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.     

Integrodifference equations in patchy landscapes

We analyze integrodifference equations (IDEs) in patchy landscapes. Movement is described by a dispersal kernel that arises from a random walk model with patch dependent diffusion, settling, and mortality rates, and it incorporates individual behavior at an interface between two patch types. Growth follows a simple Beverton–Holt growth or linear decay. We obtain explicit formulae for the critical domain-size problem, and we illustrate how different individual behavior at the boundary between two patch types affects this quantity. We also study persistence conditions on an infinite, periodic, patchy landscape. We observe that if the population can persist on the landscape, the spatial profile of the invasion evolves into a discontinuous traveling periodic wave that moves with constant speed. Assuming linear determinacy, we calculate the dispersion relation and illustrate how movement behavior affects invasion speed. Numerical simulations justify our approach by showing a close correspondence between the spread rate obtained from the dispersion relation and from numerical simulations.     

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.     

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.     

Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect

Species establishment in a model system in a homogeneous environment can be dependent not only on the parameter setting, but also on the initial conditions of the system. For instance, predator invasion into an established prey population can fail and lead to system collapse, an event referred to as overexploitation. This phenomenon occurs in models with bistability properties, such as strong Allee effects. The Allee effect then prevents easy re-establishment of the prey species. In this paper, we deal with the bifurcation analyses of two previously published predator-prey models with strong Allee effects. We expand the analyses to include not only local, but also global bifurcations. We show the existence of a point-to-point heteroclinic cycle in these models, and discuss numerical techniques for continuation in parameter space. The continuation of such a cycle in two-parameter space forms the boundary of a region in parameter space where the system collapses after predator invasion, i.e. where overexploitation occurs. We argue that the detection and continuation of global bifurcations in these models are of vital importance for the understanding of the model dynamics.     

Neutral hybridization can overcome a strong Allee effect by improving pollination quality

Small populations of plant species can be susceptible to demographic Allee effects mainly due to pollen limitation. Although sympatry with a common, co-flowering species may somewhat alleviate the problem of pollinator visitation (pollination quantity), the interspecific pollen transfer, IPT, (pollination quality) may remain a barrier to reproduction in small populations such as new introductions. However, if the two species are crosscompatible, our hypothesis is that neutral hybridization can help the small founding population overcome the Allee effect by improving the quality of pollination. We tested this hypothesis by using a novel modelling approach based on the theory of kinetic reactions wherein pollinators act as enzymes to catalyse the reaction between the two substrates: pollen and unselfed ovule. Using a single locus, two-allele genetic model, we developed a generic model that allows for hybridization between the invading and the native genotypes. Analysing the stability properties of the trivial equilibria in hybridization model as compared with the single genotype invasion model, we found that hybridization can either remove or reduce the Allee effect by making an otherwise stable trivial equilibrium unstable. Our study suggests that hybridization can be neutral but still be the key driver of a successful invasion by mediating pollen limitation. Conservation programmes should therefore account for this cryptic role that hybridization could play in plant invasions.     

Multiparametric bifurcations of an epidemiological model with strong Allee effect

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.     

Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest-neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations.     

Population consequences of movement decisions in a patchy landscape

Complex, human‐dominated landscapes provide unique challenges to animals. In landscapes fragmented by human activity, species whose home ranges ordinarily consist of continuous habitat in pristine environments may be forced to forage among multiple smaller habitat patches embedded in an inhospitable environment. Furthermore, foragers often must decide whether to traverse a heterogeneous suite of landscape elements that differ in risk of predation or energetic costs. We modeled population consequences of foraging decisions for animals occupying patches embedded in a heterogeneous landscape. In our simulations, animals were allowed to use three different rules for moving between patches: a) optimal selection resulting from always choosing the least‐cost path; b) random selection of a movement path; and c) probabilistic selection in which path choice was proportional to an animal's probability of survival while traversing the path. The resulting distribution of the population throughout the landscape was dependent on the movement rule used. Least‐cost movement rules (a) produced landscapes that contained the highest average density of consumers per patch. However, optimal movement resulted in an all‐or‐none pattern of occupancy and a coupling of occupied patches into pairs that effectively reduced the population to a set of sub‐populations. Random and probabilistic rules, (b and c), in relatively safe landscapes produced similar average densities and 100% occupancy of patches. However, as the level of risk associated with travel between patches increased, random movement resulted in an all‐or‐none occupancy pattern while occupied patches in probabilistic populations went extinct independently of the other patches. Our results demonstrate strong effects of inter‐patch heterogeneity and movement decisions on population dynamics, and suggest that models investigating the persistence of species in complex landscapes should take into account the effects of the intervening landscape on behavioral decisions affecting animal movements between patches.     

利用元胞自动机研究一类捕食食饵模型中的斑块扩散现象

利用概率元胞自动机模型对空间隐式的、食饵具Allee效应的一类捕食食饵模型进行模拟,发现随着相关参数的变化,种群的空间扩散前沿由连续的扩散波逐渐转变为一种相互隔离的斑块向外扩散,这种斑块扩散现象与以往的扩散模式有所不同。研究结果表明：(1)在斑块扩散的情况下,相关参数的微小变化会导致种群灭绝或者形成连续的扩散波,即斑块扩散发生在种群趋于灭绝和连续扩散之间;(2)当种群的空间扩散方式为斑块扩散时,种群的扩散速度会变慢,与其他扩散方式下的速度有着明显的区别。该研究结果对生物入侵控制和外来物种监测有重要的启发和指导作用。     

Population structure of a monophagous moth in a patchy landscape

1. The population structure of a monophagous noctuid moth, Abrostola asclepiadis , living on a patchily distributed perennial herb, Vincetoxicum hirundinaria is described. The study took place over 5 years at a landscape scale (about 12 km²).
2. Patch occupancy rates and population densities were studied in relation to patch size, degree of patch isolation, level of sun exposure and distance from the coast. In addition, flight tests in the laboratory were performed to estimate the potential dispersal capacity of the moth.
3. Occupancy rates were high and the likelihood of extinction depended on patch size. Small patches were less likely to be occupied than were large patches (> 10 m²). Sun-exposed patches were occupied for a lower proportion of years than were shaded patches. No distance effects could be discerned at the spatial scale of study, presumably because the insect is a strong flier.
4. Population densities in occupied patches decreased with increasing patch size. Furthermore, insect densities tended to increase with distance from the coast. Density changes in patches were synchronized.
5. The studied insect population can be described as a 'patchy population' sensu Harrison (1991) with spatially correlated population dynamics. These dynamics are superimposed on a landscape gradient.     

Speeds of invasion in a model with strong or weak Allee effects

We study an invasion model based on a reaction-diffusion equation with an Allee effect. We use a special, piecewise-linear, population growth rate. This function allows us to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effect. We show that the minimum speed for a sufficiently weak Allee may, in fact, be the same as that derived by means of linearization.     

Allee effects may slow the spread of parasites in a coastal marine ecosystem

Allee effects are thought to mediate the dynamics of population colonization, particularly for invasive species. However, Allee effects acting on parasites have rarely been considered in the analogous process of infectious disease establishment and spread. We studied the colonization of uninfected wild juvenile Pacific salmon populations by ectoparasitic salmon lice (Lepeophtheirus salmonis) over a 4-year period. In a data set of 68,376 fish, we observed 85 occurrences of precopular pair formation among 1,259 preadult female and 613 adult male lice. The probability of pair formation was dependent on the local abundance of lice, but this mate limitation is likely offset somewhat by mate-searching dispersal of males among host fish. A mathematical model of macroparasite population dynamics that incorporates the empirical results suggests a high likelihood of a demographic Allee effect, which can cause the colonizing parasite populations to die out. These results may provide the first empirical evidence for Allee effects in a macroparasite. Furthermore, the data give a rare detailed view of Allee effects in colonization dynamics and suggest that Allee effects may dampen the spread of parasites in a coastal marine ecosystem.     

The Allee Effect in Population Dynamics with a Seasonal Reproduction Pattern

The local consequences of the Allee effect in isolated populations of animal species with a seasonal reproduction pattern that nonmonotonically depends on population density are studied based on a discrete analog of the Bazykin–Ludwig model. Along with the critical population size (below which the population degenerates because of the Allee effect), the limiting population size is discovered: the population with a higher density degenerates because of overpopulation. The effect of the initial population size on possible scenarios of its development is studied in detail. It is shown that an “intermediate” population size that provides the maximum population density is unachievable in some cases.     

Population collapse to extinction: the catastrophic combination of parasitism and Allee effect

Infectious diseases are responsible for the extinction of a number of species. In conventional epidemic models, the transition from endemic population persistence to extirpation takes place gradually. However, if host demographics exhibits a strong Allee effect (AE) (population decline at low densities), extinction can occur abruptly in a catastrophic population crash. This might explain why species suddenly disappear even when they used to persist at high endemic population levels. Mathematically, the tipping point towards population collapse is associated with a saddle-node bifurcation. The underlying mechanism is the simultaneous population size depression and the increase of the extinction threshold due to parasite pathogenicity and Allee effect. Since highly pathogenic parasites cause their own extinction but not that of their host, there can be another saddle-node bifurcation with the re-emergence of two endemic equilibria. The implications for control interventions are discussed, suggesting that effective management may be possible for ?(0)?1.     

Woodland birds in patchy landscapes: the evidence base for strategic networks

Habitat creation and management within wooded networks is a potentially effective strategy to reduce ecological isolation and the deleterious effects of fragmentation. However, questions remain over the relative advantages of different approaches, e.g. buffering patches vs. increasing connectivity. Potential effects of woodland fragmentation include reduction in regional woodland cover, reduced patch size, edge effects with loss of core habitat, and increased isolation with disruption of dispersal and metapopulation dynamics. We adopt an evidence-based approach to review how each of these affects woodland birds with an emphasis on studies from the UK and use this to identify management priorities for mitigation. There is evidence for both patch area and composition effects: larger woodlands support more woodland bird species, and woods located within sparsely wooded landscapes are less valuable to specialist woodland species. Bird assemblages show a nested pattern with respect to area, and thus species found in small woods also occur in large woods but not vice versa. However, small woods may be preferred by a few edge species, while small woods also have greater variability in bird species composition. Consideration of the metapopulation dynamics of specialist species with poor dispersal shows that creating or buffering large woodlands is more efficient than a greater total area of small fragments. Connectivity appears most useful for widespread generalist species with almost continuous populations. Woodland structure and quality are of overwhelming importance: as well as mature woodland, young growth, scrub and edges are also key components. There is an urgent need to examine the relationship between nest predation and landscape structure within UK woodlands.     

Bubbling and hydra effect in a population system with Allee effect

We present a continuous time predator-prey model and predator’s growth subjected to component Allee effect. The model also includes density dependent mortality of predator. We investigate our model both analytically and numerically, and highlighted the effect of density independent mortality and Allee effect. In our system, we find that a fixed point representing the extinction of predator is always a stable point. When coexistence equilibria exists our system is bistable. We have observed that tristability is possible for our model that includes two stable co-existence fixed point. The most important phenomena which we have observed are hydra effect and cascading effect. Due to component Allee effect in predator the system shows multiple hydra effect. We discuss the phenomenon of bubbling, which indicates increasing and decreasing of amplitudes of cycles. We have presented one-parametric as well as two-parametric bifurcation diagram and also all possible bifurcations that the system could go through.     

Density dependence slows invader spread in fragmented landscapes

Patchiness is a defining characteristic of most natural and anthropogenic habitats, yet much of our understanding of how invasions spread has come from models of spatially homogeneous environments. Except for populations with Allee effects, an invader's growth rate when rare and dispersal determine its spread velocity; intraspecific competition has little to no influence. How this result might change with landscape patchiness, however, is poorly understood. We used simulation models and their analytical approximations to explore the effect of density dependence on the spread of annual plant invaders moving through heterogeneous landscapes with gaps in suitable habitat. We found that landscape patchiness and discrete invader population size interacted to generate a strong role for density dependence. Intraspecific competition greatly slowed the spread of invasions through patchy landscapes by regulating how rapidly a population could produce enough seeds to surpass habitat gaps. Populations with continuously varying density showed no such effect of density dependence. We adapted a stochastic dispersal model to approximate spread when gap sizes were small relative to the mean dispersal distance and a Markov chain approximation for landscapes with large gaps. Our work suggests that ecologists must consider reproduction at both low and high densities when predicting invader spread.     

Properties of adaptive walks on uncorrelated landscapes under strong selection and weak mutation

We examine properties of adaptive walks on uncorrelated (i.e. random) fitness landscapes starting from moderately fit genotypes under strong selection weak mutation. As an extension of Orr's model for a single step in an adaptive walk under these conditions, we show that the fitness rank of the dominant genotype in a population after the fixation of a beneficial mutation is, on average, (i+6)/4, where i is the fitness rank of the starting genotype. This accounts for the change in rank due to acquiring a new set of single-mutation neighbors after fixing a new allele through natural selection. Under this scenario, adaptive walks can be modeled as a simple Markov chain on the space of possible fitness ranks with an absorbing state at i = 1, from which no beneficial mutations are accessible. We find that these walks are typically short and are often completed in a single step when starting from a moderately fit genotype. As in Orr's original model, these results are insensitive to both the distribution of fitness effects and most biological details of the system under consideration.