics features of the extended Ricker model. We obtain domains of attraction for the trivial fixed point. We determine conditions for the existence and stability of positive fixed points and find regions where there exist no positive fixed points, two positive fixed points one of which is stable and two positive fixed points both of which are unstable. We demonstrate that the model exhibits period-doubling bifurcations and investigate the existence and stability of the cycles. We also confirm that Allee effects have stabilization effects, by different measures, through numerical simulations.     

Variable Selection in General Frailty Models Using Penalized H-Likelihood

Variable selection methods using a penalized likelihood have been widely studied in various statistical models. However, in semiparametric frailty models, these methods have been relatively less studied because the marginal likelihood function involves analytically intractable integrals, particularly when modeling multicomponent or correlated frailties. In this article, we propose a simple but unified procedure via a penalized h-likelihood (HL) for variable selection of fixed effects in a general class of semiparametric frailty models, in which random effects may be shared, nested, or correlated. We consider three penalty functions (least absolute shrinkage and selection operator [LASSO], smoothly clipped absolute deviation [SCAD], and HL) in our variable selection procedure. We show that the proposed method can be easily implemented via a slight modification to existing HL estimation approaches. Simulation studies also show that the procedure using the SCAD or HL penalty performs well. The usefulness of the new method is illustrated using three practical datasets too. Supplementary materials for the article are available online.