Allee effect makes possible patchy invasion in a predator–prey system

The dynamics of interacting ecological populations results from the interplay between various deterministic and stochastic factors and this is particularly the case for the phenomenon of biological invasion. Whereas the spread of invasive species via propagation of a population front was shown to appear as a result of deterministic processes, the spread via formation, interaction and movement of separate patches has been recently attributed to the influence of environmental stochasticity. An appropriate understanding of the comparative importance of deterministic and stochastic mechanisms is still lacking, however. In this paper, we show that the patchy invasion appears to be possible also in a fully deterministic predator–prey model as a result of the Allee effect.     

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.     

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.     

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.     

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.     

A diffusive SI model with Allee effect and application to FIV

A minimal reaction-diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist.     

Invasion triangle: an organizational framework for species invasion

Species invasion is a complex, multifactor process. To encapsulate this complexity into an intuitively appealing, simple, and straightforward manner, we present an organizational framework in the form of an invasion triangle. The invasion triangle is an adaptation of the disease triangle used by plant pathologists to help envision and evaluate interactions among a host, a pathogen, and an environment. Our modification of this framework for invasive species incorporates the major processes that result in invasion as the three sides of the triangle: (1) attributes of the potential invader; (2) biotic characteristics of a potentially invaded site; and (3) environmental conditions of the site. The invasion triangle also includes the impact of external influences on each side of the triangle, such as climate and land use change. This paper introduces the invasion triangle, discusses how accepted invasion hypotheses are integrated in this framework, describes how the invasion triangle can be used to focus research and management, and provides examples of application. The framework provided by the invasion triangle is easy to use by both researchers and managers and also applicable at any level of data intensity, from expert opinion to highly controlled experiments. The organizational framework provided by the invasion triangle is beneficial for understanding and predicting why species are invasive in specific environments, for identifying knowledge gaps, for facilitating communication, and for directing management in regard to invasive species.     

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.     

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。     

A model describing the effect of sex-reversed YY fish in an established wild population: The use of a Trojan Y chromosome to cause extinction of an introduced exotic species

A novel means of inducing extinction of an exotic fish population is proposed using a genetic approach to shift the ratio of male to females within a population. In the proposed strategy, sex-reversed fish containing two Y chromosomes are introduced into a normal fish population. These YY fish result in the production of a disproportionate number of male fish in subsequent generations. Mathematical modeling of the system following introduction of YY fish at a constant rate reveals that female fish decline in numbers over time, leading to eventual extinction of the population.     

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.     

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.     

Invasion of an intermediate predator: Fish population dynamics in a mathematical model of a trophic chain (As applied to Syamozero lake)

A mathematical model is presented for the dynamics of a spatially heterogeneous predator-prey population system; a prototype is the Syamozero lake fish community. We show that the invasion of an intermediate predator can evoke chaotic oscillations in the population densities. We also show that different dynamic regimes (stationary, nonchaotic oscillatory, and chaotic) can coexist. The “choice” of a particular regime depends on the initial invader density. Analysis of the model solutions shows that invasion of an alien species is successful only in the absence of competition between the juvenile invaders and the native species.     

Predicting the impacts of an introduced species from its invasion history: an empirical approach applied to zebra mussel invasions

1. Quantitative models of impact are lacking for the vast majority of known invasive species, particularly in aquatic ecosystems. Consequently, managers lack predictive tools to help them prioritise invasion threats and decide where they can most effectively allocate limited resources. Predictive tools would also enhance the accuracy of water quality assessments, so that impacts caused by an invader are not erroneously attributed to other anthropogenic stressors. 2. The invasion history of a species is a valuable guide for predicting the consequences of its introduction into a new environment. Regression analysis of data from multiple invaded sites can generate empirical models of impact, as is shown here for the zebra mussel Dreissena polymorpha. Dreissena's impacts on benthic invertebrate abundance and diversity follow predictable patterns that are robust across a range of habitat types and geographic regions. Similar empirical models could be developed for other invaders with a documented invasion history. 3. Because an invader's impact is correlated with its abundance, a surrogate model may be generated (when impact data are unavailable) by relating the invader's abundance to environmental variables. Such a model could help anticipate which habitats will be most affected by invasion. Lack of precision should not be a deterrent to developing predictive models where none exist. Crude predictions can be refined as additional data become available. Empirical modelling is a highly informative and inexpensive, but underused, approach in the management of aquatic invasive species.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

Sensitivity of plant functional types to climate change: classification tree analysis of a simulation model

Question: The majority of studies investigating the impact of climate change on local plant communities ignores changes in regional processes, such as immigration from the regional seed pool. Here we explore: (i) the potential impact of climate change on composition of the regional seed pool, (ii) the influence of changes in climate and in the regional seed pool on local community structure, and (iii) the combinations of life history traits, i.e. plant functional types (PFTs), that are most affected by environmental changes. Location: Fire‐prone, Mediterranean‐type shrublands in southwestern Australia. Methods: Spatially explicit simulation experiments were conducted at the population level under different rainfall and fire regime scenarios to determine the effect of environmental change on the regional seed pool for 38 PFTs. The effects of environmental and seed immigration changes on local community dynamics were then derived from community‐level experiments. Classification tree analyses were used to investigate PFT‐specific vulnerabilities to climate change. Results: The classification tree analyses revealed that responses of PFTs to climate change are determined by specific trait characteristics. PFT‐specific seed production and community patterns responded in a complex manner to climate change. For example, an increase in annual rainfall caused an increase in numbers of dispersed seeds for some PFTs, but decreased PFT diversity in the community. Conversely, a simulated decrease in rainfall reduced the number of dispersed seeds and diversity of PFTs. Conclusions: PFT interactions and regional processes must be considered when assessing how local community structure will be affected by environmental change.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

The mid-domain effect applied to elevational gradients: species richness of small mammals in Costa Rica

Aim The objective of this study was to comprehensively document and examine the alpha and gamma patterns of species richness in non-volant, small mammals (rodents, shrews and mouse opossums) along a tropical elevational gradient. These data were used to determine the support for existing hypotheses of species richness encompassing mid-domain null models, as well as climatic, and community overlap hypotheses. Location Field studies were conducted along a Caribbean slope of the Río Peñas Blancas watershed in the north-eastern region of Costa Rica between 750 and 1850 m at 10 sampling sites. Methods Species richness and abundances of small mammals were surveyed for four seasons including three temporal replicates at each of five elevational sites: late wet season (2000), early wet season (2001), and dry season (2002), and one spatial replicate at five different sites within the same elevations during the late wet season (2001). Species richness at elevations below 700 m was compiled from specimen records from 23 US national and international collections. Predictions of a null model based solely on geometric constraints were examined using a Monte Carlo simulation program, Mid-Domain Null. Results In 16,900 trap nights, 1561 individuals from 16 species were captured. Both alpha and gamma species richness peaked at mid-elevation between 1000 and 1300 m, with richness declining both at higher and lower elevations. Most of the empirical curves of species richness occur within 95% prediction curves of the mid-domain model, although deviations from the null model exist. Regression of the empirical richness on the null model predictions explained nearly half of the variation observed (r² = 0.45, P = 0.002). Main conclusions The geometric constraints of montane topography appear to influence the diversity pattern of small mammals, although climatic conditions including an intermediate rainfall and temperature regime, and distance from the persistent cloud cap also are correlated with the pattern of species richness. The predictions of productivity, and community overlap hypotheses are not supported with the empirical data.