Revealing the role of predator interference in a predator–prey system with disease in prey population

Predation on a species subjected to an infectious disease can affect both the infection level and the population dynamics. There is an ongoing debate about the act of managing disease in natural populations through predation. Recent theoretical and empirical evidence shows that predation on infected populations can have both positive and negative influences on disease in prey populations. Here, we present a predator–prey system where the prey population is subjected to an infectious disease to explore the impact of predator on disease dynamics. Specifically, we investigate how the interference among predators affects the dynamics and structure of the predator–prey community. We perform a detailed numerical bifurcation analysis and find an unusually large variety of complex dynamics, such as, bistability, torus and chaos, in the presence of predators. We show that, depending on the strength of interference among predators, predators enhance or control disease outbreaks and population persistence. Moreover, the presence of multistable regimes makes the system very sensitive to perturbations and facilitates a number of regime shifts. Since, the habitat structure and the choice of predators deeply influence the interference among predators, thus before applying predators to control disease in prey populations or applying predator control strategy for wildlife management, it is essential to carefully investigate how these predators interact with each other in that specific habitat; otherwise it may lead to ecological disaster.     

Constructive effects of environmental noise in an excitable prey–predator plankton system with infected prey

An excitable model of fast phytoplankton and slow zooplankton dynamics is considered for the case of lysogenic viral infection of the phytoplankton population. The phytoplankton population is split into a susceptible (S) and an infected (I) part. Both parts grow logistically, limited by a common carrying capacity. Zooplankton (Z) is grazing on susceptibles and infected, following a Holling-type III functional response. The local analysis of the S–I–Z differential equations yields a number of stationary and/or oscillatory regimes and their combinations. Correspondingly interesting is the behaviour under multiplicative noise, modelled by stochastic differential equations. The external noise can enhance the survival of susceptibles and infected, respectively, that would go extinct in a deterministic environment. In the parameter range of excitability, noise can induce prey–predator oscillations and coherence resonance (CR). In the spatially extended case, synchronized global oscillations can be observed for medium noise intensities. Higher values of noise give rise to the formation of stationary spatial patterns.     

Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species

In this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.     

Effects of the probability of a predator catching prey on predator–prey system stability

To understand the effect of the probability of a predator catching prey, P_catch, on the stability of the predator–prey system, a spatially explicit lattice model consisting of predators, prey, and grass was constructed. The predators and prey randomly move on the lattice space, and the grass grows according to its growth probability. When a predator encounters prey, the predator eats the prey in accordance with the probability P_catch. When a prey encounters grass, the prey eats the grass. The predator and prey give birth to offspring according to a birth probability after eating prey or grass, respectively. When a predator or prey is initially introduced or newly born, its health state is set at a high given value. This health state decreases by one with every time step. When the state of an animal decreases to less than zero, the individual dies and is removed from the system. Population densities for predator and prey fluctuated significantly according to P_catch. System stability was characterized by the standard deviation ? of the fluctuation. The simulation results showed that ? for predators increased with an increase of P_catch; ? for prey reached a maximum at P_catch = 0.4; and ? for grass fluctuated little regardless of P_catch. These results were due to the tradeoff between P_catch and the predator–prey encounter rate, which represents the degree of interaction between predator and prey and the average population density, respectively.     

Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

Evidence for the Predator Attraction Hypothesis in an amphibian predator–prey system

Many species possess damage-released chemical alarm cues that function in alerting nearby individuals to a predator attack. One hypothesis for the evolution and/or maintenance of such cues is the Predator Attraction Hypothesis, where predators, rather than prey, are the “intended” recipients of these cues. If a predator attack attracts additional predators, these secondary predators might interfere with the predation event, providing the prey with a better chance to escape. In this study, we conducted two experiments to explore this hypothesis in an amphibian predator/prey system. In Experiment 1, we found that tiger salamanders (Ambystoma mavortium) showed a foraging attraction to chemical cues from wood frog (Lithobates sylvaticus) tadpoles. Salamanders that were experienced with tadpole prey, in particular, were strongly attracted to tadpole alarm cues. In Experiment 2, we observed experimental encounters between a tadpole and either one or two salamanders. The presence of the second predator caused salamanders to increase attack speed at the cost of decreased attack accuracy (i.e., increasing the probability that the tadpole would escape attacks). We also found that the mere presence of visual and chemical cues from a second predator did not affect this speed/accuracy trade-off but did cause enough of a distraction to increase tadpole survival. Thus, our findings are consistent with the Predator Attraction Hypothesis for the evolution and/or maintenance of alarm cues.     

Effects of perturbation on the predator–prey system in a heterogeneous landscape

Environmental perturbations occur in ecosystems as the result of disturbance, which is closely related to ecosystem stability and resilience. To understand how perturbations can affect ecosystems, we constructed a spatially explicit lattice model to simulate the integrative predator–prey–grass relationships. In this model, a predator (or prey) gives birth to offspring, according to a specific birth probability, when it is able to feed on prey (or grass). When a predator or prey animal was initially introduced or newly born, its health state was set at a given high value. This state decreased by 1 with each time step. When the state of an animal decreased to zero, the animal was considered dead and was removed from the system. In this model, the perturbation was defined as the sudden death of some portion of the population. The heterogeneous landscape was characterized by a parameter, H, which controlled the degree of heterogeneity. When H ≥ 0.6, the predator population size was positively influenced by the perturbation. However, the perturbation had little effect upon the population sizes of prey or grass, regardless of the value of H.     

Viable populations in a prey–predator system

 Lotka–Volterra equations are considered a dynamical game, where the phenotypes of the predator and of the prey can vary. This differs from the usual procedure of specifying as a priori laws according to which strategies are supposed to change. The question at stake is the survival of each of the species, instead of the maximization of a given pay-off by each player, as it is commonly discussed in games. The predator needs the prey, while the prey can survive without the predator. These obvious and simplistic constraints are enough to shape the regulation of the system: notably, the largest closed set of initial conditions can be delineated, from which there exists at least one evolutionary path where the population can avoid extinction forever. To these so-called viable trajectories, viable strategies are associated, respectively for the prey or for the predator. A coexistence set can then be defined. Within this set and outside the boundary, strategies can vary arbitrarily within given bounds while remaining viable, whereas on the boundary, only specific strategies can guarantee the viability of the system. Thus, the largest set can be determined, outside of which strategies will never be flexible enough to avoid extinction. Received 2 May 1995; received in revised form 15 August 1995     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Spatiotemporal behavior of a prey–predator system with a group defense for prey

Group defense is a strategy widely employed by various species. We consider the effect of grouping on population persistence when animals join together in herds in order to provide a self-defense from predators. In literature, group defense is usually addressed in terms of individual behavioral responses. In this paper, we consider an alternative ‘mean-field’ approach which uses prey and predator densities as the dynamical variables. The model is essentially a predator–prey system but with an unconventional parametrization for the predation term. We discuss the outcomes of the ecosystem dynamics in terms of persistence and prey survival. In the spatially distributed model some specific spatio-temporal features are discovered.     

Studies of the effect of environmental factors on the rotifer predator–prey system in freshwater

The aim of this work is to evaluate the effect of environmental factors: temperature and photoperiod on the zooplankton predator–prey system. Rotifers, an important and cosmopolitan group of zooplankton in freshwater, were used in our study. We investigated the effect of temperature (20, 23, and 30°C) and of photoperiod (L:D = 12:0 and 0:12) on the predatory rotifer Asplanchna brightwelli consuming rotifer Brachionus calyciflorus as prey. Under A. brightwelli predation, populations of B. calyciflorus prey were consumed more slowly at 20 ± 1 and 30 ± 1°C as compared to 23 ± 1°C. Prey consumption by A. brightwelli increased from 0.63 ± 0.09 ind. predator⁻¹ at 20°C to a peak of 1.22 ± 0.12 ind. predator⁻¹ at 23°C, then decreased significantly to 0.93 ± 0.14 ind. predator⁻¹ at 30 ± 1°C. In addition, predation responded to temperature changing sensitively and rapidly. Statistical analysis showed that the prey consumption were significant different under altered temperature periods during 12 h. Photoperiod also significantly influenced the rate of A. brighwelli predation. B. calyciflorus suffered less predation in darkness than in light. The rate of prey consumption in light (1.06 ind. predator⁻¹) was twice the average of that in darkness (0.51 ind. predator⁻¹). Furthermore, predation rate varied under changing photoperiod but predators moved back into the light did not resume their original consumption rate. Our results demonstrate that whether the predation in rotifer successfully or not is strongly influenced by temperature and photoperiod.     

On the Neimark–Sacker bifurcation in a discrete predator–prey system

A two-parameter family of discrete models describing a predator–prey interaction is considered, which generalizes a model discussed by Murray, and originally due to Nicholson and Bailey, consisting of two coupled nonlinear difference equations. In contrast to the original case treated by Murray, where the two populations either die out or may display unbounded growth, the general member of this family displays a somewhat wider range of behaviour. In particular, the model has a nontrivial steady state which is stable for a certain range of parameter values, which is explicitly determined, and also undergoes a Neimark–Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter space and a repelling one in others.     

On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator–prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator–prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.     

Community-driven dispersal in an individual-based predator–prey model

We present a spatial, individual-based predator–prey model in which dispersal is dependent on the local community. We determine species suitability to the biotic conditions of their local environment through a time and space varying fitness measure. Dispersal of individuals to nearby communities occurs whenever their fitness falls below a predefined tolerance threshold. The spatiotemporal dynamics of the model is described in terms of this threshold. We compare this dynamics with the one obtained through density-independent dispersal and find marked differences. In the community-driven scenario, the spatial correlations in the population density do not vary in a linear fashion as we increase the tolerance threshold. Instead we find the system to cross different dynamical regimes as the threshold is raised. Spatial patterns evolve from disordered, to scale-free complex patterns, to finally becoming well-organized domains. This model therefore predicts that natural populations, the dispersal strategies of which are likely to be influenced by their local environment, might be subject to complex spatiotemporal dynamics.     

Dynamics of additional food provided predator–prey system with mutually interfering predators

Use of additional/alternative food source to predators is one of the widely recognised practices in the field of biological control. Both theoretical and experimental works point out that quality and quantity of additional food play a vital role in the controllability of the pest. Theoretical studies carried out previously in this direction indicate that incorporating mutual interference between predators can stabilise the system. Experimental evidence also point out that mutual interference between predators can affect the outcome of the biological control programs. In this article dynamics of additional food provided predator–prey system in the presence of mutual interference between predators has been studied. The mutual interference between predators is modelled using Beddington–DeAngelis type functional response. The system analysis highlights the role of mutual interference on the success of biological control programs when predators are provided with additional food. The model results indicate the possibility of stable coexistence of predators with low prey population levels. This is in contrast to classical predator–prey models wherein this stable co-existence at low prey population levels is not possible. This study classifies the characteristics of biological control agents and additional food (of suitable quality and quantity), permitting the eco-managers to enhance the success rate of biological control programs.     

Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect

Understanding dynamical complexity of alien species invasion is an important and timely issue as it is believed that identification of possible invasion scenarios may result in a more effective invasive species management and control. To address this issue, mathematical modeling is widely recognized as a convenient theoretical tool as it often helps to reveal generic tendencies in a situation when it would hardly be possible otherwise. However, although much attention has been paid to spatially continuous models of biological invasion, spatially discrete models have remained outside of the mainstream. Meanwhile, species habitat is often distinctly patchy or even fragmented, and then spatially discrete models seems to be the most appropriate modeling framework. In this paper, we consider invasion scenarios in a space- and time-discrete system described by a coupled map lattice. The alien species is assumed to be affected by the strong Allee effect and by a specialist predator. We first consider the stage of species introduction and obtain analytical conditions to distinguish between invasion success and invasion failure. We then focus on species spatial spread. By means of extensive computer simulations, we identify the main scenarios of species spread (e.g. patchy invasion and multiple traveling bands) and reveal the corresponding structure of the parameter space. Counter-intuitively, we have found that alien species can invade over the whole domain even in the case when its local persistence would not be possible.