Patterns of Patchy Spread in Deterministic and Stochastic Models of Biological Invasion and Biological Control

A few spatiotemporal models of population dynamics are considered in relation to biological invasion and biological control. The patterns of spread in one and two spatial dimensions are studied by means of extensive numerical simulations. We show that, in the case that population multiplication is damped by the strong Allee effect (when the population growth rate becomes negative for small population density), in a certain parameter range the spread can take place not via the intuitively expected circular expanding population front but via motion and interaction of separate patches. Alternatively, the patchy spread can take place in a system without Allee effect as a result of strong environmental noise. We then show that the phenomenon of deterministic patchy invasion takes place ‘at the edge of extinction’ so that a small change of controlling parameters either brings the species to extinction or restores the travelling population fronts. Moreover, we show that the regime of patchy invasion in two spatial dimensions actually takes place when the species go extinct in the corresponding 1-D system.     

Pathogens can Slow Down or Reverse Invasion Fronts of their Hosts

Infectious diseases are often regarded as possible explanations for the sudden collapse of biological invasions. This phenomenon is characterized by a host species, which firstly can successfully establish in a non-native habitat, but then spontaneously disappears again. This study proposes a reaction-diffusion model consisting of a simple SI disease with vital dynamics of Allee effect type. By way of travelling wave analysis, conditions are derived under which the invasion of the host population is slowed down, stopped or reversed as a consequence of a subsequently introduced disease. Hence, pathogens can dramatically control the rate of spread of invasive species.     

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible-infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.     

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible–infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.     

Combined impacts of Allee effects and parasitism

Despite their individual importance for population dynamics and conservation biology, the combined impacts of Allee effects and parasitism have received little attention. We built a mathematical model to compare the dynamics of populations with or without Allee effects when infected by microparasites. We show that the influence of an Allee effect takes the form of a tradeoff. The presence of an Allee effect in host populations may protect them, by reducing the range of population sizes that allow parasite spread. Yet if infection spreads, the Allee effect weakens host populations by reducing their size and by widening the range of parasite species that lead them to extinction. These results have important implications for predicting the survival of threatened populations or the success of reintroductions, and may help define size ranges within which given populations should be maintained to prevent both epidemics and Allee effects driven extinctions.     

General population growth models with Allee effects in a random environment

Allee effects on population growth are quite common in nature, usually studied through deterministic models with a specific growth rate function.In order to seek the qualitative behaviour of populations induced by such effects, one should avoid model-specific behaviours. So, we use as a basis a general deterministic model, i.e. a model with a general growth rate function, to which we add the effect on the growth rate of the random fluctuations in environmental conditions. The resulting model is the general stochastic differential equation (SDE) model that we propose here.We consider two possible cases, weak Allee effects and strong Allee effects, which lead to different qualitative behaviours of the model.We will study the model properties for both cases in terms of existence and uniqueness of the solution, extinction and stationary behaviour of the population. The two cases will be compared with each other and with the general density-dependent SDE model without Allee effects.We then consider as an example the particular case of the classic logistic model and an Allee effect version of it.     

Decelerating spread of West Nile virus by percolation in a heterogeneous urban landscape

Vector-borne diseases are emerging and re-emerging in urban environments throughout the world, presenting an increasing challenge to human health and a major obstacle to development. Currently, more than half of the global population is concentrated in urban environments, which are highly heterogeneous in the extent, degree, and distribution of environmental modifications. Because the prevalence of vector-borne pathogens is so closely coupled to the ecologies of vector and host species, this heterogeneity has the potential to significantly alter the dynamical systems through which pathogens propagate, and also thereby affect the epidemiological patterns of disease at multiple spatial scales. One such pattern is the speed of spread. Whereas standard models hold that pathogens spread as waves with constant or increasing speed, we hypothesized that heterogeneity in urban environments would cause decelerating travelling waves in incipient epidemics. To test this hypothesis, we analysed data on the spread of West Nile virus (WNV) in New York City (NYC), the 1999 epicentre of the North American pandemic, during annual epizootics from 2000-2008. These data show evidence of deceleration in all years studied, consistent with our hypothesis. To further explain these patterns, we developed a spatial model for vector-borne disease transmission in a heterogeneous environment. An emergent property of this model is that deceleration occurs only in the vicinity of a critical point. Geostatistical analysis suggests that NYC may be on the edge of this criticality. Together, these analyses provide the first evidence for the endogenous generation of decelerating travelling waves in an emerging infectious disease. Since the reported deceleration results from the heterogeneity of the environment through which the pathogen percolates, our findings suggest that targeting control at key sites could efficiently prevent pathogen spread to remote susceptible areas or even halt epidemics.     

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。     

Host-Parasitoid Dynamics and the Success of Biological Control When Parasitoids Are Prone to Allee Effects

In sexual organisms, low population density can result in mating failures and subsequently yields a low population growth rate and high chance of extinction. For species that are in tight interaction, as in host-parasitoid systems, population dynamics are primarily constrained by demographic interdependences, so that mating failures may have much more intricate consequences. Our main objective is to study the demographic consequences of parasitoid mating failures at low density and its consequences on the success of biological control. For this, we developed a deterministic host-parasitoid model with a mate-finding Allee effect, allowing to tackle interactions between the Allee effect and key determinants of host-parasitoid demography such as the distribution of parasitoid attacks and host competition. Our study shows that parasitoid mating failures at low density result in an extinction threshold and increase the domain of parasitoid deterministic extinction. When proned to mate finding difficulties, parasitoids with cyclic dynamics or low searching efficiency go extinct; parasitoids with high searching efficiency may either persist or go extinct, depending on host intraspecific competition. We show that parasitoids suitable as biocontrol agents for their ability to reduce host populations are particularly likely to suffer from mate-finding Allee effects. This study highlights novel perspectives for understanding of the dynamics observed in natural host-parasitoid systems and improving the success of parasitoid introductions.     

Regimes of biological invasion in a predator-prey system with the Allee effect

Spatiotemporal dynamics of a predator-prey system is considered under the assumption that prey growth is damped by the strong Allee effect. Mathematically, the model consists of two coupled diffusion-reaction equations. The initial conditions are described by functions of finite support which corresponds to invasion of exotic species. By means of extensive numerical simulations, we identify the main scenarios of the system dynamics as related to biological invasion. We construct the maps in the parameter space of the system with different domains corresponding to different invasion regimes and show that the impact of the Allee effect essentially increases the system spatiotemporal complexity. In particular, we show that, as a result of the interplay between the Allee effect and predation, successful establishment of exotic species may not necessarily lead to geographical spread and geographical spread does not always enhance regional persistence of invading species.     

Variation in Allee effects: evidence,unknowns, and directions forward

Allee effects, positive effects of population size or density on per-capita fitness, are of broad interest in ecology and conservation due to their importance to the persistence of small populations and to range boundary dynamics. A number of recent studies have highlighted the importance of spatiotemporal variation in Allee effects and the resulting impacts on population dynamics. These advances challenge conventional understanding of Allee effects by reframing them as a dynamic factor affecting populations instead of a static condition. First, we synthesize evidence for variation in Allee effects and highlight potential mechanisms. Second, we emphasize the “Allee slope,” i.e., the magnitude of the positive effect of density on the per-capita growth rate, as a metric for demographic Allee effects. The more commonly used quantitative metric, the Allee threshold, provides only a partial picture of the underlying forces acting on population growth despite its implications for population extinction. Third, we identify remaining unknowns and strategies for addressing them. Outstanding questions about variation in Allee effects fall broadly under three categories: (1) characterizing patterns of natural variability; (2) understanding mechanisms of variation; and (3) implications for populations, including applications to conservation and management. Future insights are best achieved through robust interactions between theory and empiricism, especially through mechanistic models. Understanding spatiotemporal variation in the demographic processes contributing to the dynamics of small populations is a critical step in the advancement of population ecology.     

How do pathogen evolution and host heterogeneity interact in disease emergence?

Heterogeneity in the parameters governing the spread of infectious diseases is a common feature of real-world epidemics. It has been suggested that for pathogens with basic reproductive number R(0)>1, increasing heterogeneity makes extinction of disease more likely during the early rounds of transmission. The basic reproductive number R(0) of the introduced pathogen may, however, be less than 1 after the introduction, and evolutionary changes are then required for R(0) to increase to above 1 and the pathogen to emerge. In this paper, we consider how host heterogeneity influences the emergence of both non-evolving pathogens and those that must undergo adaptive changes to spread in the host population. In contrast to previous results, we find that heterogeneity does not always make extinction more likely and that if adaptation is required for emergence, the effect of host heterogeneity is relatively small. We discuss the application of these ideas to vaccination strategies.     

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.     

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.     

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.     

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.     

How do toxicants affect epidemiological dynamics?

Populations are formed of their constituent interacting individuals, each with their own respective within‐host biological processes. Infection not only spreads within the host organism but also spreads between individuals. Here we propose and study a multilevel model which links the within‐host statuses of immunity and parasite density to population epidemiology under sublethal and lethal toxicant exposure. We analyse this nested model in order to better understand how toxicants impact the spread of disease within populations. We demonstrate that outbreak of infection within a population is completely determined by the level of toxicant exposure, and that it is maximised by intermediate toxicant dosage. We classify the population epidemiology into five phases of increasing toxicant exposure and calculate the conditions under which disease will spread, showing that there exists a threshold toxicant level under which epidemics will not occur. In general, higher toxicant load results in either extinction of the population or outbreak of infection. The within‐host statuses of the individual host also determine the outcome of the epidemic at the population level. We discuss applications of our model in the context of environmental epidemiology, predicting that increased exposure to toxicants could result in greater risk of epidemics within ecological systems. We predict that reducing sublethal toxicant exposure below our predicted safe threshold could contribute to controlling population level disease and infection.     

Habitat destruction and the extinction debt revisited: The Allee effect

Habitat destruction, often caused by anthropogenic disturbance, can lead to the extinction of species at an unprecedented rate. It is important, therefore, to consider habitat destruction when assessing population viability. Another factor often ignored in population viability analysis, is the Allee effect that adds to the risk of populations already on the verge of extinction. Understanding the Allee effect on species dynamics and response to habitat destruction has intrinsic value in conservation prioritization. Here, the Allee effect was considered in a multi-species hierarchical competition model. Results showed that species persistence declines dramatically due to the Allee effect, and certain species become more susceptible to habitat destruction than others. Two extinction orders emerged under habitat destruction: either the best competitor becomes extinct first or the best colonizer first. The extinction debt and order, as well as the time lag between habitat destruction and species extinction, were found to be determined by species abundance and the intensity of the Allee effect.     

Stochastic and spatial dynamics of nematode parasites in farmed ruminants

Host-parasite systems provide powerful opportunities for the study of spatial and stochastic effects in ecology; this has been particularly so for directly transmitted microparasites. Here, we construct a fully stochastic model of the population dynamics of a macroparasite system: trichostrongylid gastrointestinal nematode parasites of farmed ruminants. The model subsumes two implicit spatial effects: the host population size (the spatial extent of the interaction between hosts) and spatial heterogeneity ('clumping') in the infection process. This enables us to investigate the roles of several different processes in generating aggregated parasite distributions. The necessity for female worms to find a mate in order to reproduce leads to an Allee effect, which interacts nonlinearly with the stochastic population dynamics and leads to the counter-intuitive result that, when rare, epidemics can be more likely and more severe in small host populations. Clumping in the infection process reduces the strength of this Allee effect, but can hamper the spread of an epidemic by making infection events too rare. Heterogeneity in the hosts' response to infection has to be included in the model to generate aggregation at the level observed empirically.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.