Spatiotemporal Dynamics of the Epidemic Transmission in a Predator-Prey System

Epidemic transmission is one of the critical density-dependent mechanisms that affect species viability and dynamics. In a predator-prey system, epidemic transmission can strongly affect the success probability of hunting, especially for social animals. Predators, therefore, will suffer from the positive density-dependence, i.e., Allee effect, due to epidemic transmission in the population. The rate of species contacting the epidemic, especially for those endangered or invasive, has largely increased due to the habitat destruction caused by anthropogenic disturbance. Using ordinary differential equations and cellular automata, we here explored the epidemic transmission in a predator-prey system. Results show that a moderate Allee effect will destabilize the dynamics, but it is not true for the extreme Allee effect (weak or strong). The predator-prey dynamics amazingly stabilize by the extreme Allee effect. Predators suffer the most from the epidemic disease at moderate transmission probability. Counter-intuitively, habitat destruction will benefit the control of the epidemic disease. The demographic stochasticity dramatically influences the spatial distribution of the system. The spatial distribution changes from oil-bubble-like (due to local interaction) to aggregated spatially scattered points (due to local interaction and demographic stochasticity). It indicates the possibility of using human disturbance in habitat as a potential epidemic-control method in conservation.     

Rotifer Population Dynamics in Two Coupled Habitats: Invasion of Chaos

We study the role of interactions between habitats in rotifer dynamics. We use a simple discrete-time model to simulate the interactions between neighboring habitats with different intrinsic dynamics. Being uncoupled, one habitat shows periodical oscillations of the rotifer biomass while the other one demonstrates chaotic oscillations. As a result of the exchange of rotifer biomass, chaos replaces regular oscillations. As a result, the rotifer dynamics becomes chaotic in both habitats. We show that the invasion of chaos is followed by the synchronization of the chaotic regimes of both habitats, and this synchronization increases as coupling between the habitats is increased. We also demonstrate that the biological invasion of the rotifer species, which show chaotic dynamics, to a neighboring habitat with intrinsically regular plankton dynamics leads to the invasion of chaos and the synchronization of chaotic oscillations of the plankton biomass in both the habitats.     

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.     

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.     

Invasion dynamics of epidemic with the Allee effect

Investigating the likely success of epidemic invasion is important in the epidemic management and control. In the present study, the invasion of epidemic is initially introduced to a predator-prey system, both species of which are considered to be subject to the Allee effect. Mathematically, the invasion dynamics is described by three nonlinear diffusion-reaction equations and the spatial implicit and explicit models are designed. By means of extensive numerical simulations, the results of spatial implicit model show that the Allee effect has an opposite impact on the invasion criteria and local dynamics when that on the different species. As the intensity of the Allee effect increases, the domain of epidemic invasion reduces and the system dynamics is changed from the stable state to the limit cycle and finally becomes the chaotic state when the susceptible prey with the Allee effect, but the domain expands and the system dynamics is changed from limit cycle to a table point when the predator is subject to the Allee effect. Results from the spatial explicit model show that the strong intensity of the Allee effect can lead to the catastrophic global extinction of all species in the case of that on the susceptible prey. While the predator with the Allee effect, the increased intensity of which makes spatial species reach a stable state. Furthermore, numerical simulations reveal a certain relationship between the invasion speed and spatial patterns.     

General Allee effect in two-species population biology

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator-prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a 'phase space core' of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar systems. Finally, we show how the presence of semistable equilibria increases the number of possible Allee effect cores.     

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。     

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.     

Invasion of an intermediate predator: the dynamics of fish populations in the mathematical model of a trophic chain (as applied to the Syamozero lake)

We present a mathematical model of the dynamics of a spatially heterogeneous predator-prey population system. A prototype of the model system is the Syamozero lake fish community. We study the impact of the invader, an intermediate predator, on the dynamics of the fish community. We show that the invasion can lead to the appearance of chaotic oscillations in the population density. We show also that different dynamical regimes resulting from the invasion, i.e., stationary, non-chaotic oscillatory and chaotic ones, can coexist. The "choice" of a specific regime therewith depends on the initial invader density. Our analysis of solutions of the mathematical models shows that the successful invasion of the alien species takes place solely in the absence of the competition between the invaders and the native species.     

The stability of predator-prey systems subject to the Allee effects

In recent years, many theoreticians and experimentalists have concentrated on the processes that affect the stability of predator-prey systems. But few papers have addressed the Allee effect with focus on the their stability. In this paper, we select two classical models describing predator-prey systems and introduce the Allee effects into the dynamics of both the predator and prey populations in these models, respectively. By combining mathematical analysis with numerical simulation, we have shown that the Allee effect may be a destabilizing force in predator-prey systems: the equilibrium point of the system could be changed from stable to unstable or otherwise, the system, even when it is stable, will take much longer time to reach the stable state. We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey. It should also be pointed out that the impact of the Allee effects of predator and prey due to different mechanisms on different predator-prey systems could also vary.     

Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is considered theoretically using both analytical methods and computer simulations. The prey is assumed to be affected by the strong Allee effect. We reveal a rich variety of pattern formation scenarios. In particular, we show that, in a predator–prey system with the strong Allee effect for prey, the role of space is crucial for species survival. Pattern formation is observed both inside and outside of the Turing domain. For parameters when the local kinetics is oscillatory, the system typically evolves to spatiotemporal chaos. We also consider the effect of different initial conditions and show that the system exhibits a spatiotemporal multistability. In a certain parameter range, the system dynamics is not self-organized but remembers the details of the initial conditions, which evokes the concept of long-living ecological transients. Finally, we show that our findings have important implications for the understanding of population dynamics on a fragmented habitat.     

Dynamics of a host–parasitoid model with Allee effect for the host and parasitoid aggregation

In this paper, a discrete-time host–parasitoid model is investigated. Two biological phenomena, the Allee effect of the host population and the aggregation of the parasitism, are considered in our mathematical model. Through extensive numerical simulations, we gain some interesting findings related to Allee effect from this research. Firstly, the ranges of parameter, in which the population dynamics is chaos, are compressed when Allee effect is added. Secondly, the sensitivity to initial conditions of the host–parasitoid system decreased after adding Allee effect. Thirdly, without Allee effect, we observed two complicated dynamics, intermittent chaos and supertransients. However, when Allee effect is included, these two phenomena are replaced by another kind of phenomenon-period alternation, where chaos is eliminated. From above three novel findings, it can be concluded that dynamic complexities are alleviated by Allee effect. This conclusion is crucial in resolving the discrepancy between real population dynamics and theoretical predictions. Furthermore, the importance of this research is to help us understand the mechanisms inducing the irregular fluctuations of the natural populations.     

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.     

On the importance of dimensionality of space in models of space-mediated population persistence

Spatially explicit models have become widely used in today's mathematical ecology to study persistence of populations. For the sake of simplicity, population dynamics is often analyzed with 1-D models. An important question is: how adequate is such 1-D simplification of 2-D (or 3-D) dynamics for predicting species persistence. Here we show that dimensionality of the environment can play a critical role in the persistence of predator-prey interactions. We consider 1-D and 2-D dynamics of a predator-prey model with the prey growth damped by the Allee effect. We show that adding a second space coordinate into the 1-D model results in a pronounced increase of size of the domain in the parametric space where predator-prey coexistence becomes possible. This result is due to the possibility of formation of a number of 2-D patterns, which is impossible in the 1-D model. The 1-D and the 2-D models exhibit different qualitative responses to variations of system parameters. We show that in ecosystems having a narrow width (e.g. mountain valleys, vegetation patterns along canals in dry areas, etc.), extinction of species is more probable compared to ecosystems having a pronounced second dimension. In particular, the width of a long narrow natural reserve should be large enough to guarantee nonextinction of species via interaction of 2-D population patches.     

How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses

In Rosenzweig-MacArthur models of predator-prey dynamics, Allee effects in prey usually destabilize interior equilibria and can suppress or enhance limit cycles typical of the paradox of enrichment. We re-evaluate these conclusions through a complete classification of a wide range of Allee effects in prey and predator's functional response shapes. We show that abrupt and deterministic system collapses not preceded by fluctuating predator-prey dynamics occur for sufficiently steep type III functional responses and strong Allee effects (with unstable lower equilibrium in prey dynamics). This phenomenon arises as type III functional responses greatly reduce cyclic dynamics and strong Allee effects promote deterministic collapses. These collapses occur with decreasing predator mortality and/or increasing susceptibility of the prey to fall below the threshold Allee density (e.g. due to increased carrying capacity or the Allee threshold itself). On the other hand, weak Allee effects (without unstable equilibrium in prey dynamics) enlarge the range of carrying capacities for which the cycles occur if predators exhibit decelerating functional responses. We discuss the results in the light of conservation strategies, eradication of alien species, and successful introduction of biocontrol agents.     

Estimating Allee Dynamics before They Can Be Observed: Polar Bears as a Case Study

Allee effects are an important component in the population dynamics of numerous species. Accounting for these Allee effects in population viability analyses generally requires estimates of low-density population growth rates, but such data are unavailable for most species and particularly difficult to obtain for large mammals. Here, we present a mechanistic modeling framework that allows estimating the expected low-density growth rates under a mate-finding Allee effect before the Allee effect occurs or can be observed. The approach relies on representing the mechanisms causing the Allee effect in a process-based model, which can be parameterized and validated from data on the mechanisms rather than data on population growth. We illustrate the approach using polar bears (Ursus maritimus), and estimate their expected low-density growth by linking a mating dynamics model to a matrix projection model. The Allee threshold, defined as the population density below which growth becomes negative, is shown to depend on age-structure, sex ratio, and the life history parameters determining reproduction and survival. The Allee threshold is thus both density- and frequency-dependent. Sensitivity analyses of the Allee threshold show that different combinations of the parameters determining reproduction and survival can lead to differing Allee thresholds, even if these differing combinations imply the same stable-stage population growth rate. The approach further shows how mate-limitation can induce long transient dynamics, even in populations that eventually grow to carrying capacity. Applying the models to the overharvested low-density polar bear population of Viscount Melville Sound, Canada, shows that a mate-finding Allee effect is a plausible mechanism for slow recovery of this population. Our approach is generalizable to any mating system and life cycle, and could aid proactive management and conservation strategies, for example, by providing a priori estimates of minimum conservation targets for rare species or minimum eradication targets for pests and invasive species.     

Allee效应对具有生境恢复的集合种群的影响

Allee效应与种群的灭绝密切相关,其研究对生态保护和管理至关重要。Allee效应对物种续存是潜在的干扰因素,濒危物种更容易受其影响,可能会增加生存于生境破碎化斑块的濒危物种的死亡风险,因此研究Allee效应对种群的动态和续存的影响是必要的。从包含由生物有机体对环境的修复产生的Allee效应的集合种群模型出发,引入由其他机制形成的Allee效应,建立了常微分动力系统模型和基于网格模型的元胞自动机模型。通过理论分析和计算机模拟表明:(1)强Allee效应不利于具有生境恢复的集合种群的续存;(2)生境恢复有利于种群续存;(3)局部扩散影响了集合种群的空间结构、动态行为和稳定性,生境斑块之间的局部作用将会减缓或消除集合种群的Allee效应,有利于集合种群的续存。     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002     

Slight phenotypic variation in predators and prey causes complex predator-prey oscillations

Predator-prey oscillations are expected to show a 1/4-phase lag between predator and prey. However, observed dynamics of natural or experimental predator-prey systems are often more complex. A striking but hardly studied example are sudden interruptions of classic 1/4-lag cycles with periods of antiphase oscillations, or periods without any regular predator-prey oscillations. These interruptions occur for a limited time before the system reverts to regular 1/4-lag oscillations, thus yielding intermittent cycles. Reasons for this behaviour are often difficult to reveal in experimental systems. Here we test the hypothesis that such complex dynamical behaviour may result from minor trait variation and trait adaptation in both the prey and predator, causing recurrent small changes in attack rates that may be hard to capture by empirical measurements. Using a model structure where the degree of trait variation in the predator can be explicitly controlled, we show that a very limited amount of adaptation resulting in 10–15% temporal variation in attack rates is already sufficient to generate these intermittent dynamics. Such minor variation may be present in experimental predator-prey systems, and may explain disruptions in regular 1/4-lag oscillations.     

Long-period endogenous oscillations in fish population size: Mathematical modeling

We present a mathematical model of an aquatic community, where the size-and-age structure of hydrobiont populations is taken into account and the corresponding trophic interactions between zooplankton, peaceful fish, and predatory fish are described. We show that interactions between separate components of the aquatic community can give rise to long-period oscillations in fish population size. The period of these oscillations is on the order of decades. With this model we also show that an increase in the zooplankton growth rate may entail a sequence of bifurcations in the fish population dynamics: steady states → regular oscillations → quasicycles → dynamic chaos.